Frege’s Logic

First published Tue Feb 7, 2023

Friedrich Ludwig Gottlob Frege (b. 1848, d. 1925) is often creditedwith inventing modern quantificational logic in hisBegriffsschrift. While there has been some controversy overexactly what was novel with Frege, and what can be found in the workof contemporaries such as George Boole, Augustus DeMorgan, ErnstSchröder, Charles Sanders Peirce, and John Venn (see, e.g.,Putnam 1982 or Boolos 1994 for accounts that resist the tendency toattributeall of modern logic to Frege, and also the entry onthe algebra of logic tradition), there is no doubt that Frege’s work—especially aschampioned by Bertrand Russell and Ludwig Wittgenstein—had ahuge influence on how philosophical and mathematical logicprogressed.

While this entry is intended to provide the reader with an overview ofFrege’s logical systems as presented inBegriffsschriftandGrundgesetze, it is not intended to be a guide totranslating Frege’s logical systems into modern notation, hencethere is very little modern notation in what follows. Despite thecommon approach of “investigating” various aspects ofFrege’s logic and his logicist program via a translation of hisaxioms and theorems into modern notation, such an approach can oftenlead to misunderstandings of Frege’s actual views, since his ownnotation (in both logical systems) differs in significant ways frommodern first- and higher-order quantificational logic. As a result,anyone who is interested in understanding Frege’s logical andphilosophical views on their own terms needs to examine those views intheir native habitat—the logics and formal languages ofBegriffsschrift andGrundgesetze—and as aresult, needs to become fluent in working with Frege’s notation,deductive systems, etc., directly. This entry is, amongst otherthings, intended as a means to begin that journey.

Supplements:

1. Introduction

The story regarding Frege’s innovations in logic, and the rolethat they played in his larger logicist project is usually told alongsomething like the following lines, which focuses on his three“great books”. First, Frege invents modernquantificational logic in hisBegriffsschrift eine derarithmetischen nachgebildete Formelsprache des reinen Denkens, orConcept Script (1879a). Second, Frege criticizes thefront-running (at the time) accounts of the foundations of mathematicsinDie Grundlagen der Arithmetik: eine logisch-mathematischeUntersuchung über den Begriff der Zahl, orTheFoundation of Mathematics (1884), and he also provides aninformal account of his reduction of mathematics to logic. Third,Frege carries out the formal reconstruction of arithmetic (and beginsthe reconstruction of real and complex analysis) in the mammothGrundgesetze der Arithmetik: begriffsschriftlich abgeleitet Band I& Band II, orBasic Laws of Arithmetic (1893/1903),within the logic that he first developed inBegriffsschrift,or at least within a straightforward extension of this logic obtainedby expanding the system with value-ranges and logical laws that governthem.

If one is interested in Frege’s philosophy of mathematics, thenthis story is perhaps adequate. But if one’s interest is insteadaimed at Frege’s philosophy of logic, then the tale justsketched is woefully inadequate. The reason is simply this: Thelogical system found inGrundgesetze is in fact significantlydifferent from the system found inBegriffsschrift. There isno doubt that Frege arrived at the system given inGrundgesetze by attending to various inadequacies andlimitations of the system given inBegriffsschrift. But thedifferences between the two logics, both in terms of technical detailsand in terms of philosophical interpretation, are too substantial tobe compatible with the idea that Frege, over the course of his career,championed a single, uniform logical system which he merely extendedin various ways when his logicist project required him to do so.Attending to these differences is important, since Frege’sinvention of higher-order quantificational logic is typically markedas occurring upon the publication ofBegriffsschrift, yetmany of the distinctive—some might sayidiosyncratic—features of his logic only appear in the latersystem ofGrundgesetze

These features include the claim that sentences refer to truth values;the precise type system of objects, first-level, second-level, andthird-level functions, concepts, and relations; and the introductionof distinct first- and higher-order quantifiers that range over thesedifferent types respectively. These aspects of the later logic ofGrundgesetze are not mere additions to, or clarifications of,the earlier logic ofBegriffsschrift. On the contrary, thedifferences between the two systems entail that there are strings ofsymbols that are theorems of the logic ofBegriffsschrift,but fail to be theorems of the logic ofGrundgesetze. We willsee an important example of this—one discussed by Fregehimself—in what follows.

Before diving in, some observations on notation, terminology, andgoals are in order. First, some authors, such as Heck (2012), haveused the term “Begriffsschrift” in asystematically ambiguous way, writing it in italics(Begriffsschrift) when referring to the work, and writing itnon-italicized when referring to the (presumed single) logical systemat work in all of Frege’s writings. This approach will not workhere, hence we will continue to refer to the two systems in questionas the logic ofBegriffsschrift and the logic ofGrundgesetze respectively.

Second, some translators have rendered Frege’s“deutscher Buchstabe” as “Gothicletter”, others as “German letter”, and some haverendered “lateinischer Buchstabe” as “Latinletter”, others as “Roman letter”. Nothingphilosophical hangs on this, it is merely a matter of stylisticdifference, hence I have left these as is in translations, but willuse, in each case, the latter terminology in my discussion.

Third, we should emphasize that in both works Frege is operating witha very different conception of the methods and goals of logic than theconception usually at work within modern research on the topic. WarrenGoldfarb describes the modern conception of logic, which he calls theschematic conception, and which he argues we inherited from(among others) Tarski and Quine, as being concerned primarily withdiscovering properties of and relations holding between logicalschemata within this-or-that formal language, and secondarily withdetermining whether sentences of natural language can be translated asinstances of such a schema with this or that feature. The conceptionof logic that is at work in theBegriffsschrift andGrundgesetze, however, which Goldfarb calls theuniversalist conception, is quite different. On theuniversalist conception, logic is interested in stating and provinggeneral, universally valid logical laws applicable to any subjectmatter whatsoever. Simply put, on the modern, schematic conception oflogic, the subject matter of logic is sentences (or schemas, orpropositions, or some other type oflinguistic entity) andthe goal is to discover laws governing the properties of, andrelations between, sentences (or schemas, or propositions, etc.). OnFrege’s universalist conception, however, the goal is not todiscover universal truths about language, but rather to discoveruniversal truths about the world (Goldfarb 2010).

Finally, although Frege only provides a comprehensive presentation ofhis logic twice, inBegriffsschrift and then in the earlyparts ofGrundgesetze, he discusses aspects of his logic in anumber of other works, including “Function and Concept”,“Sense and Reference”, and “Concept andObject” (about which more below), but also in a number of lesswell-known essays in which he explicitly discusses the differencesbetween his own system(s) and the work of his contemporaries. Theseinclude (but are not necessarily limited to) “Boole’sLogical Calculus and the Concept-script” (1880/81),“Boole’s Logical Formula Language and myConcept-script” (1881), “On the Scientific Justificationof a Conceptual Notation” (1882a), “On the Aim of the‘Conceptual Notation’” (1882b), and “On Mr.Peano’s Conceptual Notation and My Own” (1897). Theseessays not only provide additional insights into the development ofFrege’s logic(s) and his philosophy of logic, but also containvigorous (and compelling) defenses of thesuperiority ofFrege’s much-maligned notation over the notation ofFrege’s contemporaries (arguments that apply equally well, inmany respects, when comparing Frege’s notation to our modernbackwards-“E” ($\exists$) andupside-down-“A” ($\forall$) notation!) Those who wish tocontinue their study of Frege’s logic beyond the materialcontained in this essay should consult not onlyBegriffsschrift andGrundgesetze, but these works aswell.

2. The Logic ofBegriffsschrift

The logic ofBegriffsschrift was formulated before Fregewrote “Function and Concept” (1891), “Sense andReference” (1892a), and “Concept and Object”(1892b). Each of these papers was aimed at solving a particularproblem in Frege’s original logic as laid out inBegriffsschrift: “Function and Concept” clarifiesthe status of concepts and relations as a species of mathematicalfunction (and contains the first appearance of a number of otherchanges to the logic, including the first published mention ofFrege’s sense/reference distinction), “Sense andReference” provides Frege with the tools to provide an adequatetreatment of identity, and “Concept and Object” addressespuzzles raised by the type distinctions at work in Frege’shigher-order logic first clearly introduced in “Function andConcept” (although the sharp concept/object distinction is alsoimplicitly assumed, and does much work, inGrundlagen[1884]). Each of these works led to significant alterations to boththe formal details and the philosophical interpretation ofFrege’s logic betweenBegriffsschrift andGrundgesetze, and we will examine a number of suchdifferences below. For the purposes of understanding the basicmechanics of the logic ofBegriffsschrift on its own,however, the most critical of these three essays is “Sense andReference”.

At the time of writingBegriffsschrift, Frege did not have aclear distinction between sense and reference. As a result, he insteadmobilizes the notion ofconceptual content which, in a sense,does the work that would be distributed between sense and reference inhis later, post “Sense and Reference” writings.

In his argument against the traditional subject-predicate analysis ofjudgements (which we return to below), Frege explicitly states thatthe logic ofBegriffsschrift is insensitive to differencesbetween judgements that express the same conceptual content:

I note that the contents of two judgements can differ in two ways:either the conclusions that can be drawn from one when combined withcertain others also always follow from the second when combined withthe same judgements, or else this is not the case. The twopropositions “At Plataea the Greeks defeated the Persians”and “At Plataea the Persians were defeated by the Greeks”,differ in the first way. Even if a slight difference in sense can bediscerned, the agreement still predominates. Now I call that part ofthe content that is thesame in both cases theconceptualcontent. Sinceonly this has significance for theBegriffsschrift, no distinction is needed betweenpropositions that have the same conceptual content. (Frege 1879a:§3)

As a result, and, interestingly, somewhat more like contemporary logicthan the later approach taken inGrundgesetze, the logicaloperators ofBegriffsschrift are operations that (in somesense) map arguments of the appropriate sort tojudgeablecontents, which can be glossed as something like possible“circumstances” or “facts” (see Dummett 1981and Currie 1984 for more discussion). Thus, in the logic ofBegriffsschrift negation is an operation that takes ajudgeable content (again, something like a possible fact orcircumstance) as input and gives another judgeable content as itsvalue, and Frege’s version of the universal quantifier is anoperator that takes a predicative conceptual content as argument andgives a judgeable content as its value.

Familiar worries regarding “negative” and“general” facts (or, more carefully,“negative” and “universal” conceptualcontents) no doubt plague the logic ofBegriffsschrift as aresult (Beaney 1997), but Frege does not address such problems, whichat any rate disappear in the post-sense/reference-distinction logic ofGrundgesetze. The important point for our purposes is thatthe use of this early notion of conceptual content, rather than themore nuanced sense/reference distinction, means that the range ofapplicability of the operators found in the logic ofBegriffsschrift is rather narrower than the range ofapplicability of the (typographically identical) operators found inthe logic ofGrundgesetze.

In Section 3 ofBegriffsschrift, Frege introduces animportant departure from the work of his contemporaries (such as Booleand Schröder), and from the logical tradition more generally: therejection of the subject-predicate analysis of propositions:

A distinction betweensubject andpredicate findsno place in my representation of a judgement. (1879a:§3)

He follows up this jettisoning of subject-predicate analyses with analternative, and more flexible, proposal later inBegriffsschrift:

Let us suppose that the circumstance that hydrogen is lighter thancarbon dioxide is expressed in our formula language. Then in place ofthe symbol for hydrogen we can insert the symbol for oxygen or thatfor nitrogen. This changes the sense in such a way that“oxygen” or “nitrogen” enters into therelation in which “hydrogen” stood before. If anexpression is thought of as variable in this way, it splits up into aconstant component, which represents the totality of relations, and asymbol which can be though of as replaceable by others and whichdenotes the object that stands in these relations. The former I callthe function, the latter its argument. The distinction has nothing todo with the conceptual content, but only with our way of grasping it.Although as viewed in the way just indicated, “hydrogen”was the argument and “being lighter than carbon dioxide”the function, we can also grasp the same conceptual content in such away that “carbon dioxide” becomes the argument and“being heavier than hydrogen” the function. (1879a:§9)

He goes on later in the same section to specify that:

If, in an expression (whose content need not be a judgeablecontent), a simple or complex symbol occurs in one or more places, andwe think of it as replaceable at all or some of its occurrences byanother symbol (but everywhere by the same symbol), then we call thepart of the expression that on this occasion appears invariant thefunction, and the replaceable part its argument. (1879a:§9)

This marks an important innovation in the logic ofBegriffsschrift: Frege has replaced the analysis of aproposition into its unique subject and predicate (a method embeddedin most work on logic since Aristotle) with the more flexible ideathat a proposition can be analyzed into anargument and afunction applied to that argument in more than one way.

A warning is in order: Frege’s distinction inBegriffsschrift between function and argument should not beconfused with the type theory of objects, first-level functions(applying to objects), second-level functions (applying to first-levelfunctions), and third-level functions (applying to second-levelfunctions) that is developed in detail inGrundgesetze.Frege, at the time of writingBegriffsschrift, does not yethave these type distinctions in place, and admits as much in theintroduction toGrundgesetze:

Moreover, the nature of functions, in contrast to objects, ischaracterized more precisely than in myBegriffsschrift.Further, from this the distinction between function of first andsecond level results. (1893/1903: x)

Thus, whatappear withinBegriffsschrift to besomething akin to first-order variables (i.e., German“$\mathfrak{a}$”,“$\mathfrak{e}$”,etc., and Roman“$x$”,“$y$”,etc.) are instead betterunderstood as variables ranging over arguments of any level (whichwould include what the later Frege would consider first- andsecond-level functions), and what appear to be second-order variables(i.e., German“$\mathfrak{f}$”,“$\mathfrak{g}$”,etc., and Roman“$f$”,“$g$”)are instead better understoodas variables ranging over functions that are appropriate to thosearguments.

While correct, even this is somewhat misleading, since anything thatcan be taken to be the argument of a judgement can also be taken to bethe function of that same judgement, and vice versa. In introducingthe concavity generalization device of the logic ofBegriffsschrift (about which more below) Frege writes:

Since the symbol $\Phi$ occurs in the expression $\Phi(A)$ and canbe thought of as replaced by other symbols $\Psi$, $X$, by meansof which other functions of the argument $A$ are then expressed,$\Phi(A)$can be regarded as a function of the argument$\Phi$. (Frege 1879a: §10)

In other words, we can parse the sentence “Hydrogen is lighterthan carbon dioxide” in such a way that “Hydrogen”is the argument and “is lighter than carbon dioxide” isthe function, but we can also parse the same sentence such that“is lighter than carbon dioxide” is the argument and“Hydrogen” (and not, as it would be inGrundgesetze, the second-level concept that we mightparaphrase as “is satisfied by hydrogen”) is thefunction—see Heck and May (2013) for more discussion. Further,Frege notes that the function/argument distinction is not a reflectionof any objective facts regarding the structure of reality (unlike thelater object/function hierarchy found inGrundgesetze) butinstead merely reflects a choice to analyze a statement in one wayrather than another:

For us the different ways in which the same conceptual content can betaken as a function of this or that argument has no importance so longas function and argument are fully determined. (1879a: §9)

Thus, anything can be an argument or a function, and the apparentlyfirst- and second-order variables ofBegriffsschrift arenothing of the sort. Instead, the difference between what appear to befirst-order variables (i.e.,“$\mathfrak{a}$”,“$\mathfrak{e}$”,etc., and Roman“$x$”,“$y$”,etc.) and what appear to besecond-order variables (i.e., German“$\mathfrak{f}$”,“$\mathfrak{g}$”,etc., and Roman“$f$”,“$g$”)is merely heuristic, servingto assist the reader in understanding the upshot of formulas withmultiple quantifiers. (The next subsection introduces the notationFrege used and anexpanded description of Frege’s notation is available.)

2.1 The Operators ofBegriffsschrift

2.1.1 The Judgement Stroke

Thejudgement stroke is perhaps the aspect of Frege’slogic, in both versions, that has been the subject of the mostcontroversy. Simply put, the judgement stroke, in the logic ofBegriffsschrift, transforms ajudgeable content intoa judgement:

A judgement will always be expressed by means of the symbol
which stands to the left of the symbol or complex of symbols whichgives the content of the judgement. If the small vertical stroke atthe left end of the horizontal one $|$ is omitted, then thejudgement will be transformed into amere complex of ideas,of which the writer does not state whether he recognizes its truth ornot. For example, let
mean the judgement “Opposite magnetic poles attract oneanother”, then
will not express this judgement, but should merely arouse in thereader the idea of the mutual attraction of opposite magnetic poles,in order, say, to draw conclusions from it and by means of these totest the correctness of the thought. In this weparaphraseusing the words “the circumstance that” or “theproposition that”. (1879a: §2)

InBegriffsschrift Frege makes it clear that not everyconceptual content is a judgeable content, and hence not everyexpression (and definitely not every name of an object) is eligible tobe the argument of the judgement stroke:

Not every content can become a judgement by placingbefore its symbol; for example, the idea “house” cannot.We therefore distinguishjudgeable andunjudgeablecontents. (1879a: §2)

He concludes his discussion of the rejection of the subject-predicateanalysis of propositions in favor of his function-argument approach,discussed above, with the following observation regarding the role ofthe judgement stroke:

Imagine a language in which the proposition “Archimedes waskilled at the capture of Syracuse” is expressed in the followingway: “The violent death of Archimedes at the capture of Syracuseis a fact”. Even here, if one wants, subject and predicate canbe distinguished, but the subject contains the whole content, and thepredicate serves only to present it as a judgement.Such alanguage would have only a single predicate for all judgements, namely“is a fact”. It can be seen that there is no questionhere of subject and predicate in the usual sense.OurBegriffsschrift is such a language and the symbolis its common predicate for all judgements. (1879a: §3)

Shortly after introducing the judgement stroke inBegriffsschrift, Frege makes the claim that it is composed oftwo parts, the horizontal stroke “ circumstance symbol ” and the vertical stroke“$|$”,which is the judgement strokeproper, and he suggests that the horizontal unites the component partsof the judgeable content that follows into a whole:

Thehorizontal stroke, from which the symbolis formed,binds the symbols that follow it into a whole, andassertion, which is expressed by means of the vertical stroke at theleft end of the horizontal, relates to this whole. The horizontalstroke may be called thecontent stroke, the vertical thejudgement stroke. The content stroke serves generally torelate any symbol to the whole formed by the symbols that follow thestroke.What follows the content stroke must always have ajudgeable content. (1879a: §2)

Note the fact that arguments of the judgement stroke are restricted tocontents that are judgeable (loosely put, to sentences, in an informalsense of the term), hence“$2$” is, in the logic ofBegriffsschrift, nota falsehood but is merely not well-formed. We will return to theimportance of this observation in understanding the differencesbetween Frege’s two logics below. But the horizontal does littleactual work in the logic ofBegriffsschrift: it never occursin isolation.

2.1.2 The Conditional Stroke

Next up is the conditional stroke. InBegriffsschrift Fregeexplains the conditional stroke as follows:

If $A$ and $B$ denote judgeable contents (§2), then there arethe following four possibilities:
$A$ is affirmed and $B$ is affirmed;
$A$ is affirmed and $B$ is denied;
$A$ is denied and $B$ is affirmed;
$A$ is denied and $B$ is denied.
now denotes the judgement thatthe third of these possibilitiesdoes not obtain, but one of the other three does. Accordingly, if
is denied, then this is to say that the third possibility does obtain;i.e., that $A$ is denied and $B$ is affirmed. (1879a: §5)

Simply put, the conditional stroke is Frege’sBegriffsschrift version of the material conditional: itcombines two conceptual contents into a single complex conceptualcontent that denotes a fact if and only if it is not the case that thefirst denotes a fact yet the second does not. In modern terminology,the consequent of the conditional (what Frege will call thesupercomponent in the logic ofGrundgesetze) occursabove the antecedent (what Frege will call thesubcomponentin the logic ofGrundgesetze). Note the explicit restrictionof the arguments of the conditional stroke to judgeable contents.

Although Frege does not discuss this as explicitly inBegriffsschrift as he does inGrundgesetze, it isworth noting that complex conditional stroke constructions can beparsed into antecedent and consequence in multiple ways. Consider:

conditional {term C} {term conditional term B term A}.

This expression is analogous (but of course not equivalent, in anyreasonable sense of equivalent) to“$C\rightarrow (B \rightarrow A )$” in modern notation. ButFrege often treats expressions of this form as instead expressingsomething more akin to the (classically) equivalent“$(C\land B) \rightarrow A$”. Inshort, Frege switches back and forth between reading the offsetformula above as a (binary) conditional with“$C$”as antecedent and

as consequent, and as a (ternary) conditional with both“$C$”and“$B$”as antecedents and“$A$”as consequent (this willbecome relevant in our discussion of his rules of inferencebelow).

2.1.3 The Negation Stroke

The third notion introduced by Frege inBegriffsschrift isthe negation stroke:

If a small vertical stroke is attached to the underside of the contentstroke, then this is intended to express the circumstance thatthe content does not obtain. Thus, for example,
means “$A$ does not obtain”. I call this small verticalstroke thenegation stroke. The part of the horizontal stroketo the right of the negation stroke is the content stroke of $A$,the part to the left of the negation stroke, on the other hand, is thecontent stroke of the negation of $A$. (1879a: §7)

Although it is not obvious that Frege thought of propositional logicas an identifiable subsystem of the logic ofBegriffsschrift(or of the logic ofGrundgesetze), negation and theconditional stroke are the only (Fregean analogues) of propositionaloperators that Frege introduces into either logic (identity does playsomething like the role of the biconditional in both the logic ofBegriffsschrift and the logic ofGrundgesetze).Frege does not offer an expressive completeness result (and it is notlikely that he was in a position, conceptually, to even state such aresult at the time of writingBegriffsschrift). But heconcludes §7 ofBegriffsschrift by gesturing in thisdirection, noting that these two operators allow us to express what wenow call inclusive disjunction, exclusive disjunction, and conjunctionas:

inclusive disjunction

not conditional {term conditional {not term B} {term A}} {not term conditional {term B} {not term A}}

exclusive disjunction

conjunction

respectively.

2.1.4 The Identity Operator

Frege next introduces what has become perhaps the most notorious partof the logic ofBegriffsschrift, his identity operator. Fregeis clearly struggling with the puzzles that he would eventually solvein “Sense and Reference” via the titular distinction, butthat notion is not yet available, and as a result he is faced with thefollowing puzzle: Given two names“$a$”and“$b$”,if“$a= b$” is true, and, if theconceptual content of names are their referents, then“$a= a$” and“$a= b$” have the same conceptualcontent. But this cannot be the case:“$a =a$” and“$a =b$” clearlydo not have the same conceptualcontent, since they imply different things.

As a result, Frege is forced to deny that“$a$”and“$b$”have the same conceptualcontent, at least in the context of identity claims. Hence, theirconceptual content cannot be their referents. Frege concludes that theconceptual content of names within identity statements are the namesthemselves, defining identity as follows (it is worth noting that heuses“$\equiv$” here, butmoves to the somewhat more standard notation“$=$”inGrundgesetze oncehe has solved these problems):

is therefore to mean:the symbol $A$ and the symbol $B$ havethe same conceptual content, so that $A$ can always be replaced by$B$ and vice versa. (1879a: §7)

This solves the problem, since in the context of a true identity claimlike“$a = b$”,“$a$”and“$b$”do not pick out the samething. Instead,“$a$”self-referentially picks out the symbol“$a$”(and similarly for“$b$”),and as a result the identityclaim“$a = b$” does notexpress (loosely put):

$a$ is the same as $b$

but instead something like (again, loosely put):

The thing picked out by“$a$”is the same as the thing picked out by“$b$”.

which has a different conceptual content from:

The thing picked out by“$a$”is the same as the thing picked out by“$a$”.

As a result, names withinBegriffsschrift formulas such as:

are forced to do double-duty: In the antecedent of the conditionalabove, the occurrences of“$a$”and“$b$”effectively denote themselves.In the consequent of this conditional, however,“$a$”and“$b$”denote more straightforwardly,to whatever objects these names actually name.

Frege is quite aware of the difficulties that arise as a result ofthis understanding of identity, beginning §8 ofBegriffsschrift with the following observation:

Identity of content differs from negation and conditionality byrelating to names, not to contents. Whilst elsewhere symbols simplyrepresent their contents, so that each combination into which theyenter merely expresses a relation between their contents, they at oncestand for themselves as soon as they are combined by the symbol foridentity of content; for this signifies the circumstances that twonames have the same content. Thus with the introduction of a symbolfor identity of content a bifurcation in the meaning of every symbolis necessarily effected, the same symbols standing one moment fortheir content, the next for themselves. (1879a: §8)

A solution to this problem would have to wait until the introductionof the sense/reference distinction.

We have so far restricted our attention to the application of theidentity symbol to names of objects. But Frege, inBegriffsschrift, never restricts the application of theidentity symbol in this manner, requiring only that its application berestricted in such a way that only judgeable contents result. Thus,within the logic ofBegriffsschrift the identity symbol canbe applied to any two arguments, rather than merely betweenobjects.

2.1.5 The Concavity for Expressing Generality

Finally, we have Frege’s device for expressing generality withinBegriffsschrift: the concavity:

In the expression of a judgement, the complex of symbols to the rightofcan always be regarded as a function of one of the symbols occurringin it.If a Gothic letter is put in place of the argument, and aconcavity containing this letter inserted in the content stroke, asin
$judgement all gothic a term \Phi(gothic a)$
then this signifies the judgement that the function is a factwhatever may be taken as its argument. Since a letter used as asymbol for a function, such as $\Phi$ in $\Phi(A)$, can beregarded as the argument of a function, it can be replaced by a Gothicletter in the manner just specified. The meaning of the Gothic letteris subject only to the obvious restrictions that the complex ofsymbols following a content stroke must still remain judgeable(§2), and that, if the Gothic letter occurs as a symbol for afunction, this circumstance must be taken into account.All otherconditions which must be imposed on what may replace a Gothic letterare to be included in the judgement. (1879a: §11)

Thus, the concavity is theBegriffsschrift version of(something like) the universal quantifier, and a formula of the form:

$all gothic a term \Phi(gothic a)$

is true (or, in the terminology Frege mobilizes inBegriffsschrift,is a fact) if and only if, for anyargument whatsoever, applying the function denoted by$\Phi(\xi)$ to that argument is true (or, again, is a fact). Notethat Frege explicitly restricts the formation of formulas in the logicofBegriffsschrift so that the concavity can only bind afunction that outputs a judgeable content when an appropriate argumentis filled in—hence, in the logic ofBegriffsschrift“ judgement all gothic a $\mathfrak{a}+1$” is not well-formed.

We can now add a bit more detail to the explication of the differencesbetween Frege’sBegriffsschrift distinction betweenfunction and argument, and the more modern-looking hierarchy ofobjects, first-level functions, second-level functions, andthird-level functions that will appear within the logic ofGrundgesetze. Properly understood:

$judgement all gothic a term \Phi(gothic a)$

withinBegriffsschrift does not assert that:

For any object $\mathfrak{a}$, $\Phi(\mathfrak{a})$ is a fact.

but instead says something like;

For any entity (of whatever “type” or “sort”)$\mathfrak{a}$such that the combination of $\Phi(\xi)$ with$\mathfrak{a}$ results in a judgeable content,$\Phi(\mathfrak{a})$ is a fact.

In understanding the quantified statement in this manner, we are freeto understand $\mathfrak{a}$ as the argument, and $\Phi(\xi)$ asthe function, or $\Phi(\xi)$ as the argument, and $\mathfrak{a}$as the function, subject only to the requirement that the combinationof $\mathfrak{a}$ and $\Phi(\xi)$ result in a judgeable content.Of course, if $\Phi(\xi)$ is a function that only maps objects tojudgeable contents, then the upshot is the same as standardfirst-order quantification. But Frege has, to emphasize this pointonce again, not yet introduced the conceptual machinery that allow himto identify such functions, nor has he ever claimed that a particularfunction must only take one type of entity (that is, take onlyobjects, or only functions, etc.) as arguments.

One clear indication that the sort of hierarchy that is mobilizedwithin the logic ofGrundgesetze in order to clean all thisup is not present—not even implicitly, in thebackground—within the logic ofBegriffsschrift is thatFrege does not introduce distinct quantifiers for distinct kinds ofentity (not even for function and argument). Although Frege usesdifferent styles of variable to suggest that some quantificationsrange over arguments, and others over functions, this is merelyheuristic—and it must be, since, as we have already seen, withinthe logic ofBegriffsschrift the function/argumentdistinction is not a metaphysical distinction found in the world, butinstead merely reflects different ways that one might parse the samestatement. Thus, the (single universal) quantifier in the logic ofBegriffsschrift ranges over both objects and functions(albeit in a rather complicated manner), and modern first- andhigher-order quantifiers do not actually appear withinBegriffsschrift in the way that they clearly appear withinthe logic ofGrundgesetze. Rather, there is a singlequantifier that ranges over objects and functions (and concepts, andrelations, etc.) alike, and the informal restrictions on whatconstructions are licit (i.e., that the result of applying function toargument must result in a judgeable content) constrain the potentialrange of each instance of this quantifier. Thus, the quantifiers ofthe logic ofBegriffsschrift bear a rather limitedresemblance to modern quantifiers (see Kemp 1995 and Heck & May2013 for arguments that this construction inBegriffsschriftdoesn’t even count as a genuine quantifier in the firstplace).

Frege has a second way of expressing generality—Roman letters.As we shall see, there is some controversy about how to understand theway in which this device is meant to be understood within the logic ofGrundgesetze. But there is no mystery regarding how they aremeant to be understood within the logic ofBegriffsschrift,since within the earlier work Frege states explicitly that the Romanletter generality device is an abbreviation for a special, andespecially important, instance of the German letter, concavity versionof universal quantification:

Only within its scope does a Gothic letter retain itsmeaning; the same Gothic letter can occur within various scopesin one judgement, without the meaning that may be ascribed to it inone scope carrying over to the others. The scope of a Gothic lettercan include another, as the example:
shows. In this casedifferent letters must be chosen;$\mathfrak{e}$ may not be replaced by $\mathfrak{a}$. It is, ofcourse permitted to replace a Gothic letter everywhere in its scope byanother particular one, provided that there are still differentletters standing where different letters stood before. This has noeffect on the content.Other substitutions are only allowable ifthe concavity immediately follows the judgement stroke, so thatthe content of the whole judgement makes up the scope of the Gothicletter. Accordingly, since this case is particularly important, Ishall introduce the following abbreviation for it.An italicletter always has as its scope the content of the whole judgement,without this needing to be signified by a concavity in the contentstroke. (1879a: §11)

Frege is quite explicit here: The occurrence of Roman (or italic)letters in a formula of the logic ofBegriffsschrift isnothing more than an abbreviation of the corresponding formula withthe Roman variables replaced by corresponding Gothic variables, andthe concavities corresponding to those variables placed immediatelyafter the judgement stroke. Thus, the formula appearing in thequotation above is, in the logic ofBegriffsschrift, merelyan abbreviation for:

judgement all gothic F all gothic G all gothic a conditional all gothic b {term gothic F (gothic a, gothic e)} {term gothic G (gothic a)}

and the Roman letter generality device is meant to remind the readerthat the restrictions on substitution (effectively protecting againsta clash of variables) that govern substitutions of German variables ingeneraldo not apply to variables bound by concavities inwhat we would call prenex position.

While this is Frege’s official understanding of Roman letters,he often treats formulas containing Roman letters as if they weresubstitution instances of the corresponding concavity-bound universalformulas—that is, as cases where the Roman letters pick outparticular functions and arguments. While this deserves more attentionthan can be given here, there is a practical reason that forces Fregeto make some such move: Since he has not introduced any name-formingoperators withinBegriffsschrift, the language does notcontain the resources to express any particular claims.

2.2 The Axioms and Rules ofBegriffsschrift

The logic ofBegriffsschrift officially contains nine axiomsand one rule, although two additional rules, used repeatedlythroughout the derivations, are explicated more informally “inpassing” by Frege. Within the derivations ofBegriffsschrift, Frege numbers formulas sequentially in termsof their occurrence within the sequence of derivations, and onlyintroduces an axiom when it is needed—hence the axioms areformulas 1, 2, 8, 28, 31, 41, 52, 54, and 58 on his numbering. I haveprovided a more convenient, conventional 1 through 9 numbering in thediscussion, and this numbering will be used in the comparison betweenthis system and the collection of axioms and rules given inGrundgesetze.

2.2.1 The Axioms

judgement conditional {term a} {term conditional term b term a}

Axiom 1 (Formula 1, 1879a: §14)

This is theBegriffsschrift analogue of:

\[A \rightarrow (B \rightarrow A)\]

although care should be taken, since Frege’s axiom, with its useof Roman letters, is a quantified formula, not a schema, and isperhaps better rendered as:

\[\forall A\; \forall B (A \rightarrow (B \rightarrow A))\]

where the quantifiers range over judgeable contents. Frege defendsthis axiom as follows:

[this axiom]… says: “The case in which $a$ is denied,$b$ is affirmed and $a$ is affirmed is excluded”. This isobvious since $a$ cannot be denied and affirmed at the same time. Wecan also express the judgement in words this way: “If aproposition $a$ holds, it holds also in the case an arbitraryproposition $b$ holds”. (1879a: §14)

Axiom 2 (Formula 2) is equally straightforward, so long as we rememberthat it is an abbreviation for a formula quantifying over judgeablecontents:

judgement conditional {term conditional {term c} {term conditional term b term a}} {term conditional {term conditional term c term b}{term conditional term c term a}}

Axiom 2 (Formula 2, 1879a: §15)

Frege’s argument that this axiom must be true stretches overfour pages, and will not be reproduced here. The argument for Axiom 1quoted above should give readers a feel for the flavor ofFrege’sBegriffsschrift justifications for particularaxioms (and similar comments apply to the later, even more complexaxioms). The intuitive validity of this axiom should be clear, sinceit is aBegriffsschrift analogue of something like:

\[\forall C\; \forall B\; \forall A(C \rightarrow(B \rightarrow A)) \rightarrow ((C \rightarrow B) \rightarrow (C\rightarrow A))\]

where the quantifiers range over judgeable contents.

The next axiom, Axiom 3 (Formula 8), allows (in combination withFrege’s version ofmodus ponens, see below) for there-arrangement of antecedents of a conditional:

judgement conditional {term conditional {term d} {term conditional term b term a}} {term conditional {term b} {term conditional term d term a}}

Axiom 3 (Formula 8, 1879a: §16)

This axiom will be replaced by a (more general) rule in the logic ofGrundgesetze.

Axiom 4 (Formula 28) (again, in combination with Frege’s versionofmodus ponens) provides a version of contraposition.

judgement conditional {term conditional term b term a} {term conditional {not term a} {not term b}}

Axiom 4 (Formula 28, 1879a: §17)

As with Axiom 3, this axiom will be replaced by a much more generalrule inGrundgesetze.

Axiom 5 (Formula 31) and Axiom 6 (Formula 41) are a pair of sorts,providing us with Frege’s axiomatic versions of double negationintroduction and double negation elimination:

judgement conditional {not not term a} {term a}

Axiom 5 (Formula 31, 1879a: §18)

judgement conditional {term a} {not not term a}

Axiom 6 (Formula 41, 1879a: §19)

These axioms, plus Frege’s version ofmodus ponens,complete what we might think of as the propositional portion of thelogic ofBegriffsschrift. Łukasiewicz proves that amodern transcription of these axioms, plus a modern transcription ofFrege’s version ofmodus ponens, are sound and completewith respect to classical logic with propositional quantifiers, and healso proves that (the modern transcription of) Axiom 3 is redundant inthe context of (modern transcriptions of) the other axioms plus,again,modus ponens (Łukasiewicz 1934). Of course, theusual warnings regarding transcribing Frege’s notations intomodern ones, and assuming that his notion of judgeable content (therange of the quantifiers that occur in Axioms 1 through 6) is the sameas a more modern notion of proposition or sentence, apply to thisresult.

Frege’s discussion of Axioms 5 and 6 also provide a niceillustration of the sorts of changes that occur in Frege’slogical thinking between the composition ofBegriffsschriftand the composition ofGrundgesetze. In the preface toBegriffsschrift Frege notes that:

I realized later that formulas (31) and (41) can be combined into thesingle one
which makes a few more simplifications possible. (1879a: Preface)

This principle, in combination with Axiom 7 and Frege’s versionofmodus ponens, does entail Axioms 5 and 6. Further, thereis every reason to think that this principle is valid on the informalsemantics Frege gives inBegriffsschrift, since (inBegriffsschrift) the variables are restricted to judgeablecontents, and there seems little reason to doubt that the judgeablecontent denoted by not not $a$ is identical to the judgeable content denoted by$a$.

All of this, however, is subject to an important caveat: thequantifiers within the logic ofBegriffsschrift areconstrained in a way that the quantifiers inGrundgesetze arenot. Within the logic ofBegriffsschrift Frege requires thatthe quantifiers be restricted so that the result of applying logicaloperators to entities in the range of the relevant quantifiers resultsinjudgeable contents. As a result, formulas that are validon theBegriffsschrift understanding of the logic are nolonger valid on theGrundgesetze understanding. Inparticular, as we shall see:

judgement not all gothic a (not not gothic a = gothic a)

is true on the intended interpretation of (the consistent,value-range-free fragment of) the logic ofGrundgesetze.

The next two axioms are relatively straightforward. Axiom 7 (Formula52) provides an axiomatic version of the indiscernibility ofidenticals:

judgement conditional {term c equiv d} {term conditional term f(c) term f(d)}

Axiom 7 (Formula 52, 1879a: §20)

A much stronger version of this axiom will make an appearance in thelogic ofGrundgesetze.

And Axiom 8 (Formula 54) provides us with self-identity:

Axiom 8 (Formula 54, 1879a: §21)

Again, this axiom applies to any argument $c$ (that is, to anythingwhatsoever), not just to objects.

Axiom 9 (Formula 58) is for our purposes a bit more interesting, notfor what it says, but instead for what it leaves out. Axiom 9 allowsus, in essence, to replace a German letter bound by a concavity in theantecedent of a conditional with a Roman letter:

judgement conditional {all gothic a term f(gothic a)} {term f(c)}

Axiom 9 (Formula 58, 1879a: §22)

Note that Frege does not provide a corresponding second-order versionof this axiom (as he does inGrundgesetze)—rather, thisaxiom should be read as covering both the first- and second-ordercase, thus expressing something along the lines of:

For any function $f$ and for any argument $c$ such that $f(c)$is a judgeable content: if, for any argument $\mathfrak{a}$ suchthat $f(\mathfrak{a})$ is a judgeable content, $f(\mathfrak{a})$is a fact, then $f(c)$ is a fact.

rather than:

For any (first-level) function $f$ and for any object $c$: if, forany object $\mathfrak{a}$, $f(\mathfrak{a})$ is a fact, then$f(c)$ is a fact.

2.2.2 The Rules of Inference

On to the rules of inference. In the preface toBegriffsschrift, Frege claims that he only uses one mode ofinference:

The restriction, in §6, to a single mode of inference isjustified by the fact that inlaying the foundations of suchaBegriffsschrift the primitive elements must be as simple aspossible if perspicuity and order are to be achieved. (1879a: Preface)

The rule in question is a version ofmodus ponens, whichFrege explains as follows:

From the explanation given in §5 it is clear that from the twojudgements
and
the new judgement$A$ follows. Of our four cases enumerated above, the third isexcluded by
and the second and fourth by$B$, so that only the first is left. (1879a: §6)

The four cases Frege is referring to are the four possiblecombinations of $A$ and $B$ being, or failing to be, facts givenin the explanation of the conditional stroke, quoted above.

At first glance, this appears to be the familiar rule ofmodusponens, but in fact it is a good bit more complicated. Fregeroutinely applies the rule to pairs of formulas that contain Romanletters. Remembering that inBegriffsschrift Roman lettersare abbreviations for prenex concavity quantifiers, the simple casethat Frege uses in his explication of this rule is really shorthandfor the transition from:

judgement all gothic a all gothic e conditional {term gothic a} {term gothic e}

and

to:

where the quantifiers range over judgeable contents. As a result, thisrule, as he applies is throughoutBegriffsschrift, is not thepropositional rulemodus ponens at all. Instead, it is ananalogue of something like the following (in modern notation):

\[\begin{aligned}~ & \forall a_1\forall a_2\dots \forall a_n\forall b_1\forall b_2\dots \forall b_m\forall c_1\forall c_2\dots\forall c_k\\&(\Phi_1(a_1, a_2, \dots a_n, b_1, b_2, \dots b_m) \rightarrow\Phi_2(a_1, a_2, \dots a_n, c_1, c_2, \dots c_k))\\[P_2]~ & \forall a_1\forall a_2\dots \forall a_n\forall b_1\forall b_2\dots \forall b_m (\Phi_1(a_1, a_2, \dots a_n, b_1,b_2, \dots b_m))\\\hline\\[C]~ & \forall a_1\forall a_2\dots \forall a_n\forall c_1\forall c_2\dots \forall c_k(\Phi_2(a_1, a_2, \dots a_n, c_1,c_2, \dots c_k))\end{aligned}\]

There are two things to notice about the rule, understood in thisway.

First off, once we recognize the role played by the (German-letterbound) quantifiers abbreviated by Roman letters in any instance of therule, it is clear that Frege’s defense of the rule is utterlyinadequate. Frege’s argument would be adequate for a particularsubstitution instance of the rule, where no Roman letters appeared.But it doesn’t address the much more general principle thatcodifies the manner in which he in fact applies the rule throughoutBegriffsschrift.

Second, this is not, in fact, a single rule at all, but is instead aschema for infinitely many rules: one for each triple of sequences ofvariables $a_1, a_2, a_n$; $b_1, b_2,\dots b_m$; and $c_1,c_2,\dots c_k$, where not only the number of $a_i$s, $b_i$s, and$c_i$s in each instance can vary, but their type (where, remember,these are argument variables and function variables, not objectvariables and function variables) can vary as well. It does not seemunlikely that the strongly schematic flavor of this rule would havebothered Frege, given that his interest was in giving particularlogical principles from which we could derive particular universallogical truths. As a result, it is perhaps unsurprising that Fregewill give a completely different (albeit somewhat unclear) account ofthe role of Roman letters inGrundgesetze.

Although Frege claims that this version ofmodus ponens ishis only rule of inference in the preface toBegriffsschrift,he modifies this claim later in the work, noting that:

In logic, following Aristotle, a whole series of modes of inferenceare enumerated; I use just this one—at least in all cases wherea new judgement is derived from more than one single judgement.(1879a: §6)

The two rules that Frege has in mind, which involve the transitionfrom a single judgement to a single judgement, are rules that we shallcall concavity-introduction and a rule of substitution (Frege does notgive them names).

Frege explains the rule of concavity-introduction as follows:

An italic letter may always be replaced by a Gothic letter thatdoes not yet occur in the judgement, the concavity being insertedimmediately after the judgement stroke. E.g. instead of:
one can put:

if $a$ occurs only in the argument-places of $X(a)$:It isalso clear that from:

$judgement conditional term A term \Phi(a)$

one can derive:

$judgement conditional {term A} {all gothic a term \Phi(gothic a)}$

if $A$ is an expression in which $a$ does not occur, and if$a$ stands only in the argument place of $\Phi(a)$. (1879a:§11)

Frege then gives a second example that takes advantage of the factthat conditional stroke constructions can be parsed into antecedent(s)and consequent in more than one way:

Similarly, from:
$judgement conditional {term B} {term conditional {term A} {term \Phi(a)}}$
we can deduce:
$judgement conditional {term B} {term conditional {term A} {all gothic a term \Phi(gothic a)}}$
(1879a: §11)

Frege’s concavity-introduction rule operates as follows: Givenany formula containing a Roman letter, we can infer any propositionthat uniformly replaces the Roman letter with a German letter andinserts a concavity containing the same German letter immediately infront ofsome consequent that contains all occurrences of thenew German letter (recalling that formulas can be parsed intoconsequent and antecedent in more than one way), or the concavity canbe placed in front or the entire formula (after the judgement stroke).The new German letter must be chosen so that it does not“conflict” with other German letters already present inthe original proposition. Looking at a more complex example, if“$A$”and“$B$”are any formulas notcontaining the Roman letter“$x$”,and“$\Phi(\xi)$”and“$\Psi(\xi)$”do not contain“$\mathfrak{a}$”,then from:

$judgement conditional {term A} {term conditional {term B} {term conditional term \Phi(x) term \Psi(x)}}$

we can infer any of:

$judgement conditional {term A} {term conditional {term B} {all gothic a term conditional {term \Phi(gothic a)} {term \Psi(gothic a)}}$

$judgement conditional {term A} {all gothic a term conditional {term B} {term conditional {term \Phi(gothic a)} {term \Psi(gothic a)}}}$

$judgement all gothic a conditional {term A} {term conditional {term B} {term conditional {term \Phi(gothic a)} {term \Psi(gothic a)}}}$

but not:

$judgement conditional {term A} {term conditional {term B} {term conditional {term \Phi(gothic a)} {all gothic a term \Psi(gothic a)}}}$

This rule does not really involve “introducing” aconcavity within the logic ofBegriffsschrift, since theRoman letter being replaced is, of course, an abbreviation of aninstance of the concavity. Instead, this rule is a means formoving a concavity from one location to another. The name“concavity introduction” is thus used here to emphasizethe connection between this rule and the syntactically similar rulefound in the logic ofGrundgesetze (where Roman letters are,as we shall see, not an abbreviation of corresponding initialconcavities, but are a second, completely separate device forachieving universal generality, and hence the rule does involveintroducing a concavity not present beforehand).

Frege’s final rule of inference inBegriffsschrift is arule of substitution: Once one has proven a particular formula, onemay, in later derivations, make use, not only of the proven formulaitself, but also of the result of carrying out any uniformsubstitution of expressions for Roman letters occurring in theoriginal—again, subject to the requirement that the formula andall relevant sub-formulas are judgeable contents. Jean van Heijenoortargues, in his introductory essay onBegriffsschrift, thatFrege applies his rule of substitution in illicit ways—ways thatwill lead to contradictions. See the supplementary essayThe Supposed Contradiction inBegriffsschrift for discussion.

The third section ofBegriffsschrift introduces definitionsof what have become known as the weak and strongancestralsof a relation, and proves a powerful induction theorem based on thesenotions. A careful examination of this construction is beyond thescope of this essay—the reader is encouraged to consult theentry onFrege’s theorem for more details.

The most important aspect of the third section ofBegriffsschrift, for our purposes at least, is the new bit ofnotation that Frege introduces: the definition stroke“ definition stroke: like the judgement symbol except two vertical lines on the left ”. The definition stroke occurs initially in a formula ofthe form:

$definition term \Phi equiv \Psi$

where“$\Psi$” is thedefiniendum, and“$\Phi$”is thedefiniens.After giving an example of the definition stroke in action (formula69, the notion of $F$ being hereditary in thef-sequence),Frege explains the definition stroke inBegriffsschrift asfollows:

This sentence is different from those considered previously sincesymbols occur in it which have not been defined before; it itselfgives the definition. It does not say, “The right side of theequation has the same content as the left side”; but,“They are to have the same content”. This sentence istherefore not a judgement; and consequently, to use the Kantianexpression, alsonot a synthetic judgement. […]
Although originally (69) is not a judgement, still it is readilyconverted into one; for once the meaning of the new symbols isspecified, it remains fixed from then on; and therefore formula (69)holds also as a judgement, but as an analytic one, since we can onlyget out what was put into the new symbols. This dual role of theformula is indicated by the doubling of the judgement stroke. (1879a:§24)

The use of the notion of sameness of conceptual content (i.e.,“$\equiv$”)in the definitions ofBegriffsschrift automatically entails that thedefiniens anddefiniendum have the same content(that is, denote the same fact or circumstance), whereas the improvedunderstanding of identity mobilized inGrundgesetze onlyentails, as a matter of logic, that thedefiniendum anddefiniens denote the same object. Hence, in his informalexplications Frege explicitly stipulates that withinGrundgesetze definitions the expressions occurring on bothsides of the identity symbol have not only the same reference but alsothe same sense.

3. The Logic ofGrundgesetze

The reader who is interested in the evolution of Frege’s logicalideas between the writing ofBegriffsschrift and the writingofGrundgesetze should consult the short supplementary essayThe Period BetweenBegriffsschrift andGrundgesetze. Here, we will jump directly into the formal system contained in thelatter work.

One of the main, and most obvious, differences between the logic ofBegriffsschrift and the logic ofGrundgesetze, otherthan the addition of value-ranges, is the fact that Frege now has afully worked out, rigorous type theory in place. The most fundamentaldistinction is that between objects, which are saturated, andfunctions (including concepts as a special case), which are not, andthus require being “completed” via application to one ormore arguments.

Two particularly important types of function are concepts andrelations. Aconcept is a unary function such that, for anyargument (of the appropriate type), the value of the function appliedto that argument is a truth-value. Arelation is a functionwith two (or more) arguments such that, for any pair (orn-tuple) of arguments (again, of the appropriate type), thevalue of the function applied to that pair is a truth-value (Frege1893/1903: §4, see also 1893/1903: §22).

Frege also subdivides functions in terms of thekinds ofargument that they take. Thus, a function is afirst-levelfunction if and only if it takes an object or objects (and henceonly takes an object or objects) as argument(s); a function is asecond-level function if and only if it takes a first-levelfunction or functions (and hence only takes a first-level function orfunctions) as argument(s); and a function is athird-levelfunction if and only if it takes a second-level function orfunctions (and hence only takes a second-level function or functions)as argument(s) (Frege 1893/1903: §21 through §23, see also§26).

In what follows we will divide up our discussion of the logic ofGrundgesetze into three sections, where the first considersthose notations that occur in the earlier logic ofBegriffsschrift (although often with rather differentunderstandings), the second presents those notations that are novel tothe logic ofGrundgesetze, and the third presents the axioms(now called Basic Laws) and rules of inference of the logic ofGrundgesetze.

3.1 The “Old” Operators ofGrundgesetze

3.1.1 The Judgement Stroke

As was the case inBegriffsschrift, the judgement stroke ofGrundgesetze transforms expressions into judgements. Unlikethe earlier system, however, in the logic ofGrundgesetze thejudgement stroke does not attach to expressions that name facts orcircumstances, but instead attaches to expressions that name objects(that is, proper names):

Above it is already stated that within a mere equation no assertion isyet to be found; with“$2 + 3 =5$” only a truth-value is designated, without its beingsaid which one of the two it is. Moreover, if I wrote“$(2+ 3 = 5) = (2 = 2)$” andpresupposed that one knows that $2 = 2$ is the True, even then Iwould not thereby have asserted that the sum of 2 and 3 is 5; rather Iwould only have designated the truth-value of: that“$2+ 3 = 5$” refers to the same as“$2 = 2$”. We are therefore inneed of another special sign in order to be able to assert somethingas true. To this end, I let the signprecede the name of the truth value, in such a way that, e.g., in:
it is asserted that the square of 2 is 4. I distinguish thejudgement from thethought in such a way that that Iunderstand by ajudgement the acknowledgement of the truth ofathought. (Frege 1893/1903: §5)

Thus, the application of the judgement stroke to aGrundgesetze expression lacking the judgement stroke assertsthat the expression in question is a name of the True, wheretheTrue is the object denoted by true sentences (andtheFalse is the object denoted by false sentences). An expression ofthe form:

$judgement \Phi$

now no longer says that “$\Phi$ is a fact”, but insteadsays something like “$\Phi$ is (i.e., is identical to) theTrue”. It is worth noting that this simple account of thejudgement stroke will be complicated somewhat in our discussion ofFrege’s newGrundgesetze understanding of Romanletters.

InGrundgesetze Frege once again suggests that the judgementstroke proper (1893/1903: §5), as well as the negation stroke(1893/1903: §6), the conditional stroke (1893/1903: §12),and the concavity (1893/1903: §8), can be understood asconsisting merely of the actual vertical “stroke” or line(or curve with variable, in the case of the concavity), with theattached horizontal portion(s) of the notation understood as separateoccurrences of the horizontal. In actual practice withinGrundgesetze the judgement stroke never appears without theattached horizontal. Unlike in theBegriffsschrift, however,the horizontal strokedoes occur relatively frequently on itsown, as an operator distinct from the judgement stroke, theconditional stroke, and the negation stroke.

In the logic ofGrundgesetze, thehorizontal strokeis a unary function symbol that attaches to names of objects, and itnames a function that always outputs a truth-value, regardless of thekind of object input:

I regard it as a function-name such that:
$circumstance \Delta$
is the True when $\Delta$ is the True, and is the False when$\Delta$ is not the True. Accordingly,
$circumstance \xi$
is a function whose value is always a truth-value, or a conceptaccording to our stipulation. (1893/1903: §5)

In other words, if the horizontal is prefixed to the name of atruth-value“$\Delta$”, thenthe resulting complex name:

$circumstance \Delta$

names the same truth-value as is named by“$\Delta$”.If, however,“$\Delta$”does not name atruth-value, then“ circumstance $\Delta$” names the False.

Of particular importance here is the fact that, within the logic ofGrundgesetze, the horizontal stroke is not limited in itsapplication to judgeable contents: the horizontal stroke can bemeaningfully applied to any name whatsoever. As a result ofFrege’s insistence that all functions be defined on allarguments of the appropriate type, the function denoted by thehorizontal stroke must be defined on all objects. Thus, when appliedto an object that is not a truth-value, it outputs the False.

As a result, although Frege requires that the judgement stroke beattached to a truth-value name in the explication of the judgementstroke quoted above, we can, within the logic of Grundgesetze, achievethe effect of attaching the judgement stroke to any name $\Delta$whatsoever by attaching the judgement stroke to“ circumstance $\Delta$”, obtaining:

$judgement (circumstance \Delta)$

If the horizontal stroke is a part of the judgement stroke, and giventhat (as will be discussed below) multiple horizontals can be fusedinto a single horizontal, then the above judgement is equivalent to:

$judgement \Delta$

or would be, were the latter well-formed. Thus“ judgement $2$” (or, at least,“$($ circumstance $2)$” is a well-formed judgement in the logic ofGrundgesetze, albeit an incorrect one.

3.1.2 The Negation Stroke

Within the logic ofGrundgesetze, thenegationstroke is a unary function symbol that attaches to names ofobjects—in Frege’s terminology it names a first-levelconcept. Like the horizontal, the negation stroke transforms anyproper name into a truth-value name:

We do not need a specific sign to declare a truth-value to be theFalse, provided we have a sign by means of which every truth-value istransformed into its opposite, which in any case is indispensable. Inow stipulate:
The value of the function
$not \xi$
is to be the false for every argument for which the value of thefunction
$circumstance \xi$
is the True, and it is to be the True for all other arguments.(1893/1903: §6)

Thus, if the negation stroke is prefixed to the name of a truth-value“$\Delta$”, then“ not $\Delta$” names the True if“$\Delta$”names the False, andnames the False if“$\Delta$”names the True. If, however,“$\Delta$”does not name atruth-value, then“ not $\Delta$” names the True.

Frege’s treatment of the negation stroke as a total function hasodd consequences. For example, if“$\Delta$”is the name of any objectother than a truth value, then “ not $\Delta$” names the True, and hence we have:

$judgement not \Delta$

Frege explicitly notes that“ judgement not $2$” is a correct judgement (1893/1903: §6).

3.1.3 The Conditional Stroke

Within the logic ofGrundgesetze, Frege’sconditional stroke is a binary function symbol that attachesto names of objects—in Frege’s terms the conditionalstroke names a first-level relation:

Next, in order to be able to designate subordination of concepts andother important relations, I introduce the function with twoarguments:
$conditional term \zeta term \xi$
by means of the specification that its value shall be the False if theTrue is taken as the $\zeta$-argument, while any object that is notthe True be taken as the $\xi$-argument; that in all other cases thevalue of the function shall be the True. (1893/1903: §12)

The conditional stroke is a total function: Given any two proper names“$\Delta$” and“$\Gamma$”:

$conditional term \Delta term \Gamma$

is a name of the True if either“$\Delta$”fails to name the True(i.e., either names the False or does not name a truth-value), or“$\Gamma$” names the True; andit names the False otherwise. Hence, for any name“$\Delta$”whatsoever:

$conditional term 2 term \Delta$

is a name of the true, and hence:

$judgement conditional term 2 term \Delta$

is a correct judgement in the logic ofGrundgesetze

Frege calls the lower component of a conditional (what modern readerswould term the antecedent) thesubcomponent of theconditional, and the upper component (what modern readers would callthe consequent) thesupercomponent. As noted in ourdiscussion of the logic ofBegriffsschrift, however,conditional stroke constructions can be parsed into supercomponent andsubcomponents(s) in multiple ways. For example, given proper names“$\Delta$”,“$\Gamma$”,“$\Theta$”,“$\Lambda$”,and“$\Xi$”,we can parse the complexexpression:

$conditional {term \Delta} {term conditional {term \Gamma} {term conditional {term \Theta} {term conditional term \Lambda term \Xi}}}$

as having any of:

$conditional {term \Gamma} {term conditional {term \Theta} {term conditional term \Lambda term \Xi}}$

$conditional {term \Theta} {term conditional term \Lambda term \Xi}$

$conditional term \Lambda term \Xi$

or“$\Xi$” as supercomponent(with one, two, three, or four subcomponents on each reading,respectively). Although Frege introduces the conditional stroke as ifit is a simple binary first-level function from pairs of objects totruth-values, in his manipulation of the conditional stroke (andespecially in the rules of inference for reasoning with theconditional stroke) he treats the conditional stroke much more like akind of open-endedn-ary function name that takes a singleargument as supercomponent, but which can take any (finite) number ofarguments as its subcomponents. Since many of Frege’s rules ofinference are formulated in terms of adding, eliminating, orrepositioning supercomponents and subcomponents, this systematicambiguity has profound implications for how proofs are constructedwithinGrundgesetze.

The multiple-subcomponent reading of complex conditional strokeconstructions has two consequence worth mentioning at this point.First, Frege notes that, on the reading of:

$conditional {term \Delta} {term conditional term \Lambda term \Theta}$

where $\Delta$ and $\Lambda$ are the two subcomponents, eachsubcomponent plays exactly the same role as the other—the“ordering” of subcomponents does not matter (1893/1903:§12). Thus, this expression names the same truth-value as:

$conditional {term \Lambda} {term conditional term \Delta term \Theta}$

Frege introduces a rule of inference (one that can be applied withinderivations without comment) that allows arbitrary re-ordering ofsubcomponents. This rule amounts to aGrundgesetze analogueof Axiom 3 from the logic ofBegriffsschrift.

Along similar lines:

$conditional {term \Delta} {term conditional term \Delta term \Theta}$

names the same truth-value as:

$conditional term \Delta term \Theta$

Frege introduces a rule of inference that allows one to move from thefirst formula to the second, and allows the “fusion” ofidentical subcomponents generally (which, likewise, can be appliedwithin derivations without comment). Frege goes on to note that theserules of inference generalize to conditional stroke constructions withany number of subcomponents.

3.1.4 Equivalences for the Judgement Stroke

Now that we have considered what we might naturally (although, as wehave already noted, rather anachronistically) consider to be thepropositional fragment of the logic ofGrundgesetze, we needto return to the horizontal stroke. Frege suggests that the negationstroke, the conditional stroke, and the judgement stroke can beunderstood as consisting merely of the actual vertical“strokes” or lines involved in their formalization, withthe attached horizontal portions of their notation understood asseparate occurrences of the horizontal (see (1893/1903: §5,§6, and §12). As a result, for any name“$\Delta$”,all of:

$\Delta$,
($\Delta$),
($\Delta$), and
(($\Delta$))

name the same truth-value (1893/1903: §6), and, for any names“$\Delta$” and“$\Gamma$”,all of:

$conditional \Delta term \Gamma$

(a)

$circumstance (conditional term \Delta term \Gamma)$

(b)

$conditional term (circumstance \Delta) term \Gamma$

(c)

$conditional {term \Delta} {term (circumstance \Gamma)}$

(d)

$conditional {term (circumstance \Delta )} {term (circumstance \Gamma)}$

(e)

$circumstance (conditional {term \Delta} {term (circumstance \Gamma)})$

(f)

$circumstance (conditional {term (circumstance \Delta)} {term \Gamma})$

(g)

$circumstance (conditional {term (circumstance \Delta)} {(circumstance Gamma)})$

(h)

name the same truth-value (1893/1903: §12). Frege’s callsthese equivalences, and the transformations that result from replacingone of the expressions above with another, equivalent formulation, thefusion of horizontals. Like the permutation of subcomponentsand the fusing of identical subcomponents, Frege allows one to fuse(and “un-fuse”) horizontals withinGrundgesetzederivations (1893/1903: §48) without comment.

The careful reader might wonder why Frege chose the particularfunctions he in fact chose. Could he not have defined negation so that“ not $\Delta$” named the Trueif“$\Delta$” was not the nameof a truth-value? We are now in a position to provide an answer tothis question: The fact that the negation stroke and the conditionalstroke must fuse with the horizontal entails that the particulardefinitions of the negation stroke and the conditional stroke providedby Frege are the only ones possible, given the way in which he definesthe horizontal stroke (Berg & Cook 2017).

3.1.5 The Equality Sign

We now arrive at another dramatic difference between the logic ofBegriffsschrift and the logic ofGrundgesetze. Nowthat he has introduced the sense/reference distinction, Frege canexplain the difference in content between“$a= a$” and“$a= b$” (where both are true) interms of a difference in sense. Hence, in the logic ofGrundgesetze Frege’sequality sign is definedas one would expect:

We have already used the equality-sign rather casually to formexamples but it is necessary to stipulate something more preciseregarding it.
\[\text{‘}\Gamma =\Delta\text{’}\]
refers to the True, if $\Gamma$ is the same as $\Delta$; in allother cases it is to refer to the False. (1893/1903: §7)

Note that he has shifted to using the traditional identity symbol“$=$”, rather than the specialsymbol“$\equiv$” that heintroduced for sameness of conceptual content inBegriffsschrift. A nice overview of the development ofFrege’s sense/reference distinction, with particular attentionpaid to the role that the distinction plays in the later logic ofGrundgesetze, can be found in Kremer (2010).

While the definition of the equality-sign is now straightforward,Frege’s use of it is somewhat different from the way in whichthe equality symbol is used in modern predicate logic. Frege uses theequality-sign when making everyday equality claims such as“$2+ 2 = 4$”, but he also uses thesame symbol flanked by truth-value names in order to express the claimthat the truth-value names are names of the sametruth-value—that is, that the expressions in question areequivalent. This explains an apparent oversight in Frege’sdiscussion of defined propositional operators. Frege does notexplicitly provide a definition of the material biconditional withinGrundgesetze, although he could have easily defined thematerial biconditional along standard lines as the conjunction of twoconditionals:

$not conditional {term conditional term \Delta term \Gamma} {not term conditional term \Gamma term \Delta}$

If“$\Delta$” and“$\Gamma$”are both truth-valuenames, then this defined notion and the function named byFrege’s primitive equality-sign output the same value. If one orboth of“$\Delta$” and“$\Gamma$”are proper names but nottruth-value names, however, then the value of the complex materialconditional applied to these arguments can differ from the value ofthe equality-sign applied to them. For example:

not conditional {term conditional term 1 term 2} {not term conditional term 2 term 1}

is a name of the True, while “1 = 2” is a name of theFalse.

Frege notes that the identity sign, in combination with thehorizontal, allows us to construct a function that maps thetruth-values to the True, and every other object to the False(1893/1903: §5):

$\xi = circumstance \xi$

This truth-value concept helps us to sort out a technical issuerelated to one of the deep differences between the logic ofBegriffsschrift and the logic ofGrundgesetze.

Recall that Frege noted in the preface toBegriffsschriftthat he could have added:

to the logic, and this principle would have simplified thepresentation, as it implies both Axiom 5 and Axiom 6. We also notedthat this claim is correct with respect to the logic ofBegriffsschrift, since the quantifiers in question arerestricted to judgeable contents. Things stand differently, however,within the logic ofGrundgesetze, since on theGrundgesetze interpretation this same formula (with“sameness” understood purely syntactically, subject to thereplacement of“$\equiv$” with“$=$”) is false. Let“$\Delta$”be the name of any objectthat is not a truth value. Then “ not $\Delta$” is a name of the True, hence“ not not $\Delta$” is a name of the False, and“$\Delta$”and“ not not $\Delta$” do not name the same object. So“ not not $\Delta= \Delta$” is a name of the False.

We can use the truth-value concept just discussed to construct acorrect judgement within the logic ofGrundgesetze thatcaptures the intuitive import of the principle Frege considered addingtoBegriffsschrift:

judgement conditional {term (a = circumstance a)} {term (a = not not a)}

In short, if“$\Delta$” namesa truth value, then“$\Delta$”and the double negation of“$\Delta$”name the same object.

3.1.6 Two Forms of Universal Quantification

Next up are the quantifiers. In the logic ofGrundgesetze,unlike (strictly speaking) in the logic ofBegriffsschrift,Frege mobilizes two distinct forms of universal quantification. Thefirst of these is the simplest: the concavity. The concavity (withassociated German letter) is a unary second-level concept, mappingfirst-level functions to truth-values:

[…] let:
$all gothic a term \Phi(gothic a)$
refer to the True if the value of the function $\Phi(\xi)$ is theTrue for every argument, and otherwise the False. (Frege 1893/1903:§8)

Frege includes no restriction that“$\Phi(\xi)$”must be the name of aconcept, hence“ judgement all gothic a $\mathfrak{a} + 1$” is a well-formed judgement withinthe logic ofGrundgesetze, albeit an incorrect one. In fact,he could not have coherently imposed any such restriction, sincesecond-level functions must be defined for all first-level functionsin exactly the same manner as first-level functions must be definedfor all objects.

In the logic ofGrundgesetze, the concavity, like thenegation stroke and the conditional stroke, “fuses” withhorizontals. In other words, all of the following name the sametruth-value (i.e., are equivalent):

$all gothic a term \Phi(gothic a)$ ,
$circumstance (all gothic a term \Phi(gothic a))$ ,
$all gothic a term (circumstance \Phi(gothic a))$ , and
$circumstance (all gothic a term (circumstance \Phi(gothic a)))$

and Frege also allows the fusing or un-fusing of horizontals connectedto the concavity to be carried out without comment within derivations(1893/1903: §8).

Frege explicitly introduces notation for second-order quantificationvia the concavity, using the now-familiar method: identifyingwhich (in this case third-level) function such a quantifier names. If“$\mathcal{F}_\beta$” is asecond-level function name (and where the subscripted occurrence of“$\beta$” binds object-leveloccurrences of“$\beta$” thatoccur in the argument to which“$\mathcal{F}_\beta$”is applied),then:

$all gothic f term \mathcal{F}_\beta(gothic f (\beta))$

names the truth-value of the claim that, for every first-levelfunction $\Phi(\xi)$, the result of applying the function named by“$\mathcal{F}_\beta$” to$\Phi(\xi)$ is the True (1893/1903: §24). Thus, unlike thelogic ofBegriffsschrift, the logic ofGrundgesetzeinvolves distinct quantifiers for first- and second-orderquantification, rather then merely introducing a single quantifierthat sometimes ranges over functions and at other times ranges overthe arguments of those functions depending on context.

Frege does not introduce notation for third- or higher orderconcavity-quantification, since from a practical perspective he neverneeds it: instead, he uses his value-range operator to“reduce” the level of various constructions withinGrundgesetze (as described below) so that they are in therange of his first- and second-order concavity.

Finally, we have the definition stroke. As inBegriffsschrift, Frege uses thedefinition stroke“ definition symbol ” to indicate when a sentence is a definition:

In order to introduce new signs by means of those already known, wenow require thedouble-stroke of definition which appears asa double judgement-stroke combined with a horizontal:
and which is used instead of the judgement stroke where something isto be defined rather than judged. By means of adefinition weintroduce a new name by determining that it is to have the same senseand the same reference as a name composed of already known signs. Thenew sign thereby becomes co-referential with the explaining sign; thedefinition thus immediately turns into a proposition. Accordingly, weare allowed to cite a definition just like a proposition replacing thedefinition stroke by a judgement stroke. (1893/1903: §27)

There is little difference between this explanation of the definitionstroke and the one given inBegriffsschrift, but it is worthnoting that the judgements of equality entailed byGrundgesetze definitions only assert that“$a$”and“$b$”have the same referent, notthat they have the sense. Thus, Frege explicitly stipulates that thedefiniens anddefiniendum of aGrundgesetzedefinition have the samesense.

3.2 The New Operators ofGrundgesetze

3.2.1 A Device for Generality: Roman Letters

3.2.1.1 The Basics of Roman Letters

We begin our discussion of the new notions that did not occur withinthe logic ofBegriffsschrift with the second means forexpressing generality withinGrundgesetze: theRomanletter generality device. At first glance, this might besurprising, since Roman letters were used as a device for expressinggenerality withinBegriffsschrift, as discussed above. But inBegriffsschrift they were (officially, at least) merely anabbreviation for instances of the concavity that occurred initially inthe formula in question. In the logic ofGrundgesetze,however, Roman letters are a completely new, independent device ofgeneralization.

Restricting our attention to the simplest case, where the lower-caseRoman letter“$x$”“indicates” (in Frege’s terminology) an object, andwhere“$\Phi(\xi)$” is anyfirst-level function name :

$judgement \Phi(x)$

is a correctGrundgesetze proposition if and only if thefunction named by“$\Phi(\xi)$”outputs the True forevery possible argument, and it is incorrect otherwise (1893/1903:§17).

The astute reader will have noticed that (following Frege) ourexplanation of the Roman letter generality device does not follow thegeneral pattern utilized in our discussion of the horizontal, negationstroke, conditional stroke, or concavity. In short, we have notidentified a function to which theexpression“$\Phi(x)$”refers (where“$x$”is an occurrence of the Romanletter generality device), but instead have only explained whenjudgements involving the Roman letter generality device arecorrect.

Frege never gives a function-identifying definition of this sort forthe Roman letter generality device—that is, he never identifiesa particular second-level function which is denoted by this particularlogical device. The reasons for this are simple: Were he to do so, (i)it would presumably be the same function as is picked out by instancesof the concavity of the same “order”; (ii) it would beapplicable to sub-expressions within aGrundgesetze formula,but it is not; and (iii) it would not have theflexibility ofscope that it in fact has. As a result, expressions containingRoman letters are not names:

I shall callnames only those signs or combinations of signsthat refer to something. Roman letters, and combinations of signs inwhich these occur, are thus notnames as they merelyindicate. A combination of signs which contains Romanletters, and which always results in a proper name when every Romanletter is replaced by a name, I will call aRomanobject-marker. In addition, a combination of signs which containsRoman letters and which always results in a function-name when everyRoman letter is replaced by a name, I will call aRomanfunction-marker orRoman marker of a function.(1893/1903: §17)

Given the fact that the Roman letter generality device seems to be ofa very different character than the other logical notions found inGrundgesetze, the reader might wonder (a) why Frege includedit at all, and (b) how, exactly, we should understand it. The answerto the first question is relatively simple, the answer to the second,less so. Frege explains the Roman letter generality device asfollows:

Let us now see how the inference called “Barbara” in logicfits in here. From the two propositions:
“All square roots of 1 are fourth roots of 1”
and:
“All fourth roots of 1 are eighth roots of 1”
we can infer:
“All square roots of 1 are eighth roots of 1”
If we now write the premises thus:
$judgement all gothic a conditional {term gothic a^2 = 1} {term gothic a^4 = 1}$
$judgement all gothic a conditional {term gothic a^4 = 1} {term gothic a^8 = 1}$
then we cannot apply our modes of inference; however, we can if wewrite the premises as follows:
$judgement conditional {term x^2 = 1} {term x^4 = 1}$
$judgement conditional {term x^4 = 1} {term x^8 = 1}$
Here we have the case of §15. Above we attempted to expressgenerality in this way using aRoman letter, but abandoned itbecause we observed that the scope of generality would not beadequately demarcated. We now address this concern by stipulating thatthescope of aRoman letter is to include everythingthat occurs in the proposition apart from the judgement stroke.Accordingly, one can never express the negation of a generality bymeans of a Roman letter, although we can express the generality of anegation. An ambiguity is thus no longer present. Nevertheless, it isclear that the expression of generality with German letters andconcavity is not rendered superfluous. Our stipulation regarding thescope of aRoman letter is only to demarcate its narrowestextent and not its widest. It thus remains permissible to let thescope extend to multiple propositions so that the Roman letters aresuitable to serve in inferences in which the German letters, withtheir strict demarcation of scope, cannot serve. So, when ourpremisses are
$judgement conditional {x^2 = 1}{x^4 = 1}$
and
$judgement conditional {x^4 = 1}{x^8 = 1}$
then in order to make the inference to the conclusion
$judgement conditional {x^2 = 1}{x^8 = 1}$
we temporarily expand the scope of “$x$” to include bothpremises and conclusion, although each of these propositions holdseven without this extension. (1893/1903: §17)

There are a number of important things to note about this passage. Thefirst is that this novel treatment of the Roman letter generalitydevice (in comparison to the logic ofBegriffsschrift) ismotivated by exactly the puzzle regardingmodus ponens thatwe raised earlier in this essay. Here the inference in question is aversion ofhypothetical syllogism (explicated in §15 ofGrundgesetze, and about which more below), but the issue isthe same. The rule in question allows us, for any expressions$\Delta$, $\Gamma$, and $\Theta$, to move from:

$judgement conditional term \Delta term \Gamma$

$judgement conditional term \Gamma term \Theta$

to:

$judgement conditional term \Delta term \Theta$

But if, as was the case in the logic ofBegriffsschrift, theRoman letter involving formulas:

judgement conditional term x^2 = 1 term x^4 = 1

judgement conditional term x^4 = 1 term x^8 = 1

are merely abbreviations of:

$judgement all gothic a conditional {term gothic a ^2 = 1} {term gothic a^4 = 1}$

$judgement all gothic a conditional {term gothic a ^4 = 1} {term gothic a^8 = 1}$

then we do not, strictly speaking, have an instance of the premisesfor this rule, and thus cannot advance to the desired (and correct)conclusion. Thus, we need an alternative understanding of the Romanletter generality device.

Frege suggests that, when we carry out the relevant instance ofhypothetical syllogism, we temporarily extend the scope ofthe Roman letter“$x$” so thatit includes both premises and the conclusion. As a result, the Romanletter“$x$”“indicates” the same object (whatever object this mightbe) uniformly throughout all threeGrundgesetze propositions,and we can apply hypothetical syllogism.

3.2.1.2 Higher Order Quantification and Roman Letters

Frege also allows both second- and third-order quantification to beexpressed via the Roman letter generality device. Thus,if“$\Delta$” is the name ofan object, then:

$judgement f(\Delta)$

is a correctGrundgesetze proposition if and only if, for anyfirst-level function $f$, the result of applying $f$ to the objectnamed by“$\Delta$” is theTrue. Likewise, if“$\Phi(\zeta)$”is a first-levelfunction name, then:

$judgement M_\beta(\Phi(\beta))$

is a correctGrundgesetze proposition if and only if, forevery second-level function $\mathcal{F}$, the result of applying$\mathcal{F}$ to the function named by“$\Phi(\xi)$”is the True(1893/1903: §25). Although Frege does explicitly providetreatment of third-order quantification via the Roman generalitydevice (unlike his treatment of the concavity, which is limited tofirst- and second-order), he does not provide any notation for fourth-or higher-order quantification of either sort.

3.2.1.3 How Roman Letters Work

We now move on to the second question: How, exactly, is the Romanletter device to work? How are we to understand Frege’s ideathat expressions involving the Roman letter generality device“indicate” but do not “name” truth-values, andhow are we to understand the idea that their scope must contain theentirety of the judgement in which they occur (other than, possibly,the judgement stroke), but could be expanded to include more than oneformula at once? The right way to work out the answers to thesequestions is a matter of rather substantial controversy. Landinisuggests that Frege is gesturing at the idea of variable assignments(Landini 2012), an idea that would not be developed fully until Tarski(1933); while Heck suggests instead that Frege intended the Romanletters to be understood substitutionally in terms of auxiliary names(that is, “extra” names not included in theobject-language-level vocabulary), where, for example, an expressionof the form:

$judgement \Phi(x)$

(or multiple such expressions taken together in inference) indicatesthe True if and only if“$\Phi(n)$”is a name of the True,no matter what object the auxiliary name“$n$”denotes (Heck 2012). For amodern version of this kind of treatment of quantifiers, see Mates(1972). No attempt will be made here to settle this debate.

3.2.2 The Value-Range Operator

The most notorious primitive notion inGrundgesetze isFrege’s value-range operator, given its central role inRussell’s paradox and the collapse of Frege’s logicistproject. The value-range symbol, or “smooth breathing”,names a second-level function from first-level functions to objects.Given any first-level function name“$\Phi(\xi)$”,the object named bythe application of the unary second-level“smooth-breathing” operator to“$\Phi(\xi)$”:

\[ἐ(Φ(\varepsilon))\]

is the value-range of the function named by“$\Phi(\xi)$”.Unlike the otherprimitive function symbols in the logic ofGrundgesetze,Frege does not give an explicit definition of the function picked outby the value-range operator (for good reason, since there is, thanksto Cantor’s theorem, no such function!), instead explicating thenotion in terms of an informal version of Basic Law V:

I use the words:
“The function $\Phi(\xi)$ has the samevalue-range asthe function $\Psi(\xi)$”
always as co-referential with the words:
“the function $\Phi(\xi)$ and $\Psi(\xi)$ always have thesame value for the same argument”.
(1893/1903: §3)

One of the most important applications of the value-range operator isits application to first-level concepts, and the resulting objects,which Frege callsextensions, can be thought of, from amodern perspective and loosely speaking, as akin to the graphs of thecharacteristic functions of these concepts. Extensions do“behave” logically very similarly to (naive) sets, but thesensitive (or merely sensible) reader should be wary of attributingtoo much of our own modern views about sets ontoGrundgesetzeextensions. For an in-depth examination of the development ofFrege’s thought regarding the nature of extensions, see Burge(1984).

Frege identifies another sub-class of objects that can be constructedusing the value-range operator that do not correspond to anythingwidely used within modern mathematics: double value-ranges. Given anybinary first-level function name“$\Phi(\xi,\zeta)$”, we form the double value-range of (thefunction named by)“$\Phi(\xi, \zeta)$” by applying thevalue-range operator to“$\Phi(\xi,\zeta)$” (binding the argument place marked by“$\xi$”),obtaining the unaryfirst-level function name“$ἐ(\Phi(\varepsilon,\zeta))$”. We now obtain the double value-range of“$\Phi(\xi,\zeta)$” by applying thevalue-range function a second time, to“$ἐ(\Phi(\varepsilon,\zeta))$”, to obtain“$ἀἐ(\Phi(\varepsilon,\alpha))$”, which names the double value-range of thefunction named by“$\Phi(\xi,\zeta)$” (1893/1903: §36).

The need for double value-ranges provides one very practicalexplanation for the fact that the higher-order quantifiers ofGrundgesetze range over functions generally rather thanmerely over concepts and relations as in modern systems. Given abinary relation symbol“$\Phi(\zeta,\xi)$”, the result of the first step—that is, thereferent of“$ἐ(\Phi(\varepsilon,\xi))$”—is not a concept but a function that mapsobjects to value-ranges. Thus, the introduction of value-rangesrequires that Frege accept not just concepts and relations, butfunctions more generally, into his higher-order ontology—seeLandini (2012: ch. 4) for further discussion.

Frege only defines value-ranges for first-level functions. Of course,he could have extended the notion in order to obtain object-levelanalogues of second- and third-level functions directly. But there isno need, since we can “reduce” second- and higher-levelfunctions to first-level functions via repeated applications of thevalue-range operator on first-level functions. For example, given asecond-level concept name“$\mathcal{F}_\beta$”mappingfirst-level functions to truth-values, we can construct anobject-level analogue by first constructing the name of the conceptthat holds of an object if and only if that object is the value-rangeof a first-level function that the concept named by“$\mathcal{F}_\beta$”maps to theTrue:

$not all gothic f conditional {term \xi = ἐ(f(\varepsilon))} {not term \mathcal{F}_\beta(f(\beta))}$

The object-level analogue of the second-level concept named by“$\mathcal{F}_\beta$”is then thevalue-range of this first-level concept:

$ἀ (not all gothic f conditional {term \alpha= ἐ(f(\varepsilon))} {not term \mathcal{F}_\beta(f(\beta))})$

This maneuver is quite general. Any time one might, withinGrundgesetze, desire an object-level analogue of a second- orthird-level function, one can use this trick to construct such anobject.

3.2.3 The Backslash Operator for Definite Descriptions

The final primitive notation of the logic ofGrundgesetze isthebackslash. The backslash is a unary first-level functionmapping objects to objects:

[…] we can help ourselves by introducing the function:
\[\backslash \xi\]
with the specification to distinguish two cases:
if, for the argument, there is an object $\Delta$ such that$ἐ(\Delta = \varepsilon)$ is the argument, then the value ofthe function $\backslash\xi$ is to be $\Delta$ itself.
if, for the argument, there is no object $\Delta$ such that$ἐ(\Delta = \varepsilon)$ is the argument, then the argumentitself is to be the value of the function $\backslash\xi$.
(1893/1903: §11)

A bit of terminology is helpful: Given any proper name“$\Delta$”,let us call the objectnamed by“$ἐ(\Delta =\varepsilon)$” thesingleton-extension of theobject named by“$\Delta$”.Then:

“$\backslash\Gamma$” is co-referential with“$\Delta$”if“$\Gamma$”names thesingleton-extension of the object named by“$\Delta$”
“$\backslash\Gamma$” is co-referential with“$\Gamma$”otherwise.

Put even more simply, the backslash is a“singleton-stripping” device.

Frege utilizes the backslash as kind of a definite descriptionoperator. In modern treatments, a definite description operator“$\iota$”attaches to predicatesand, given a predicate“$\Phi(x)$”,“$\iotax(\Phi(x))$” denotes the unique object that satisfies thepredicate“$\Phi(\xi)$” (ifthere is such). Frege, however, in keeping with the strategy ofreducing levels via successive applications of the value-rangeoperator, defines his definite description operator as applying, notto concepts, but rather to their value-ranges. Thus, where“$\Phi(\xi)$”is a concept name“$\backslash\Gamma$” denotesthe unique object that is mapped to the True by the concept named by“$\Phi(\xi)$”, if there issuch, and denotes the object named by“$ἐ(\Phi(\varepsilon))$”otherwise.

3.3 The Axioms ofGrundgesetze

A notable difference between the logic ofBegriffsschrift andthe logic ofGrundgesetze is that the former depends on manyaxioms, but very few rules of inference, while the latter depends onfewer axioms but more rules. The logic ofGrundgesetzecontains a mere six axioms, now calledBasic Laws, includingthe completely new axioms required to deal with value ranges and thebackslash operator. This is not, however, mere reorganization.Instead, in the logic ofGrundgesetze Frege replaces a numberof the axioms found inBegriffsschrift with correspondingrules that are much more flexible, and hence much more powerful, intheir application. For an insightful discussion of what theGrundgesetze-era Frege takes to be characteristics of a BasicLaw (as opposed to any other logical truth derivable from the BasicLaws and rules of inference) and their relation to proof, see Pedriali(2019).

3.3.1 Basic Law I

Frege’s Basic Law I looks familiar, since, syntactically atleast, it is justAxiom 1 from the logic ofBegriffsschrift:

Basic Law I (1893/1903: §18)

InGrundgesetze Frege justifies Basic Law I as follows:

We will now set up some general laws for Roman letters which we willhave to make use of later. According to §12:
$judgement conditional {term \Gamma} {term conditional term \Delta term \Gamma}$
would be the False only if $\Gamma$ and $\Delta$ were the Truewhile $\Gamma$ was not the True. This is impossible; accordingly:
(1893/1903: §18)

Given the discussion above (and Frege’s careful use of“$\Gamma$was not the True” ratherthan“$\Gamma$ was the False”in the justification of the Basic Law), it is no surprise that thisexpression is a name of the True even when the objects named by“$\Gamma$”and“$\Delta$”are not truth-values. Thereader is encouraged to verify, for example, that:

conditional {term 2} {term conditional term 3 term 2}

is a name of the True. Frege notes that:

is a special instance of the formulation of Basic Law I above,obtained by replacing“$b$”with“$a$” and then fusingequal subcomponents (1893/1903: §18). Given its obvious utility,Frege lists this as a second version of Basic Law I, one that we canuse as a primitive axiom without explicitly deriving it.

3.3.2 Basic Law II

Basic Law II also looks familiar, as it is theGrundgesetzeanalogue of Axiom 9 from the logic ofBegriffsschrift. Nowthat Frege has a clear conception of the hierarchy of object,first-level function, second-level function, and third-level functionin place, and now that he has distinct quantifiers that range,respectively, over different “levels” of entity fromwithin this hierarchy, he is careful to formulate both a“first-order” version and a “second-order”version of the Basic Law in question. The first version, Basic Law IIais:

judgement conditional {all gothic a term f(gothic a)} {term f(a)}

Basic Law IIa (1893/1903: §20)

Frege describes this Basic Law as expressing the thought that“what holds of all objects, also holds of any” (1893/1903:§20). Basic Law IIa, combined with the generalized version ofmodus ponens used inGrundgesetze (and discussedbelow) provides a means for inferring a Roman letter generality from agenerality formulated using the concavity. Given a concavityproposition of the form“ judgement all gothic a $\Phi(\mathfrak{a})$”, we can invoke an instance ofBasic Law IIa:

$judgement conditional {all gothic a term \Phi(gothic a)} {term \Phi(a)}$

and combine these, usingmodus ponens, to conclude“ judgement $\Phi(a)$”.

The second-order version of Basic Law II is called Basic Law IIb:

$judgement conditional {all gothic f term M_\beta(gothic f(\beta))} {term M_\beta(f(\beta))}$

Basic Law IIb (1893/1903: §25)

Note that the occurrence of“$\mathfrak{f}$”in thesubcomponent, and“$f$” in thesupercomponent, are distinct variables (the former is a German letter,the latter a Roman letter).

3.3.3 Basic Law III

Basic Law III, theGrundgesetze principle governing theequality-sign, appears at first glance to be a slight variant of theindiscernibility of identicals:

judgement conditional {term g(a=b)} {term g(all gothic f term conditional {gothic f(b)} {term gothic f(a)})}

Basic Law III (1893/1903: §20)

If we replace the Roman letter“$g$”by the horizontal, and theapply fusion of horizontals, we do indeed obtain theGrundgesetze version of the indiscernibility of identicals:

judgement conditional {term a=b} {all gothic f term conditional term gothic f (b) term gothic f(a)}

Basic Law III is a good bit more powerful than this, however.

Basic Law III states that for any unary first-level function name“$\Phi(\xi)$” and any propernames“$\Delta$” and“$\Gamma$”,it is not the case thatthe application of the function named by“$\Phi(\xi)$”to the truth-valuenamed by“$\Delta = \Gamma$”is the True, while application of the function named by“$\Phi(\xi)$”to the truth-valuenamed by:

$all gothic f conditional term gothic f(\Gamma) term gothic f(\Delta)$

is something other than the True. Thus, this axiom amounts to theclaim that one can always replace an equality with the correspondinguniversally quantified expression anywhere in a proposition (i.e., asthe argument of any function $\Phi(\xi)$).

The negation stroke is, of course, one of the functions that can besubstituted for“$g$”. Thus,the following is a substitution instance of Basic Law III:

judgement conditional {not term (a=b)} {not term (all gothic f conditional term gothic f(b) term gothic f(a))}

which (although we have not yet discussed the generalizedcontraposition rule required) is equivalent to:

judgement conditional {all gothic f term conditional term gothic f(b) term gothic f(a)} {term (a=b)}

Thus, Basic Law III implies aGrundgesetze analogue of theidentity of indiscernibles. As a result, Frege has no need for ananalogue ofBegriffsschrift’s Axiom 8 within the logicofGrundgesetze, since he proves:

by first proving:

judgement all gothic f conditional {term gothic f(a)} {term gothic f(a)}

and then applying Basic Law III (and hisGrundgesetze versionofmodus ponens). He calls this principle of self-identityIIIe (1893/1903: §50).

3.3.4 Basic Law IV

Basic Law IV is:

judgement conditional {not term (circumstance a) = (note b)} {term (circumstance a) = (circumstance b)}

Basic Law IV (1893/1903: §18)

This principle might appear to be nothing more than aGrundgesetze analogue of a familiar principle of classicalpropositional logic:

\[(\neg (\Delta\leftrightarrow \neg \Gamma)) \rightarrow (\Delta \leftrightarrow\Gamma)\]

As usual, however, we should be careful not to read this axiom as onlyapplying to truth-value names. Instead, instances of Basic Law IV namethe True no matter what names are substituted in for“$a$”and“$b$”.Thus:

judgement conditional {not term (circumstance 2) = (not 3)} {term (circumstance 2) = (circumstance 3)}

is a name of the True (recall that both“ circumstance $2$” and“$3$” are names of the False,“ not $3$” names the True, and hence both the supercomponentand the subcomponent of this instance of Basic Law IV are names of theTrue).

Unlike most of the (non-value-range involving) axioms and rules of thelogic ofGrundgesetze, Basic Law IV has no direct analoguewithin the logic ofBegriffsschrift (but see the discussionof Axioms 5 and 6 below), and its purpose seem to be, in part, toenforce the intended interpretation of the horizontal and negation incases where the arguments are not truth-values. Nevertheless, giventhat all occurrences of Roman letters within Basic Law IV are prefixedby the horizontal, the practical import of this principle is much thesame as the classical analogue: Given any two truth-values, if thefirst is not equal to the negation of the second, then they arethemselves equal.

Given that Basic Law IV, in effect, forces the logic ofGrundgesetze to be bivalent, we might wonder what happened tothe axioms that played this role in the logic ofBegriffsschrift—Axioms 5 and 6.Grundgesetzeanalogues of both Axiom 5 and Axiom 6 are proven very early in thederivations ofGrundgesetze, but, as was the case with BasicLaw II, Frege in fact identifies and proves versions that are muchmore general. The theorems in question are:

judgement conditional {term f(not not a)} {term f(circumstance a)}

IVc

judgement conditional {term f(circumstance a)} {term (not not a)}

IVd

which he calls IVc and IVd respectively (1893/1903: §51).

3.3.5 Basic Law V

It is somewhat surprising how little fanfare accompanies Frege’sintroduction of the now notorious Basic Law V. Frege merely notesthat:

[…] a value-range equality can always be converted into thegenerality of an equality, andvice versa. (1893/1903:§20)

and then states the axiom:

$judgement (ἐf(\varepsilon) = ἀg(\alpha)) = (all gothic a term f(gothic a) = g(gothic a)))$

Basic Law V (1893/1903: §20)

TheGrundgesetze formulation of Basic Law V is a good bitmore general than well-known modern “translations” of thislaw within higher-order logic such as:

\[\forall X\;\forall Y(\S(X) = \S(Y) \leftrightarrow \forall z(X(z)\leftrightarrow Y(z)))\]

where the quantifiers range over concepts or properties. TheGrundgesetze formulation of Basic Law V entails not only thatevery concept has a corresponding extension, but in addition that anyfunction whatever (concept or not) has a value-range. Thus, theGrundgesetze formulation of Basic Law V also capturessomething akin to:

\[\forall f\;\forall g(\S(f) = \S(g) \leftrightarrow \forall z(f(z) = g(z)))\]

Since within modern treatments of higher-order logic concepts are adifferent sort from functions, however, rather than being a sub-classof the class of functions as inGrundgesetze, Frege’sBasic Law V is, strictly speaking, not equivalent to either of thesemodern formulations.

3.3.6 Basic Law VI

The final axiom ofGrundgesetze, Basic Law VI, governs thebehavior of the backslash:

$judgement a =backslash ἐ(a = \varepsilon)$

Basic Law VI (1893/1903: §18)

This axiom makes explicit one part of the informal definition of thebackslash operator discussed earlier. If“$\Gamma$”is a name of thesingleton extension of the object named by“$\Delta$”,then the result ofapplying the backslash to“$\Gamma$”names the same object asdoes“$\Delta$”. But Basic LawVI tells us nothing regarding what results when we apply the backslashto a name that isnot the name of a singleton extension. Wemight wonder why Frege did not add a second axiom governing this case,such as:

$judgement conditional {all gothic a not term b = ἐ(gothic a = \varepsilon)} {term backslash b = b}$

The absence of this principle from the deductive system ofGrundgesetze leads to a striking insight into Frege’smethodology. The question at issue is whether Frege would have takenhis logic to be complete—that is, would he have taken it toprove every logical truth expressible in the language ofGrundgesetze. Michael Dummett suggests that:

No doubt Frege would have claimed his axioms, taken together with theadditional informal stipulations not embodied in them, as yielding acomplete theory: to impute to him an awareness of the incompletenessof higher-order theories would be an anachronism. (1981: 423)

The stipulations Dummett has in mind are the identification of thetruth values with their singleton extensions at the conclusion of the“Permutation Argument” in §10 ofGrundgesetze. What the lack of a second instance of Basic LawVI reveals, however, is that Frege would have been quite aware of theincompleteness of the logic ofGrundgesetze (were it notinconsistent), since he has failed to include a principle that isclearly true on the intended interpretation. Imputing to him anawareness of an obvious gap in the logical principles he actuallyincludes in his logic (given the informal semantic elucidations heprovides regarding the intended interpretation of his logic) is fardifferent from claiming that he was aware of thein principleincompleteness of second-order logic, of course.

Interestingly, Heck points out (2012: ch. 2) that Frege claims laterinGrundgesetze that a proposition “is, it seems,unprovable” (Frege 1893/1903: §114). Frege is careful toclaim only that the principle is unprovable, and hence that he willnot claim that isis true (i.e., he will not write it as ajudgement with“ judgement symbol ”, but he also refrains from claiming that it is false.Heck suggests that Frege was at this point recognizing the potentialincompleteness of his system (or, at least, of the consistentsub-system relevant to the derivations at issue in §114). Whatprevents Frege from proving the claim in question, and what allowsDedekind to prove a similar principle in his own treatment ofarithmetic (Dedekind 1888), is that the latter has an informal versionof the axiom of choice at his disposal, while Frege has included noformal version of choice within the logic ofGrundgesetze(Heck 2012: ch. 2). This is an important observation, but nothing thissophisticated is needed in order to conclude that Frege would likelyhave thought that his system was incomplete.

Frege had no need for an axiom covering the application of thebackslash to non-singleton extensions in his derivation of arithmeticwithin the formal system ofGrundgesetze (and we can presumehe also had no need of it in his envisioned derivation of real andcomplex analysis). Thus, he did not need to add it to his stock ofBasic Laws. Frege’s formal system only needs to include thoselogical principles that are required for his reconstructions ofarithmetic and analysis—his project did not require a logic thatwas proof-theoretically complete in the modern sense, and as a resultwe should not be surprised that the logic he did formulate would haveappeared (to Frege, to his readers, and to us) to be rather obviouslyincomplete (had Basic Law V not rendered it inconsistent, and hencetrivially complete).

3.4 The Rules of Inference ofGrundgesetze

Despite frequent claims that Frege’s logic is unwieldy anddifficult to use, the rules of inference that Frege formulatesdemonstrate the opposite: the resulting system, which takes advantageof the systematic ambiguity present in Frege’s way of readingconditionals, is in many ways more powerful and more elegant thanmodern deductive systems.

First, we remind the reader of the three rules of inference that havealready been discussed, and which can be applied in derivations withinthe logic ofGrundgesetze without any comment or label. Theseare the permutation of subcomponents, the fusion of identicalsubcomponents, and the fusion of horizontals. In addition to these,Frege introduces six rules of inference.

3.4.1 Generalized Modus Ponens

The first isGeneralized Modus Ponens (the term“Generalized Modus Ponens” is not Frege’s, he callsthis inference “Inferring(a)”), which Frege describesthusly:

If a subcomponent of a proposition differs from a secondproposition only in lacking the judgement-stroke, then one may infer aproposition which results from the first by suppressing thatsubcomponent. (1893/1903: §14)

Simply put, if one has proven aGrundgesetze conditional, andone has also proven a subcomponent of that conditional, then one mayinfer the result of deleting that subcomponent from the conditional.Assume that we have proven theGrundgesetze proposition:

$judgement conditional {term \Delta} {term conditional term \Gamma term \Theta}$

Then, if we also have“ judgement $\Delta$”, then we can conclude:

$judgement conditional term \Gamma term \Theta$

On this application, we are parsing the conditional in question suchthat“$\Delta$” is therelevant subcomponent, and:

$judgement conditional term \Gamma term \Theta$

the supercomponent. If, on the other hand, we have“ judgement $\Gamma$”, then we would instead conclude:

$judgement conditional term \Delta term \Theta$

treating both“$\Delta$” and“$\Gamma$” as subcomponents,and“$\Theta$” assupercomponent. Applications of Generalized Modus Ponens are indicatedby a solid horizontal line. This is the first instance of atransition sign (1893/1903: §14).

This rule is an incredibly powerful generalization of the simpleversion ofmodus ponens found inBegriffsschrift, orincluded in modern deductive systems that use linear notation.Consider how many steps would be needed to transition from:

\[A_1 \rightarrow (A_2 \rightarrow (A_3 \rightarrow(A_4 \rightarrow (A_5 \rightarrow (A_6 \rightarrow (A_7 \rightarrow (A_8\rightarrow B)))))))\]

and $A_8$ to:

\[A_1\rightarrow (A_2 \rightarrow (A_3 \rightarrow (A_4 \rightarrow (A_5\rightarrow (A_6 \rightarrow (A_7 \rightarrow B))))))\]

in a typical natural deduction system. Within the logic ofGrundgesetze, Frege can carry out the analogous deduction inone step. Similar comments regarding the power of Frege’s rulesof inference apply to the other rules that take advantage of the factthatGrundgesetze formulas can be parsed into subcomponent(s)and supercomponent in more than one way (that is, to GeneralizedHypothetical Syllogism, Generalized Contraposition, and GeneralizedDilemma).

Frege allows for multiple, simultaneous applications of thegeneralized form of GeneralizedModus Ponens. Thus, if wehave proven:

$judgement conditional {term \Delta} {term conditional term \Gamma term \Theta}$

and we then prove both“ judgement $\Delta$”, and“$\Gamma$”, we may eliminate both subcomponentssimultaneously, marking the transition with a double horizontal line(1893/1903: §14).

3.4.2 Generalized Hypothetical Syllogism

The next rule of inference isGeneralized HypotheticalSyllogism (This is again novel terminology—Frege calls thisrule “Inferring(b)”):

If the same combination of signs occurs in one proposition assupercomponent and in another as subcomponent, then a proposition maybe inferred in which the supercomponent of the latter features assupercomponent and all subcomponents of both, save that mentioned,feature as subcomponents. However, subcomponents that occur in bothneed only be written once. (1893/1903: §15)

This is a generalized, and more powerful, version of familiar rulehypothetical syllogism that takes advantage of the“equal” status of subcomponents: Given aGrundgesetze proposition, and a second proposition whosesupercomponent is a subcomponent of the first, we can infer theproposition that results from replacing the relevant subcomponent inthe first proposition with the subcomponents of the secondproposition. For example, if we have derived both of:

$judgement conditional {term \Delta} {term conditional term \Gamma term \Theta}$

and

$judgement conditional {term \Gamma} {term conditional term \Sigma term \Delta}$

then we can combine these to obtain:

$judgement conditional {term \Sigma} {term conditional term \Gamma term \Theta}$

Note that we have combined the two occurrences of“$\Gamma$”into a single occurrence.Applications of Generalized Hypothetical Syllogism are indicated bynew type of transition sign, a dashed horizontal line“—– – –”, and multiplesimultaneous applications of Generalized Hypothetical Syllogism areindicated by the double dashed horizontal line “= = = =”.Generalized Hypothetical Syllogism is a much more powerful rule-basedform of Axiom 2 from the logic ofBegriffsschrift.

3.4.3 Generalized Contraposition

The third rule of inference isGeneralized Contraposition(Frege calls this rule merely “contraposition”).

One may permute a subcomponent with a supercomponent provided onesimultaneouslyreverses the truth-value of each. (1893/1903:§15)

Generalized contraposition allows us to “switch” thesupercomponent of aGrundgesetze proposition withany subcomponent of it, provided one “simultaneouslyreverses the truth-value of each”. Recalling the factthat Frege describes negation as “a sign by means of which everytruth-value is transformed into its opposite” (1893/1903:§6), this amounts to either adding a single negation to, orremoving a single negation from (if at least one negation is present),each of the formulas in question. Thus, if we have derived:

$judgement conditional {term \Delta} {term conditional {not term \Gamma} {term \Theta}}$

then Generalized Contraposition would allows us to obtain any of thefollowing:

$judgement conditional {not term \Theta} {term conditional {not term \Gamma} {not term \Delta}}$

(a)

$judgement conditional {term \Delta} {term conditional {not term \Theta} {not not term \Gamma}}$

(b)

$judgement conditional {term \Delta} {term conditional {not term \Theta} {term \Gamma}}$

(c)

$judgement conditional {not term conditional {not term \Gamma} {term \Theta}} {not term \Delta}$

(d)

Applications of generalized contraposition are represented using thetransition sign“$\Large\times$”. Generalized Contraposition is a much morepowerful rule-based version of Axiom 4 from the logic ofBegriffsschrift.

3.4.4 Generalized Dilemma

The fourth rule of inference isGeneralized Dilemma (Fregecalls this “Inferring(c)”):

If two propositions agree in their supercomponents while asubcomponent of the one differs from a subcomponent of the other onlyby a prefixed negation stroke, then we can infer a proposition inwhich the common supercomponent features as supercomponent, and allsubcomponents of both propositions with the exception of the twomentioned feature as subcomponents. In this, subcomponents which occurin both propositions need only be written down once. (1893/1903:§16)

Thus, if we have derivedGrundgesetze propositions of theform:

$judgement conditional {term \Delta} {term conditional term \Gamma term \Theta}$

(a)

$judgement conditional {term \Sigma} {term conditional {not term \Delta} {term \Theta}}$

(b)

we can infer:

$judgement conditional {term \Gamma} {term conditional {term Sigma term \Theta}}$

Generalized Dilemma is indicated by the dot-dashed line“$\cdot\!-\!\cdot\!-\!\cdot\!-\!\cdot$”.

Generalized Dilemma is the only official axiom or rule found in thelogic ofGrundgesetze, other than those that explicitlyinvolve value-ranges, that does not have some clear (even if muchweaker) analogue in the logic ofBegriffsschrift, (assumingwe count Axioms 5 and 6 as at least very loose analogues of Basic LawIV—see above), although an analogue of the rule is of coursederivable in the earlier system (to put it in contemporaryterms). After noting, in the Preface toBegriffsschrift, thathe has limited himself to as few axioms and rules of inference aspossible, Frege writes that:

This does not rule out,later, transitions from severaljudgements to a new one, which are possible by this single mode ofinference only in an indirect way, being converted into direct onesfor the sake of abbreviation. In fact, this may be advisable for laterapplications. In this way, then, further modes of inference wouldarise. (1879a: Preface)

In the Foreword toGrundgesetze, however, Frege notes arather notable shift in strategy:

In order to attain more flexibility and to avoid excessive length, Ihave allowed myself tacit use of permutation of subcomponents(conditions) and fusion of equal subcomponents, and have not reducedthe modes of inference and consequence to a minimum. Anyone acquaintedwith my little bookBegriffsschrift will gather from it howhere too one could satisfy the strictest demands, but also that thiswould result in a considerable increase in extent. (1893/1903:Foreword)

Presumably it is Generalized Dilemma, in particular, that Frege has inmind here.

3.4.5 Concavity Introduction

Frege’s next rule of inference, which we will (following ourdiscussion of the analogous rule in the logic ofBegriffsschrift) callConcavity Introduction (Fregecall this rule “Transformation of a Roman letter into a Germanletter”), governs the interactions between the two devices forexpressing generality withinGrundgesetze, the Roman lettergenerality device and the concavity:

A Roman letter may be replaced whenever it occurs in a propositionby one and the same German letter. At the same time, the latter has tobe placed above a concavity in front of one such supercomponentoutside of which the Roman letter does not occur. If in thissupercomponent the scope of a German letter is contained and the Romanletter occurs within this scope, then the German letter that is to beintroduced for the latter must be distinct from the former.(1893/1903: §17)

Mechanically speaking, this is the same rule as the version ofConcavity Introduction given in the logic ofBegriffsschrift.The transition symbol used to label applications of concavityintroduction is“$\raise1ex{\underparen{\hspace{1.5em}}}$”.

3.4.6 Roman Letter Elimination

While Concavity Introduction allows us to move from a generalityexpressed with Roman letters to a generality expressed with theconcavity, and Basic Law II (combined with generalizedmodusponens) allows us to move from a generality expressed with theconcavity to a generalization expressed with Roman letters, as of yetwe have no way to move from a proposition expressing a generality (ofeither sort) to an particular instance (the logic ofBegriffsschrift contains no names, hence this lack was not ashortcoming in that earlier system). Frege rectifies this byintroducing what we shall call theRoman Letter Eliminationrule (Frege calls this rule “Citing Propositions”), whichhe describes as follows:

When citing a proposition by its label, we may effect a simpleinference by uniformly replacing a Roman letter within the propositionby the same proper name or the same Roman object-marker.
Likewise, we may replace all occurrences in a proposition of a Romanfunction letter“$f$”,“$g$”,“$h$”,“$F$”,“$G$”,“$H$”by the same name or Romanmarker of a first-level function with one or two arguments, dependingon whether the Roman letter indicates a function with one or twoarguments.
When we cite law (IIb), we may replace both occurrences of“$M_\beta$” by the same name or Roman marker of asecond-level function with one argument of the second kind.(1893/1903: §48)

Roman Letter Elimination allows for two sorts of application. Thefirst, and simplest, involves uniformly replacing a particular Romanletter with a proper name. But the Roman letter elimination rule alsoallows us to replace a single Roman letter with a Roman object marker.Hence, using this rule we can also obtain“ judgement $\Phi($ not $y)$” from“$\Phi(x)$” by replacing the Roman letter“$x$”with the Roman object marker“ not $y$”. Likewise, given aGrundgesetzeproposition of the form “$f(\Delta)$” we can infer either“ judgement not $\Delta$” (by replacing the Roman letter“$f$”with the function name“” or“$g($ not $\Delta)$” (by replacing the Roman letter“$f$”with the Roman function marker“ not $g$ not ”). One of the most visible and most importantapplications of the Roman letter elimination rule is when introducing“instances” of axioms, since Frege does not require thatwe write the axiom explicitly. Instead, we can cite any“instance” of the axiom that results from applying theRoman letter elimination rule (one or more times) to the axiomitself.

There is no transition symbol for this inference, since it is appliedwhen a previously derived formula is used as a premise in anapplication of one of the other rules listed above.

This concludes our survey of the logical systems found inFrege’sBegriffsschrift andGrundgesetze. Butit is far from the conclusion of Frege’s thinking on logic.Although Frege eventually abandoned his logicist project, and thoseparts of his logical system that led to the contradiction (Basic LawsV and VI, and the notion of value ranges more generally) in light ofthe Russell paradox, he continued to teach courses on logic for manyyears. The reader interested in Frege’s later views shouldconsult the supplementary essayFrege’s Logic After the Paradox.

Bibliography

Frege’s Writings

1879a,Begriffsschrift, eine der arithmetischen nachgebildeteFormelsprache des reinen Denkens, Halle a. S.: Louis Nebert; allquotes from this work are from the translation in T.W. Bynum (ed. andtrans.),Frege: Conceptual Notation and Related Articles,Oxford: Clarendon, 1972.
1879b, “Anwendungen der Begriffsschrift”,Sitzungsberichte der Jenaische Zeitschrift für Medicin undNaturwissenschaft, 13: 29–33.
1880/1881, “Booles rechnende Logik und dieBegriffsschrift”, unpublished manuscript. Translated as“Boole’s Logical Calculus and the Concept Script”,in LW79: 9–46.
1881, “Booles logische Formelsprache und meineBegriffsschrift”, unpublished manuscript. Translated as“Boole’s Logical Formula Language and myConcept-script”, in LW79: 47–52.
1882a, “Über die wissenschaftliche Berechtigung einerBegriffsschrift”,Zeitschrift für Philosophie undphilosophische Kritik, 81: 48–56. Translated as “Onthe Scientific Justification of a Conceptual Notation” in By72:83–89.
1882b, “Über den Zweck der Begriffsschrift”,Lecture at the 27 January 1882 meeting of the Jenaischen Gesellschaftfür Medizin und Naturwissenschaft. Translated as “On theAim of the ‘Conceptual Notation’” in By72:90–100.
1884,Die Grundlagen der Arithmetik: eine logischmathematische Untersuchung über den Begriff der Zahl,Breslau: W. Koebner. Translated asThe Foundations of Arithmetic:A Logico-Mathematical Enquiry into the Concept of Number, J. L.Austin (trans.), second revised edition, Oxford: Blackwell, 1953.
1891, “Funktion und Begriff” (“Function andConcept”),Vortrag, gehalten in der Sitzung vom 9. Januar1891 der Jenaischen Gesellschaft für Medizin undNaturwissenschaft, Jena: Hermann Pohle.
1892a, “Über Sinn und Bedeutung” (“On Senseand Reference”),Zeitschrift für Philosophie undphilosophische Kritik, 100: 25–50.
1892b, “Über Begriff und Gegenstand”’(“Concept and Object”),Vierteljahresschrift fürwissenschaftliche Philosophie, 16: 192–205.
1893/1903,Grundgesetze der Arithmetik: begriffsschriftlichabgeleitet, Jena: Verlag Hermann Pohle, volume 1 in 1893 andvolume 2 in 1903. Translated in ER13.
1897, “Über die Begriffsschrift des Herrn Peano undmeine eigene”,Berichte über die Verhandlungen derKöniglich Sächsischen Gesellschaft der Wissenschaften zuLeipzig: Mathematisch-physische Klasse, 48: 361–378.Translated as “On Mr. Peano’s Conceptual Notation and MyOwn” in BD84: 234–248.

Translations

[Be97]The Frege Reader, Michael Beaney (ed.),Oxford ; Malden, MA: Blackwell Publishers, 1997.
[BD84]Collected Papers on Mathematics, Logic, andPhilosophy, Brian McGuinness (ed.), Oxford/New York: B.Blackwell, 1984.
[By72]Conceptual Notation, and Related Articles, TerrellWard Bynum (ed./tran.), Oxford: Clarendon Press, 1972.
[ER13]Gottlob Frege: Basic Laws of Arithmetic: Derived UsingConcept-Script. Volumes I & II, Philip A. Ebert and MarcusRossberg (eds/trans), Oxford: Oxford University Press, 2013.
[LW79]Posthumous Writings, Hans Hermes, FriedrichKambartel, and Friedrich Kaulbach (eds.). Peter Long and Roger White(trans.), Chicago: University of Chicago Press, 1979.
[RA04]Frege’s Lectures on Logic: Carnap’s StudentNotes, 1910-1914, Gottfried Gabriel (ed.). Erich H. Reck andSteve Awodey (trans/eds), Chicago, IL: Open Court, 2004.
Van Heijenoort, Jean (ed.), 1967a,From Frege to Gödel: ASource Book in Mathematical Logic, 1879–1931, (Source Booksin the History of the Sciences), Cambridge, MA: Harvard UniversityPress.

Secondary Literature

Beaney, Michael, 1997, “Introduction”, in Be97,1–46.
––– (ed.), 2013,The Oxford Handbook of theHistory of Analytic Philosophy, Oxford: Oxford University Press.doi:10.1093/oxfordhb/9780199238842.001.0001
Berg, Eric D. and Roy T. Cook, 2017, “The PropositionalLogic of Frege’sGrundgesetze: Semantics andExpressiveness”,Journal for the History of AnalyticalPhilosophy, 5: article 6. doi:10.15173/jhap.v5i6.2920
Boolos, George, 1994, “1879?”, in Clark and Hale (eds)1994: 31–48 (ch. 2).
Burge, Tyler, 1984, “Frege on Extensions of Concepts, from1884 to 1903”,The Philosophical Review, 93(1):3–34. doi:10.2307/2184411
Clark, Peter and Bob Hale (eds.), 1994,Reading Putnam,Cambridge, MA: Blackwell Publishers.
Cook, Roy T., 2013, “How to ReadGrundgesetze”, appendix to ER13: A1–A42.
Currie, Gregory, 1984, “Frege’s MetaphysicalArgument”,The Philosophical Quarterly, 34(136):329–342. Reprinted Wright 1984: 144–157.doi:10.2307/2218764
Dedekind, Richard, 1888,Was sind und was sollen dieZahlen?, Braunschweig: Friedrich Vieweg.
Dummett, Michael A. E., 1981,The Interpretation ofFrege’s Philosophy, London: Duckworth.
Ebert, Philip A. and Marcus Rossberg (eds.), 2019,Essays onFrege’s: “Basic Laws of Arithmetic”, Oxford:Oxford University Press.
Gabbay, Dov M. and John Woods (eds.), 2004,Handbook of theHistory of Logic, Volume 3, the Rise of Modern Logic: From Leibniz toFrege, Amsterdam/Boston: Elsevier.
Goldfarb, Warren, 2010, “Frege’s Conception ofLogic”, in Potter and Ricketts 2010: 63–85.doi:10.1017/CCOL9780521624282.003
Heck, Richard Kimberly, 2012,Reading Frege’s“Grundgesetze”, Oxford: Clarendon Press. (originallypublished under Richard G. Heck)
–––, 2019, “Formal Arithmetic BeforeGrundgesetze”, in Ebert and Rossberg 2019:497–537 (ch. 18).
Heck, Richard Kimberly and Robert May, 2013, “The FunctionIs Unsaturated”, in Beaney 2013: 825–850. (originallypublished under Richard Kimberley Heck and Robert May)doi:10.1093/oxfordhb/9780199238842.013.0028
–––, 2020, “The Birth of Semantics”,Journal for the History of Analytical Philosophy, 8: article6. doi:10.15173/jhap.v8i6.3970
Jones, Emily Elizabeth Constance, 1890,Elements of Logic as aScience of Propositions, Edinburgh: T. & T. Clark.
Jourdain, Philip, 1912, “The Development of Theories ofMathematic Logic and the Principles of Mathematics”,TheQuarterly Journal of Pure and Applied Mathematics, 43:219–314.
Kemp, Gary, 1995, “Truth in Frege’s ‘Law ofTruth’”,Synthese, 105(1): 31–51.doi:10.1007/BF01064102
Kremer, Michael, 2010, “Sense and Reference: The Origins andDevelopment of the Distinction”, in Potter and Ricketts 2010:220–292. doi:10.1017/CCOL9780521624282.007
Landini, Gregory, 2012,Frege’s Notations: What They Areand How They Mean, (History of Analytic Philosophy), Houndmills,UK/New York: Palgrave Macmillan.
Łukasiewicz, Jan, 1934, “Z Historii LogikiZdãn”,Przeglad Filozoficzny, 37: 369–377.
Mates, Benson, 1972,Elementary Logic, second edition,New York: Oxford University Press.
Mostowski, Andrzej, 1957, “On a Generalization ofQuantifiers”,Fundamenta Mathematics, 44(1):12–36.
Pedriali, Walter B., 2019, “When Logic Gives Out: Frege onBasic Logical Laws”, in Ebert and Rossberg 2019: 57–89(ch. 3).
Potter, Michael D. and Tom Ricketts (eds.), 2010,TheCambridge Companion to Frege, Cambridge, UK/New York: CambridgeUniversity Press. doi:10.1017/CCOL9780521624282
Putnam, Hilary, 1982, “Peirce the Logician”,Historia Mathematica, 9(3): 290–301.doi:10.1016/0315-0860(82)90123-9
Russell, Bertrand, 1902, “Letter to Frege”, printed invan Heijenoort (ed.) 1967a: 124–125.
Tarski, Alfred, 1933, “The Concept of Truth in the Languagesof the Deductive Sciences” (Polish),Prace TowarzystwaNaukowego Warszawskiego, Wydzial III Nauk Matematyczno-Fizycznych34, Warsaw. Reprinted in Zygmunt (ed.), 1995,Alfred Tarski, PismaLogiczno-Filozoficzne, 1 Prawda, Warsaw: Wydawnictwo Naukowe PWN,13–172. English translation in Tarski’sLogic,Semantics, Metamathematics: Papers from 1923 to 1938, 2ndedition, John Corcoran (ed.), Indianapolis: Hackett PublishingCompany, 1983, 152–278.
van Heijenoort, Jean, 1967b, “Introduction toBegriffsschrift”, in van Heijenoort (ed.) 1967:1–5.
Wright, Crispin (ed.), 1984,Frege: Tradition andInfluence, Oxford/New York: B. Blackwell.

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Frege’s Logic

1. Introduction

2. The Logic ofBegriffsschrift

2.1 The Operators ofBegriffsschrift

2.1.1 The Judgement Stroke

2.1.2 The Conditional Stroke

2.1.3 The Negation Stroke

2.1.4 The Identity Operator

2.1.5 The Concavity for Expressing Generality

2.2 The Axioms and Rules ofBegriffsschrift

2.2.1 The Axioms

2.2.2 The Rules of Inference

3. The Logic ofGrundgesetze

3.1 The “Old” Operators ofGrundgesetze

3.1.1 The Judgement Stroke

3.1.2 The Negation Stroke

3.1.3 The Conditional Stroke

3.1.4 Equivalences for the Judgement Stroke

3.1.5 The Equality Sign

3.1.6 Two Forms of Universal Quantification

3.2 The New Operators ofGrundgesetze

3.2.1 A Device for Generality: Roman Letters

3.2.1.1 The Basics of Roman Letters

3.2.1.2 Higher Order Quantification and Roman Letters

3.2.1.3 How Roman Letters Work

3.2.2 The Value-Range Operator

3.2.3 The Backslash Operator for Definite Descriptions

3.3 The Axioms ofGrundgesetze

3.3.1 Basic Law I

3.3.2 Basic Law II

3.3.3 Basic Law III

3.3.4 Basic Law IV

3.3.5 Basic Law V

3.3.6 Basic Law VI

3.4 The Rules of Inference ofGrundgesetze

3.4.1 Generalized Modus Ponens

3.4.2 Generalized Hypothetical Syllogism

3.4.3 Generalized Contraposition

3.4.4 Generalized Dilemma

3.4.5 Concavity Introduction

3.4.6 Roman Letter Elimination

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