Logic is usually thought to concern itself only with features thatsentences and arguments possess in virtue of their logical structuresorforms. The logical form of a sentence or argument isdetermined by its syntactic or semantic structure and by the placementof certain expressions called “logical constants.”^[1] Thus, for example, thesentences

Every boy loves some girl.

and

Some boy loves every girl.

are thought to differ in logical form, even though they share acommon syntactic and semantic structure, because they differ in theplacement of the logical constants “every” and “some”. By contrast, thesentences

Every girl loves some boy.

and

Every boy loves some girl.

are thought to have the same logical form, because “girl” and “boy”are not logical constants. Thus, in order to settle questions aboutlogical form, and ultimately about which arguments are logically validand which sentences logically true, we must distinguish the “logicalconstants” of a language from its nonlogical expressions.

While it is generally agreed that signs for negation, conjunction,disjunction, conditionality, and the first-order quantifiers shouldcount as logical constants, and that words like “red”, “boy”, “taller”,and “Clinton” should not, there is a vast disputed middle ground. Isthe sign for identity a logical constant? Are tense and modal operatorslogical constants? What about “true”, the epsilon of set-theoreticmembership, the sign for mereological parthood, the second-orderquantifiers, or the quantifier “there are infinitely many”? Is there adistinctive logic of agency, or of knowledge? In these border areas ourintuitions from paradigm cases fail us; we need something moreprincipled.

However, there is little philosophical consensus about the basis forthe distinction between logical and nonlogical expressions. Until thisquestion is resolved, we lack a proper understanding of the scope andnature of logic, and of the significance of the distinction between the“formal” properties and relations logic studies and related butnon-formal ones. For example, the sentence

If Socrates is human and mortal, then he ismortal.

is generally taken to be a logical truth, while the sentence

If Socrates is orange, then he is colored.

is not, even though intuitively both are true, necessary, knowablea priori, and analytic. What is the significance of thedistinction we are making between them in calling one but not the other“logically true”? A principled demarcation of logical constants mightoffer an answer to this question, thereby clarifying what is at stakein philosophical controversies for which it matters what counts aslogic (for example, logicism and structuralism in the philosophy ofmathematics).

This article will discuss the problem of logical constants andsurvey the main approaches to solving or resolving it.

• 1. Syncategorematic terms
• 2. Grammatical criteria
• 3. A Davidsonian approach
• 4. Topic neutrality
• 5. Permutation invariance
• 6. Inferential characterizations
  • 6.1 Semantic value determination
  • 6.2 Sense determination
• 7. Pragmatic demarcations
• 8. Problem or pseudoproblem?
• Further reading
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Syncategorematic terms

The most venerable approach to demarcating the logical constantsidentifies them with the language’ssyncategorematic signs:signs that signify nothing by themselves, but serve to indicate howindependently meaningful terms are combined. This approach was naturalin the context of the “term logics” that were dominant until thenineteenth century. All propositions were thought to be composed out ofpropositions of subject-predicate form by means of a small number ofconnectives (“and”, “or”, “if …then”, and so on). In this framework,words divide naturally into those that can be used as subjects orpredicates (“categorematic” words) and those whose function is toindicate the relation between subject and predicate or between twodistinct subject-predicate propositions (“syncategorematic” words). Forexample, “Socrates”, “runs”, “elephant”, and “large” are categorematicwords, while “only”, “every”, “necessarily”, and “or” aresyncategorematic. (For a more detailed account of the distinction, seeKretzmann 1982, 211–214.) The syncategorematic words were naturallyseen as indicating thestructure orform of theproposition, while the categorematic words supplied its “matter.” Thusthe fourteenth-century logician Buridan writes:

I say that in a proposition (as we’re speaking here ofmatter and form), we understand by the “matter” of the proposition orconsequentia the purely categorical terms, i.e. subjects andpredicates, omitting the syncategorematic terms that enclose them andthrough which they are conjoined or negated or distributed or forced toa certain mode of supposition. All the rest, we say, pertains to theform. (Buridan 1976, I.7.2)

The Fregean revolution in our conception oflogical form made this way of demarcating the logical constantsproblematic. Whereas the term logicians had seen every proposition ascomposed of subject and predicate terms linked together bysyncategorematic “glue,” Frege taught us to see sentences andpropositions as built up recursively by functional application andfunctional abstraction (for a good account, see Dummett 1981, ch. 2).To see the difference between the two approaches, consider thesentence

A term logician would have regarded \(\refp{moby}\) as composed of a subjectterm (“boat”) and a predicate term (“thing smaller than Moby Dick”)joined together in a universal affirmative categorical form. Frege, bycontrast, would have regimented \(\refp{moby}\) as

which he would have analyzed as the result of applying thesecond-level function^[2]

to the first level functions

and

(The Greek letters \(\xi\), \(\Phi\), and \(\Psi\) here indicate thefunctions’ argument places: lowercase Greek letters indicate placesthat can be filled by proper names, while uppercase Greek lettersindicate places that must be filled by function expressions like\(\refp{function-boat}\)and \(\refp{function-smaller}\).) He would have regarded \(\refp{function-smaller}\)as itself the result of“abstracting” on the place occupied by “Shamu” in

which in turn is the result of applying the function

to “Shamu” and “Moby Dick”. Frege showed that by describingsentences and propositions this way, in terms of theirfunction/argument composition, we can represent the logical relationsbetween them in a far more complete, perspicuous, and systematic waythan was possible with the old subject/predicate model of propositionalform.

However, once we have thrown out the old subject/predicate model, wecan no longer identify the categorematic terms with the subject andpredicate terms, as the medievals did. Nor can we think of thesyncategorematic terms as the expressions that have no “independent”significance, or as the “glue” that binds together the categorematicterms to form a meaningful whole. Granted, there is a sense in whichthe “logical” function \(\refp{second-level}\) is the glue that bindstogether \(\refp{function-boat}\) and \(\refp{function-smaller}\)to yield \(\refp{moby-frege}\). But in the very same sense, the function\(\refp{smaller-than}\) is the gluethat binds together “Shamu” and “Moby Dick” to yield\(\refp{shamu-smaller}\). If we countall functional expressions as syncategorematic on the grounds that theyare “incomplete” or “unsaturated” and thus not “independentlymeaningful,” then the syncategorematic expressions will include notjust connectives and quantifiers, but ordinary predicates. On the otherhand, if we count all the functional expressions as categorematic, thenthe syncategoremata will be limited to variables, parentheses, andother signs that serve to indicate functional application andabstraction. In neither case would the distinction be useful fordemarcating the logical constants. An intermediate proposal would be tocount first-level functions as categoremata and second-level functionsas syncategoremata. That would make \(\refp{second-level}\) syncategorematic and\(\refp{function-boat}\) and \(\refp{function-smaller}\) categorematic. However, notevery second-level function is(intuitively) “logical.” Consider, for example, the second-levelfunction

Granted, standard logical languages do not have a simple expressionfor this function, but there is no reason in principle why we could notintroduce such an expression. Conversely, not every first-levelfunction is (intuitively) nonlogical: for example, the identityrelation is usually treated as logical.

In sum, it is not clear how the distinction between categorematicand syncategorematic terms, so natural in the framework of a termlogic, can be extended to a post-Fregean function/argument conceptionof propositional structure. At any rate, none of the natural ways ofextending the distinction seem apt for the demarcation of the logicalconstants. Carnap concedes that the distinction between categorematicand syncategorematic expressions “seems more or less a matter ofconvention” (1947, 6–7). However, the idea that logical constants aresyncategoremata does not wither away entirely with the demise of termlogics. Its influence can still be felt in Wittgenstein’s insistencethat the logical constants are like punctuation marks (1922,§5.4611),^[3] in Russell’s claim that logical constantsindicate logical form and not propositional constituents (1992, 98;1920, 199), and in the idea (found in Quine and Dummett) that thelogical constants of a language can be identified with its grammaticalparticles.

2. Grammatical criteria

Quine and Dummett propose that the logical constants of a languageare its grammaticalparticles—the expressions by meansof which complex sentences are built up, step by step, from atomicones—while non-logical expressions are the simple expressions ofwhich atomic sentences are composed (see Quine 1980, Quine 1986,Dummett 1981, 21–2, and for discussion, Føllesdal 1980 andHarman 1984). On this conception, “[l]ogic studies the truth conditionsthat hinge solely on grammatical constructions” (Quine 1980, 17).^[4]This criterion yields appropriate results when applied to the languageof first-order logic (FOL) and other standard logical languages. In FOL(without identity),all singular terms and predicates areparadigm nonlogical constants, andall operators andconnectives are paradigm logical constants.^[5]

However, this nice coincidence of intuitively logical expressions andgrammatical particles in FOL cannot be taken as support for theQuine/Dummett proposal, because FOL wasdesigned so that itsgrammatical structure would reflect logical structure. It is easyenough to design other artificial languages for which the grammaticalcriterion gives intuitively inappropriate results. For example, takestandard FOL and add a variable-binding operator “¢” whoseinterpretation is “there is at least one cat such that ….” Thegrammatical criterion counts “¢” as a logical constant, butsurely it is not one.

Moreover, there are alternative ways of regimenting the grammar ofFOL on which the standard truth-functional connectives are notgrammatical particles, but members of a small lexical category (Quine1986, 28–9). For example, instead of recognizing four grammaticaloperations that form one sentence from two sentences (one that takes\(P\) and \(Q\) and yields \(\cq{P \vee Q}\), onethat takes \(P\) and \(Q\) and yields \(\cq{P \and Q}\), and soon), we could recognize a single grammatical operation that forms onesentence from two sentences and one connective. On this way ofregimenting the grammar of FOL, \(\dq{\and}\) and \(\dq{\vee}\) would not count as grammatical particles.

The upshot is that a grammatical demarcation of logical constants willnot impose significant constraints on what counts as a logicalconstant unless it is combined with some principle for limiting thelanguages to which it applies (excluding, for example, languages withthe operator “¢”) and privileging some regimentations of theirgrammars over others (excluding, for example, regimentations thattreat truth-functional connectives as members of a small lexicalcategory). Quine’s own approach is to privilege the language bestsuited for the articulation of scientific theories and the grammarthat allows the most economical representation of the truth conditionsof its sentences. Thus the reason Quine holds that logic shouldrestrict itself to the study of inferences that are truth-preservingin virtue of their grammatical structures is not that he thinks thereis something special about the grammatical particles (in an arbitrarylanguage), but rather that he thinks we should employ a language inwhich grammatical structure is a perspicuous guide to truthconditions: “what we call logical form is what grammatical formbecomes when grammar is revised so as to make for efficient generalmethods of exploring the interdependence of sentences in respect oftheir truth values” (1980, 21).

Instead of applying the grammatical criterion to artificiallanguages like FOL, one might apply it to natural languages likeEnglish. One could then appeal to the work of empirical linguists for afavored grammatical regimentation. Contemporary linguists posit astructural representation called LF that resolves issues of scope andbinding crucial to semantic evaluation. But the question remains whichlexical items in the LF should count as logical constants. GeneralizingQuine’s proposal, one might identify the logical constants with membersof small, “closed” lexical categories: for example, conjunctions anddeterminers. However, by this criterion prepositions in English wouldcount as logical constants (Harman 1984, 121). Alternatively, one mightidentify the logical constants with members offunctionalcategories (including tense, complementizers, auxiliaries,determiners, and pronouns), and the nonlogical constants with membersofsubstantive categories (including nouns, verbs, adjectives,adverbs, and prepositions) (for this terminology, see Chomsky 1995, 6,54 and Radford 2004, 41). If a distinction that plays an important rolein a theory of linguistic competence should turn out to coincide (inlarge part) with our traditional distinction between logical andnonlogical constants, then this fact would stand in need ofexplanation. Why should we treat inferences that are truth-preservingin virtue of their LF structures and functional words differently fromthose that are truth-preserving in virtue of their LF structures andsubstantive words? Future work in linguistics, cognitive psychology andneurophysiology may provide the materials for an interesting answer tothis question, but for now it is important that the question be asked,and that we keep in mind the possibility of a sceptical answer.

3. A Davidsonian approach

The Quinean approach identifies the logical constants as theexpressions that play a privileged, “structural” role in a systematicgrammatical theory for a language. An alternative approach, due toQuine’s studentDonald Davidson, identifies the logical constants as the expressions that play aprivileged, “structural” role in a systematic theory ofmeaning for a language. A Davidsonian theory of meaning takesthe form of aTarskian truth theory. Thus, it contains two kinds of axioms:base clauses thatspecify the satisfaction conditions of atomic sentences,^[6] andrecursiveclauses that specify the satisfaction conditions of complexsentences in terms of the satisfaction conditions of their proper parts.^[7] For example:

Davidson suggests that “[t]he logical constants may be identified asthose iterative features of the language that require a recursiveclause in the characterization of truth or satisfaction” (1984, 71).(In our example, “or” and “some”.)

This criterion certainly gives reasonable results when applied tostandard truth theories like the one above (although the sign foridentity once more gets counted as nonlogical). But as Davidson goes onto observe, “[l]ogical form, in this account, will of course berelative to the choice of a metalanguage (with its logic) and a theoryof truth” (1984, 71). Different truth theories can be given for thesame language, and they can agree on the truth conditions of wholesentences while differing in which expressions they treat in therecursive clauses. Here are two examples (both discussed further inEvans 1976).

1. We might have recursive clauses for “large” and other gradableadjectives, along these lines:

For all assignments \(a\), terms \(\tau\), and sentences \(\phi\), \(\cq{\tau \text{ is a large } \phi }\) is satisfied by \(a\) iff Ref(\(\tau\), \(a\)) is a large satisfier of \(\phi\) on \(a\). (cf. Evans 1976, 203)

We would in this case have to use a metalanguage with a strongerlogic, one that provides rules for manipulating “large satisfier of\(\phi\) on \(a\).” (As Evans notes, all we would really need inorder to derive T-sentences would be a rule licensing the derivationof\(\cq{\tau \text{ is a large satisfier of } \phi \text{ on } a}\)from\(\cq{\phi \equiv \psi}\) and\(\cq{\tau \text{ is a larger satisfier of } \psi \text{ on } a}\).) Butsuch a metalanguage cannot be ruled out without begging the questionabout the logicality of “large.”

2. We might assign values to “and”, “or”, and the othertruth-functional connectives in thebase clauses, allowing usto get by with a single generic recursive clause for truth-functionalconnectives:

This approach requires a stronger metatheory than the usualapproach, since it requires quantification over truth functions. But itis not clear why this is an objection. It is still possible to deriveT-sentences whose right sides are no more ontologically committed thanthe sentences named on their left sides, like

So it is hard to see how the use of functions here is any moreobjectionable than Davidson’s own appeal to sequences or assignments ofvalues to variables.

In sum, the problem with Davidson’s truth-theoretic proposal is muchlike the problem discussed above with Quine’s grammatical proposal.Without further constraints on the theory of meaning (or, in Quine’scase, the grammar), it does not yield a definite criterion for logicalconstancy. I do not mean to suggest that either Davidson or Quine wasdeluded on this score. As we saw above, Quine appeals to pragmaticconsiderations to pick out a favored language and grammaticalregimentation. No doubt Davidson would do the same, arguing (forexample) that the advantages of using a simple and well-understoodlogic in the metalanguage outweigh any putative advantages of treating“large” and the like in recursive clauses. (For a recent defense of aDavidsonian criterion against Evans’s objections, see Lepore and Ludwig2002.)

4. Topic neutrality

Logic, it seems, is not about anything in particular; relatedly, it isapplicable everywhere, no matter what we are reasoning about. So it isnatural to suppose that the logical constants can be marked out as the“topic-neutral” expressions (Ryle 1954, 116; Peacocke 1976, 229; Haack1978, 5–6; McCarthy 1981, 504; Wright 1983, 133; Sainsbury 2001,365). We have reason to care about the topic-neutral expressions, andto treat them differently from others, because we are interested inlogic as auniversal canon for reasoning, one that isapplicable not just to reasoning about this or that domain, but to allreasoning.

Unfortunately, the notion of topic neutrality is too vague to be ofmuch help when it comes to the hard cases for which weneed aprinciple of demarcation. Take arithmetic, for instance. Is ittopic-neutral? Well, yes: anything can be counted, so the theorems ofarithmetic will be useful in any field of inquiry. But then again, no:arithmetic has its own special subject matter, the natural numbers andthe arithmetical relations that hold between them. The same can besaid about set theory: on the one hand, anything we can reason aboutcan be grouped into sets; on the other hand, set theory seems to beabout a particular corner of the universe—the sets—andthus to have its own special “topic.” The general problem of whichthese two cases are instances might be called theantinomy oftopic-neutrality. As George Boolos points out, the antinomy canbe pressed all the way to paradigm cases of logical constants: “itmight be said that logic is not so ‘topic-neutral’ as itis often made out to be: it can easily be said to be about the notionsof negation, conjunction, identity, and the notions expressed by‘all’ and ‘some’, among others …”(1975, 517). It is plausible to think that the source of the antinomyis the vagueness of the notion of topic neutrality, so let us considersome ways in which we might make this notion more precise.

Gilbert Ryle, who seems to have coined the expression“topic-neutral”, gives the following rough criterion:

We may call English expressions “topic-neutral” if aforeigner who understood them, but only them, could get no clue at allfrom an English paragraph containing them what that paragraph wasabout. (1954, 116)^[8]

There are, I suppose, a few paradigm cases of such expressions:“is”, for instance, and “if”. But the criterion gives little help whenwe venture beyond these clear-cut cases. The problem is that one mightanswer the question “what is this paragraph about?” at many differentlevels of generality. Suppose I understand English badly, and I hearsomeone say:

blah blah blahand not blah blah blahbecauseit blah blah blahto be blah blah blahand wasalways blah blah blah.But every blah blahisblah blah,although a few blah blahmight beblah.

Do I have any clue as to what the paragraph is about? Well, surely Ihavesome clue. “Because” reveals that the passage is aboutcausal or explanatory relations. “It” reveals that the passage is aboutat least one object that is not known to be a person. The tenseoperator “was always” reveals that it is about events that occur intime. “Might be” reveals that it is about the realm of the possible (orthe unknown), and not just the actual (or the known). Finally, “every”and “a few” reveal that it is about discrete, countable objects.Perhaps some of these words are not topic-neutral and should not beincluded in the domain of logic, but we certainly don’t want to ruleoutall of them. And Ryle’s criterion gives no guidance aboutwhere to draw the line. One might even suspect that there is no line,and that topic neutrality is a matter of degree, truth-functionalexpressions being more topic-neutral than quantifiers, which are moretopic-neutral than tense and modal operators, which are moretopic-neutral than epistemic expressions, and so on (Lycan 1989).

The problem with Ryle’s account is its reliance on vague andunclarified talk of “aboutness.” If we had a precise philosophicalaccount of what it is for a statement to beabout a particularobject or subject matter, then we could define a topic-neutralstatement as one that is not about anything—or, perhaps, one thatis about everything indifferently. Here we might hope to appeal toNelson Goodman’s classic account of “absolute aboutness,” which impliesthat logical truths are not absolutely about anything (1961, 256), orto David Lewis’s (1988) account of what it is for a proposition to beabout a certain subject matter, which implies that logical truths areaboutevery subject matter indifferently. However, neitheraccount is appropriate for our purpose. On Goodman’s account, “what astatement is absolutely about will depend in part upon what logic ispresupposed,” and hence upon which expressions are taken to be logicalconstants (253–4), so it would be circular to appeal to Goodman’saccount of aboutness in a demarcation of the logical constants. OnLewis’s account,all necessarily true propositions turn out tobe topic-neutral. But if there is any point to invoking topicneutrality in demarcating logic, it is presumably to distinguish thelogical truths from a wider class of necessary propositions, some ofwhich are subject matter-specific. If we are willing to broaden thebounds of logic to encompass all necessary propositions (or,alternatively, all analytic sentences), then we might as well demarcatelogic as the realm of necessary truth (alternatively, analytic truth).It is only if we want todistinguish the logical from thegenerically necessary, or to demarcate logic without appealing to modalnotions at all, that we need to invoke topic neutrality. And in neitherof these cases will Lewis’s criterion of aboutness be of service.

We rejected Ryle’s criterion for topic-neutrality because itappealed to an unclarified notion of aboutness. We rejected Goodman’sexplication of aboutness because it assumed that the line between logicand non-logic had already been drawn. And we rejected Lewis’s accountof aboutness because it did not distinguish logical truths from otherkinds of necessary truths. How else might we cash out the idea thatlogic is “not about anything in particular”? Two approaches have beenprominent in the literature.

The first starts from the idea that what makes an expressionspecific to a certain domain or topic is its capacity todiscriminate between different individuals. For example, themonadic predicate “is a horse”, the dyadic predicate “is taller than”,and the quantifier “every animal” all distinguish between Lucky Feet,on the one hand, and the Statue of Liberty, on the other:

• “Lucky Feet is a horse” is true; “The Statue of Liberty is a horse”is false.
• “The Statue of Liberty is taller than Lucky Feet” is true; “LuckyFeet is taller than the Statue of Liberty” is false.
• The truth of “Every animal is healthy” depends on whether LuckyFeet is healthy, but not on whether the Statue of Liberty ishealthy.

On the other hand, the monadic predicate “is a thing”, the dyadicpredicate “is identical with”, and the quantifier “everything” do notdistinguish between Lucky Feet and the Statue of Liberty. In fact, theydo not distinguish betweenany two particular objects. As faras they are concerned, one object is as good as another and might justas well be switched with it. Expressions with this kind of indifferenceto the particular identities of objects might reasonably be said to betopic-neutral. As we will see in the next section, this notion of topicneutrality can be cashed out in a mathematically precise way asinvariance under arbitrary permutations of a domain. It is in thissense that the basic concepts of arithmetic and set theory are nottopic-neutral, since they distinguish some objects (the empty set, thenumber 0) from others.

The second approach locates the topic neutrality of logic in itsuniversal applicability. On this conception, logic is useful for theguidance and criticism of reasoning about any subjectwhatsoever—natural or artefactual, animate or inanimate, abstractor concrete, normative or descriptive, sensible or merelyconceptual—because it is intimately connected somehow with thevery conditions for thought or reasoning. This notion of topicneutrality is not equivalent to the one just discussed. It allows thata science with its own proprietary domain of objects, like arithmeticor set theory, might still count as topic-neutral in virtue of itscompletely general applicability. Thus, Frege, who took arithmetic tobe about numbers, which he regarded as genuine objects, could stillaffirm its absolute topic neutrality:

…the basic propositions on which arithmetic is basedcannot apply merely to a limited area whose peculiarities they expressin the way in which the axioms of geometry express the peculiarities ofwhat is spatial; rather, these basic propositions must extend toeverything that can be thought. And surely we are justified inascribing such extremely general propositions to logic. (1885, 95, inFrege 1984; for further discussion, see MacFarlane 2002)

The tradition of demarcating the logical constants as expressionsthat can be characterized by purely inferential introduction andelimination rules can be seen as a way of capturing this notion ofcompletely general applicability. For, plausibly, it is the fact thatthe logical constants are characterizable in terms of notionsfundamental to thought or reasoning (for example, valid inference) thataccounts for their universal applicability.

The antinomy with which we started can now be resolvedby disambiguating. Arithmetic and set theory make distinctions amongobjects, and so are not topic-neutralin the first sense, but they mightstill be topic-neutral in the second sense, by virtue of their universalapplicability to reasoning about any subject.We are still faced with a decision about which ofthese notions of topic neutrality is distinctive of logic. Let us postpone thisproblem, however, until we have had a closer look at both notions.

5. Permutation invariance

A number of philosophers have suggested that what is distinctive oflogical constants is their insensitivity to the particular identitiesof objects, or, more precisely, theirinvariance underarbitrary permutations of the domain of objects (Mautner 1946;Mostowski 1957, 13; Scott 1970, 160–161; McCarthy 1981, 1987; Tarski1986; van Benthem 1989; Sher 1991, 1996; McGee 1996).

Let us unpack that phrase a bit. Apermutation of acollection of objects is a one-one mapping from that collection ontoitself. Each object gets mapped to an object in the collection(possibly itself), and no two objects are mapped to the same object.For example, the following mapping is a permutation of the first fiveletters of the alphabet:

And the function \(f(x) = x + 1\) is apermutation of the set of integers onto itself. (Note, however, that apermutation need not be specifiable either by enumeration, as in ourfirst example, or by a rule, as in our second.)

The extension of a predicate is invariant under a permutation of the domain ifreplacing each of its members with the object towhich the permutation maps it leaves us with the same set westarted with. Thus, for example, the extension of “is aletter between \(\textrm{A}\) and \(\textrm{E}\)” is invariant under the permutation ofletters described above. By contrast, theextension of “is a vowel between \(\textrm{A}\) and \(\textrm{E}\)”,the set \(\{\textrm{A}, \textrm{E}\}\),is not invariant under this permutation, which transforms itto a different set, \(\{\textrm{C}, \textrm{D}\}\).

We can make the notion of permutation invariance more preciseas follows. Given a permutation \(p\) of objects on a domain \(D\),we define a transformation \(p^*\) ofarbitrary types in the hierarchy:

• if \(x\) is an object in \(D\), \(p^*(x) = p(x)\).
• if \(x\) is a set, then \(p^*(x) =\{ y : \exists z (z \in x \and y = p^*(z))\}\)(that is, the set of objects to which\(p^*\) maps members of \(x\)).
• if \(x\) is an ordered \(n\)-tuple\(\langle x_1, \dots, x_n \rangle\), then\(p^*(x) = \langle p^*(x_1), \dots, p^*(x_n) \rangle\)(that is, the \(n\)-tuple of objects to which \(p^*\) maps\(x_1, \dots, x_n\)).^[9]

These clauses can be applied recursively to definetransformations of sets of ordered tuples in \(D\) (the extensionsof two-place predicates), setsof sets of objects in \(D\) (the extensions of unary first-order quantifiers),and so on. (For an introduction tothe type theoretic hierachy, seethe entry onType Theory.) Where \(x\) is anitem in this hierarchy, we say that \(x\) isinvariantunder a permutation \(p\) just in case \(p^*(x) = x\).To return to our example above, the set \(\{\textrm{A}, \textrm{B}, \textrm{C}, \textrm{D}, \textrm{E}\}\) isinvariant under all permutations of the letters \(\textrm{A}\) through \(\textrm{E}\):no matter how we switch these letters around, we end up with the sameset. But it is not invariant under all permutations of the entirealphabet. For example, the permutation that switches the letters \(\textrm{A}\)and \(\textrm{Z}\), mapping all the other letters to themselves, transforms \(\{\textrm{A}, \textrm{B}, \textrm{C}, \textrm{D}, \textrm{E}\}\) to \(\{\textrm{Z}, \textrm{B}, \textrm{C}, \textrm{D}, \textrm{E}\}\). The set containing all the letters,however, is invariant under all permutations of letters. So isthe set of all sets containing at least two letters, and the relationof identity, which holds between each letter and itself.

So far we have defined permutation invariance for objects, tuples,and sets, but not for predicates, quantifiers, or other linguisticexpressions. But it is the latter, not the former, that we need to sortinto logical and nonlogical constants. The natural thought is that anexpression should count as permutation-invariant just in case itsextension on each domain of objects is invariant under all permutationsof that domain. (As usual, the extension of a name on a domain is theobject it denotes, the extension of a monadic predicate is the set ofobjects in the domain to which it applies, and the extension of an\(n\)-adic predicate is the set of \(n\)-tuples of objects inthe domain to which it applies.) As it stands, this definition does notapply to sentential connectives, which do not haveextensions in the usual sense,^[10] but it can be extended to cover them in a natural way (followingMcGee 1996, 569). We can think of the semantic value of an \(n\)-aryquantifier or sentential connective \(C\) on a domain \(D\)as a function from \(n\)-tuples of sets of assignments (of valuesfrom \(D\) to the language’s variables) to sets ofassignments. Where the input to the function is the \(n\)-tupleof sets of assignments that satisfy \(\phi_1, \dots, \phi_n\),its output is the set of assignmentsthat satisfies \(C\phi_1 \dots \phi_n\).(Check your understanding by thinking about how this works forthe unary connective \(\exists x\).)We can then define permutationinvariance for these semantic values as follows. Where \(A\) is aset of assignments and \(p\) is a permutation of a domain \(D\), let\(p^\dagger(A) = \{ p \circ a : a \inA\}\).^[11] Then if \(e\) is the semantic value of an \(n\)-placeconnective or quantifier (in the sense defined above), \(e\) isinvariant under a permutation \(p\) just in case for any\(n\)-tuple \(\langle A_1, \dots, A_n \rangle\) ofsets of assignments, \(p^\dagger (e(\langle A_1, \dots, A_n\rangle)) =e(\langle p^\dagger (A_1), \dots, p^\dagger (A_n)\rangle\)).And a connective or quantifieris permutation-invariant just in case its semantic value on each domain ofobjects is invariant under all permutations of that domain.

It turns out that this condition does not quite suffice to weed outall sensitivity to particular features of objects, for itallows that a permutation-invariant constant might behave differentlyon domains containing different kinds of objects. McGee (1996, 575)gives the delightful example ofwombat disjunction, whichbehaves like disjunction if the domain contains wombats and likeconjunction otherwise. Sher’s fix, and McGee’s, is to consider not justpermutations—bijections of the domain onto itself—butarbitrary bijections of the domain onto another domain of equal cardinality.^[12] For simplicity, we will ignore thiscomplication in what follows and continue to talk of permutations.

Which expressions get counted as logical constants, on thiscriterion? The monadic predicates “is a thing” (which applies toeverything) and “is not anything” (which applies to nothing), theidentity predicate, the truth-functional connectives, and the standardexistential and universal quantifiers all pass the test. So do thestandard first-order binary quantifiers like “most” and “the” (see the entry ondescriptions). Indeed, because cardinality is permutation-invariant, everycardinality quantifier is included, including “there are infinitelymany”, “there are uncountably many”, and others that are notfirst-order definable. Moreover, the second-order quantifiers count aslogical (at least on the standard semantics, in which they range overarbitrary subsets of the domain), as do all higher-orderquantifiers. On the other hand, all proper names are excluded, as arethe predicates “red”, “horse”, “is a successor of”, and “is a memberof”, as well as the quantifiers “some dogs” and “exactly two naturalnumbers”. So the invariance criterion seems to accord at leastpartially with common intuitions about logicality or topic neutrality,and with our logical practice. Two technical results allow us to be abit more precise about the extent of this accord: Lindenbaum andTarski (1934–5) show that all of the relations definable in thelanguage ofPrincipia Mathematica arepermutation-invariant. Moving in the other direction, McGee (1996)shows that every permutation-invariant operation can be defined interms of operations with an intuitively logical character (identity,substitution of variables, finite or infinite disjunction, negation,and finite or infinite existential quantification). He alsogeneralizes the Lindenbaum-Tarski result by showing that everyoperation so definable is permutation invariant.

As Tarski and others have pointed out, the permutation invariancecriterion for logical constants can be seen as a natural generalizationof Felix Klein’s (1893) idea that different geometries can bedistinguished by the groups of transformations under which their basicnotions are invariant. Thus, for example, the notions of Euclideangeometry are invariant under similarity transformations, those ofaffine geometry under affine transformations, and those of topologyunder bicontinuous transformations. In the same way, Tarski suggests(1986, 149), thelogical notions are just those that areinvariant under the widest possible group of transformations: the groupofpermutations of the elements in the domain. Seen in thisway, the logical notions are the end point of a chain of progressivelymore abstract, “formal,” or topic-neutral notions defined by theirinvariance under progressively wider groups of transformations of adomain.^[13]

As an account of the distinctive generality of logic, then,permutation invariance has much to recommend it. It is philosophicallywell-motivated and mathematically precise, it yields results thataccord with common practice, and it gives determinate rulings aboutsome borderline cases (for example, set-theoretic membership). Best ofall, it offers hope for a sharp and principled demarcation of logicthat avoids cloudy epistemic and semantic terms like “about”,“analytic”, and “a priori”.^[14]

A limitation of the permutation invariance criterion (as it has beenstated so far) is that it applies only to extensional operators andconnectives. It is therefore of no help in deciding, for instance,whether the necessity operator in S4 modal logic or theHoperator (“it hasalways been the case that”) in temporal logic are bonafide logical constants, and these are among the questions that wewanted a criterion to resolve. However, the invariance criterion can beextended in a natural way to intensional operators. The usual strategyfor handling such operators semantically is to relativize truth notjust to an assignment of values to variables, but also to a possibleworld and a time. In such a framework, one might demand that logicalconstants be insensitive not just to permutations of the domain ofobjects, but to permutations of the domain of possible worlds and thedomain of times (see Scott 1970, 161, McCarthy 1981, 511–13, vanBenthem 1989, 334). The resulting criterion is fairly stringent: itcounts the S5 necessity operator as a logical constant, but not the S4necessity operator or theH operator in temporal logic.The reason is that the latter two operators are sensitive tostructure on the domains of worlds andtimes—the “accessibility relation” in the former case, therelation of temporal ordering in the latter—and this structure isnot preserved by all permutations of these domains.^[15] (See the entries onmodal logic andtemporal logic.)

One might avoid this consequence by requiring only invariance underpermutations that preserve the relevant structure on these domains(accessibility relations, temporal ordering). But one would then befaced with the task of explaining whythis structure deservesspecial treatment (cf. van Benthem 1989, 334). And if we are allowed tokeep some structure on the domain of worlds or times fixed, thequestion immediately arises why we should not also keep some structureon the domain ofobjects fixed: for example, the set-theoreticmembership relation, the mereological part/whole relation, or thedistinction between existent and nonexistent objects (see theentry onfree logics).Whatever resources we appeal to in answering this questionwill be doing at least as much work as permutation invariance in theresulting demarcation of logical constants.

It may seem that the only principled position is to demandinvariance underall permutations. But even that positionneeds justification, especially when one sees that it is possible toformulate even stricter invariance conditions. Feferman (1999) definesa “similarity invariance” criterion that counts the truth-functionaloperators and first-order existential and universal quantifiers aslogical constants, but not identity, the first-order cardinalityquantifiers, or the second-order quantifiers. Feferman’s criteriondraws the line between logic and mathematics much closer to thetraditional boundary than the permutation invariance criterion does.Indeed, one of Feferman’s criticisms of the permutation invariancecriterion is that it allows too many properly mathematical notions tobe expressed in purely logical terms. Bonnay (2008) argues fora different criterion, invariance under potential isomorphism, whichcounts finite cardinality quantifiers and the notion of finiteness aslogical, while excluding the higher cardinalityquantifiers—thus “[setting] the boundary betweenlogic and mathematics somewhere between arithmetic and set theory”(37; see Feferman 2010, §6, for further discussion).Feferman (2010) suggests that instead of relying solelyon invariance, we might combine invariance under permutationswith a separateabsoluteness requirement, which captures theinsensitivity of logic to controversial set-theoretic theses likeaxioms of infinity. He shows that the logical operations thatare both permutation-invariant and absolutely definablewith respect to Kripke–Platek set theory without anaxiom of infinity are just those definable in first-order logic.

There is another problem that afflicts any attempt to demarcate thelogical constants by appeal to mathematical properties like invariance.As McCarthy puts it: “the logical status of an expression is notsettled by the functions it introduces, independently of how thesefunctions arespecified” (1981, 516). Consider a two-placepredicate \(\dq{\approx}\), whose meaning is given by the followingdefinition:

According to the invariance criterion, \(\dq{\approx}\) is a logicalconstant just in case its extension on every domain is invariant underevery permutation of that domain. On a domain \(D\) containing notwo objects with exactly the same mass, \(\dq{\approx}\) has the sameextension as \(\dq{=}\)—the set \(\{ \langle x, x \rangle :x \in D\}\)—and as we have seen, this extensionis invariant under every permutation of the domain. Hence, if there isno domain containing two objects with exactly the same mass,\(\dq{\approx}\) counts as a logical constant, and\(\dq{\forall x (x \approx x)}\) as a logicaltruth.^[16]But itseems odd that the logical status of \(\dq{\approx}\) and\(\dq{\forall x (x \approx x)}\) should depend on a matter ofcontingent fact: whether there are distinct objects with identicalmass. Do we really want to say that if we lived in a world in which notwo objects had the same mass, \(\dq{\approx}\) would be a logical constant?^[17]

A natural response to this kind of objection would be to requirethat the extension of a logical constant on everypossibledomain of objects be invariant under every permutation of that domain,or, more generally, that a logical constant satisfy the permutationinvariance criterion as a matter ofnecessity. But this wouldnot get to the root of the problem. For consider the unary connective\(\dq{\#}\), defined by the clause

Assuming that Kripke (1971; 1980) is right that water isnecessarilyHO,\(\dq{\#}\) has the same extension as\(\dq{\neg}\) in every possible world, and so satisfies the permutationinvariance criterion as a matter of necessity (McGee 1996, 578). But intuitively, it doesnot seem that \(\dq{\#}\) should be counted a logical constant.^[18]

One might evade this counterexample by appealing to anepistemic modality instead of a metaphysical one. This isMcCarthy’s strategy (1987, 439). Even if it is metaphysically necessarythat water is HO, there arepresumably epistemically possible worlds, or information states, inwhich water is not HO. So if werequire that a logical constant be permutation invariant as a matter ofepistemic necessity (ora priori), \(\dq{\#}\) does not count as alogical constant. But even on this version of the criterion, aconnective like \(\dq{\%}\), defined by

would count as a logical constant (Gómez-Torrente2002, 21), assuming that it is epistemically necessary that there areno male widows. It may be tempting to solvethis problem byappealing to a distinctivelylogical modality—requiring,for example, that logical constants have permutation-invariantextensions as a matter oflogical necessity. But we would thenbe explicating the notion of a logical constant in terms of an obscureprimitive notion of logical necessity which we could not, on pain ofcircularity, explicate by reference to logical constants. (McCarthy1998, §3 appeals explicitly to logical possibility and notices thethreat of circularity here.)

McGee’s strategy is to invoke semantic notions instead of modalones: he suggests that “[a] connective is a logical connective if andonly if it follows from the meaning of the connective that it isinvariant under arbitrary bijections” (McGee 1996, 578). But thisapproach, like McCarthy’s, seems to count \(\dq{\%}\) as a logical constant.And, like McCarthy’s, it requires appeal to a notion that does not seemany clearer than the notion of a logical constant: the notion offollowing (logically?) from the meaning of the connective.

Sher’s response to the objection is radically different from McGee’sor McCarthy’s. She suggests that “logical terms are identified withtheir (actual) extensions,” so that \(\dq{\#}\), \(\dq{\%}\),and \(\dq{\neg}\) are justdifferent notations for the same term. More precisely: if theseexpressions are used the way a logical constant must be used—asrigid designators^[19] of their semantic values—then they can be identified with theoperation of Boolean negation and hence with each other. “Quaquantifiers, ‘the number of planets’ and ‘9’ areindistinguishable” (Sher 1991, 64). But it is not clear what Sher canmean when she says that logical terms can be identified with theirextensions. We normally individuate connectivesintentionally, by the conditions for grasping them or therules for their use, and not by the truth functions they express. Forexample, we recognize a difference between \(\dq{\and}\), defined by

and \(\dq{@}\), defined by

even though they express the same truth function. The distinctionbetween these terms is not erased, as Sher seems to suggest, if we usethem as rigid designators for the truth functions they express. (That“Hesperus”, “Phosphorus”, and “the planetI actually saw near the horizon on the morning of November 1, 2004”all rigidly designate Venusdoes not entail that they have the same meaning.)Thus Sher’s proposal can only beunderstood as astipulation that if one of a pair ofcoreferential rigid designators counts as a logical constant, the otherdoes too. But it is not clear why we should accept this stipulation. Itcertainly has some counterintuitive consequences: for example, that\(\dq{P \vee \#P}\) is a logicaltruth, at least when \(\dq{\#}\) is used rigidly (see Gómez-Torrente2002, 19, and the response in Sher 2003).

It is hard not to conclude from these discussions that thepermutation invariance criterion gives at best a necessary conditionfor logical constancy. Its main shortcoming is that it operates at thelevel of reference rather than the level of sense; it looks at thelogical operations expressed by the constants, but not at theirmeanings. An adequate criterion, one might therefore expect, wouldoperate at the level of sense, perhaps attending to the way wegrasp the meanings of logical constants.

6. Inferential characterizations

At the end of the section on topic neutrality, we distinguished twonotions of topic neutrality. The first notion—insensitivity tothe distinguishing features of individuals—is effectivelycaptured by the permutation invariance criterion. How might we capturethe second—universal applicability to all thought or reasoning,regardless of its subject matter? We might start by identifying certainingredients that must be present in anything that is to count asthought or reasoning, then class as logical any expression that can beunderstood in terms of these ingredients alone. That would ensure aspecial connection between the logical constants and thought orreasoning as such, a connection that would explain logic’s universalapplicability.

Along these lines, it has been proposed that the logical constantsare just those expressions that can be characterized by a set of purelyinferential introduction and elimination rules.^[20] To grasp the meaningof the conjunction connective \(\dq{\and}\), for example, it is arguablysufficient to learn that it is governed by the rules:\begin{equation*}\frac{A, B}{A \and B}\quad\frac{A \and B}{A}\quad\frac{A \and B}{B}\end{equation*}Thus the meaning of \(\dq{\and}\) can be grasped by anyone who understandsthe significance of the horizontal line in an inference rule. (Contrast\(\dq{\%}\) from the last section, which cannot be grasped by anyone who doesnot understand what a male is and what a widow is.) Anyone who iscapable of articulate thought or reasoning at all should be able tounderstand these inference rules, and should therefore be in a positionto grasp the meaning of \(\dq{\and}\). Or so the thought goes.^[21]

To make such a proposal precise, we would have to make a number ofadditional decisions:

• We would have to decide whether to use natural deduction rules orsequent rules. (See the entry onthe development of proof theory.)

• If we opted to use sequent rules, we would have to decide whether ornot to allow “substructure” (see the entry onsubstructural logics) and whether to allow multiple conclusions in the sequents. We wouldalso have to endorse a particular set of purely structural rules(rules not involving any expression of the language essentially).

• We would have to specify whether it is introduction or eliminationrules, or both, that are to characterize the meanings of logicalconstants.^[22] (In a sequent formulation, we would have todistinguish between right and left introduction and eliminationrules.)

• We would have to allow for subpropositional structure in our rules,in order to make room for quantifier rules.

• We would have to say when an introduction or elimination rule countsas “purely inferential,” to exclude rules like these:\begin{equation*}\frac{a~\text{is red}}{Ra}\quad\frac{A, B, \text{water is}~H_{2}O}{A * B}\end{equation*}The strictest criterion would allow only rules in which every sign,besides a single instance of the constant being characterized, iseither structural (like the comma) or schematic (like \(\dq{A}\)).But although this condition is met by the standard rules forconjunction, it is not met by the natural deduction introduction rulefor negation, which must employ either another logical constant(\(\dq{\bot}\)) or another instance of the negation sign than the one beingintroduced. Thus one must either relax the condition for being “purelyinferential” or add more structure (see especially Belnap 1982).

Different versions of the inferential characterization approach makedifferent decisions about these matters, and these differences affectwhich constants get certified as “logical.” For example, if we usesingle-conclusion sequents with the standard rules for the constants,we get the intuitionistic connectives, while if we usemultiple-conclusion sequents, we get the classical connectives (Kneale1956, 253). If we adopt Došen’s constraints on acceptable rules(Došen 1994, 280), the S4 necessity operator gets counted as alogical constant, while if we adopt Hacking’s constraints, it doesn’t(Hacking 1979, 297). Thus, if we are to have any hope of deciding thehard cases in a principled way, we will have to motivate all of thedecisions that distinguish our version of the inferentialcharacterization approach from the others. Here, however, we will avoidgetting into these issues of detail and focus instead on the basicidea.

The basic idea is that the logical constants are distinguished fromother sorts of expressions by being “characterizable” in terms ofpurely inferential rules. But what does “characterizable” mean here? AsGómez-Torrente (2002, 29) observes, it might be taken to requireeither the fixation of reference (semantic value) or the fixation ofsense:

Let us consider these two versions of the inferentialcharacterization approach in turn.

6.1 Semantic value determination

Hacking shows that, given certain background semantic assumptions(bivalence, valid inference preserves truth), any introduction andelimination rules meeting certain proof-theoretic conditions(subformula property, provability of elimination theorems for Cut,Identity, and Weakening) will uniquely determine a semantics for theconstant they govern (Hacking 1979, 311–314). It is in this sense thatthese rules “fix the meaning” of the constant: “they are such that ifstrong semantic assumptions of a general kind are made, then thespecific semantics of the individual logical constants is therebydetermined” (313).

The notion of determination of semantic value in a well-definedsemantic framework is, at least, clear—unlike the general notionof determination of sense. However, as Gómez-Torrente pointsout, by concentrating on the fixation of reference (or semantic value)rather than sense, Hacking opens himself up to an objection not unlikethe objection to permutation-invariance approaches we considered above(see also Sainsbury 2001, 369). Consider the quantifier \(\dq{W}\),which means “not for all not …, if all are not male widows, and forall not …, if not all are not male widows” (Gómez-Torrente2002, 29). (It is important here that \(\dq{W}\) is a primitive signof the language, not one introduced by a definition in terms of\(\dq{\forall}\), \(\dq{\neg}\), “male”, and“widow”.) Since there are no malewidows, \(\dq{W}\) has the same semantic value as our ordinaryquantifier \(\dq{\exists}\). (As above, we can think of the semantic value ofa quantifier as a function from sets of assignments to sets ofassignments.) Now let \(R\) be the standard introduction andelimination rules for \(\dq{\exists}\), and let \(R'\) be theresult of substituting \(\dq{W}\) for \(\dq{\exists}\) in these rules.Clearly, \(R'\) is no less “purely inferential” than\(R\). And if \(R\) fixes a semantic value for\(\dq{\exists}\), then \(R'\) fixes a semantic value—the verysame semantic value—for \(\dq{W}\). So if logical constants areexpressions whose semantic values can be fixed by means of purelyinferential introduction and elimination rules, \(\dq{W}\) counts asa logical constant if and only if \(\dq{\exists}\) does.

Yet intuitively there is an important difference between theseconstants. We might describe it this way: whereas learning the rules\(R\) is sufficient to impart a full grasp of \(\dq{\exists}\), one couldlearn the rules \(R'\) without fully understanding what ismeant by \(\dq{W}\). To understand \(\dq{W}\) one must know aboutthe human institution of marriage, and that accounts for our feelingthat \(\dq{W}\) is not “topic-neutral” enough to be a logicalconstant. However, this difference between \(\dq{W}\) and \(\dq{\exists}\)cannot be discerned if we talk only of reference or semantic value; itis a difference in thesenses of the two expressions.

6.2 Sense determination

The idea that introduction and/or elimination rules fix thesense of a logical constant is often motivated by talk of therules asdefining the constant. Gentzen remarks that the naturaldeduction rules “represent, as it were, the‘definitions’ of the symbols concerned, and the eliminations are nomore, in the final analysis, than the consequences of these definitions”(1935, §5.13; 1969, 80).However, a genuine definition would permit the constant to beeliminated from every context in which it occurs (see the entryonDefinitions), and introduction andelimination rules for logical constants do not, in general, permitthis. For example, in an intuitionistic sequent calculus, there is nosequent (or group of sequents) not containing \(\dq{\rightarrow}\)that is equivalent to the sequent \(\dq{A \rightarrow B \vdash C}\).For this reason, Kneale(1956, 257) says only that we can “treat” the rules as definitions,Hacking (1979) speaks of the rules “not as defining but only as characterizingthe logical constants,” and Došen (1994) says that the rulesprovide only an “analysis,” not a definition.^[23]

However, even if the rules are not “definitions,” there may still besomething to say for the claim that they “fix the senses” of theconstants they introduce. For it may be that a speaker’s grasp of themeaning of the constants consists in her mastery of these rules: herdisposition to accept inferences conforming to the rules as“primitively compelling” (Peacocke 1987, Hodes 2004). (A speaker findsan inference form primitively compelling just in case she finds itcompelling and does not take its correctness to require externalratification, e.g. by inference.) If the senses of logical constantsare individuated in this way by the conditions for their grasp, we candistinguish between truth-functionally equivalent constants withdifferent meanings, like\(\dq{\vee}\), \(\dq{\ddagger}\), and \(\dq{\dagger}\),as defined below:

To understand \(\dq{\vee}\) one must findthe standard introduction rules primitivelycompelling:

To understand \(\dq{\ddagger}\) one must find thefollowing elimination rule primitively compelling:

Finally, to grasp the sense of \(\dq{\dagger}\) one must find theseintroduction rules primitively compelling:

\(\dq{\vee}\) and \(\dq{\ddagger}\) will count aslogical constants, because their sense-constitutive rules are purelyinferential, while \(\dq{\dagger}\) will not, because its rules are not.(In the same way we can distinguish \(\dq{\exists}\) from\(\dq{W}\).) Note that appropriately rewritten versions of \(\refp{or-intro}\)willhold for \(\dq{\ddagger}\) and \(\dq{\dagger}\); the difference isthat one can grasp \(\dq{\ddagger}\) and \(\dq{\dagger}\) (but not\(\dq{\vee}\)) without finding these rulesprimitivelycompelling (Peacocke 1987, 156; cp. Sainsbury 2001, 370–1).

Some critics have doubted that the introduction and eliminationrules for the logical constants exhaust the aspects of the use of theseconstants that must be mastered if one is to understand them. Forexample, it has been suggested that in order to grasp the conditionaland the universal quantifier, one must be disposed to treat certainkinds of inductive evidence as grounds for the assertion ofconditionals and universally quantified claims (Dummett 1991, 275–8;Gómez-Torrente 2002, 26–7; Sainsbury 2001, 370–1). It is notclear that these additional aspects of use can be captured in “purelyinferential” rules, or that they can be derived from aspects of usethat can be so captured.

It is sometimes thought that Prior’s (1960) example of aconnective “tonk,”

whose rules permitinferring anything from anything, decisively refutesthe idea that the senses of logical constants are fixedby their introduction and/or elimination rules. But althoughPrior’s example(anticipated in Popper 1946–7, 284) certainly shows thatnot all sets ofintroduction and elimination rules determine a coherent meaning for alogical constant, it does not show thatnone do,or that the logical constants arenot distinctive in having their meanings determined in this way.For some attempts to articulate conditions underwhich introduction and elimination rules do fix a meaning, see Belnap(1962), Hacking (1979, 296–8), Kremer (1988, 62–6),and Hodes (2004, 156–7).

Prawitz (1985; 2005) argues thatany formally suitable introduction rule can fix the meaning for alogical constant. On Prawitz’s view, the lesson we learn from Prior is that we cannotalso stipulate an elimination rule, but mustjustify any proposed elimination rule by showing that there is a procedurefor rearranging any direct proof of the premisesof the elimination rule into a direct proof of the conclusion.Thus, we can stipulatethe introduction rule for “tonk”, but must thencontent ourselves with the strongest elimination rule for which such a procedureis available:

Other philosophers reject Prawitz's (and Gentzen's) view that the introductionrules have priority in fixing the meanings of constants, butretain the idea that the introduction and elimination rules that fix the meaning ofa constant must be inharmony:the elimination rules must not permit us to infer more from a compound sentencethan would be justified by the premises of the corresponding introduction rules(Dummett 1981, 396; Tennant 1987, 76-98).(For analyses of various notions of harmony, and their relation to notions likenormalizability and conservativeness, see Milne 1994, Read 2010, and Steinberger 2011.)

7. Pragmatic demarcations

The proposals for demarcating logical constants that we haveexamined so far have all beenanalytical demarcations. Theyhave sought to identify some favored property (grammaticalparticlehood, topic neutrality, permutation invariance,characterizability by inferential rules, etc.) as a necessary andsufficient condition for an expression to be a logical constant. Afundamentally different strategy for demarcating the constants is tostart with ajob description for logic and identify theconstants as the expressions that are necessary to do that job. Forexample, we might start with the idea that the job of logic is to serveas a “framework for the deductive sytematization of scientifictheories” (Warmbrod 1999, 516), or to characterize mathematicalstructures and represent mathematical reasoning (Shapiro 1991), or to“[express] explicitlywithin a language the features of theuse of that language that confer conceptual contents on the states,attitudes, performances, and expressions whose significances aregoverned by those practices” (Brandom 1994, xviii). Let us calldemarcations of this kindpragmatic demarcations.

There are some very general differences between the two kinds ofdemarcations. Unlike analytical demarcations, pragmatic demarcationsare guided by what Warmbrod calls a “requirement of minimalism”:

…logical theory should be as simple, as modest in itsassumptions, and as flexible as possible given the goal of providing aconceptual apparatus adequate for the project of systematization. Inpractice, the minimalist constraint dictates that the set of termsrecognized as logical constants should be as small as possible.(Warmbrod 1999, 521)

Or, in Harman’s pithier formulation: “Count as logic only as much asyou have to” (Harman 1972, 79). Warmbrod uses this constraint to arguethat the theory of identity is not part of logic, on the grounds that it isnot needed to do the job he has identified for logic: “[w]e cansystematize the same sets of sentences by recognizing only thetruth-functional connectives and first-order quantifiers as constants,treating ‘=’ as an ordinary predicate, and adopting appropriate axiomsfor identity” (521; cf. Quine 1986, 63, 1980, 28). On similar grounds,both Harman and Warmbrod argue that modal operators should not beconsidered part of logic.^[24] Their point is not that identity ormodal operators lack some feature that the first-order quantifiers andtruth-functional operators possess, but merely that, since wecan get by without taking these notions to be part of ourlogic, we should. Warmbrod and Tharp even explore the possibility oftaking truth-functional logic to be the whole of logic and viewingquantification theory as a non-logical theory (Warmbrod 1999, 525;Tharp 1975, 18), though both reject this idea on pragmatic grounds.

While pragmatic demarcations seek to minimize what counts as logic,analytical demarcations are inclusive. They count as logicalany expression that has the favored property. It is simplyirrelevant whether an expression isrequired for a particularpurpose: its logicality rests on features that it has independently ofany use to which we might put it.

Relatedly, pragmatic approaches tend to be holistic. Because it iswhole logicalsystems that can be evaluated as sufficient orinsufficient for doing the “job” assigned to logic, properties ofsystems tend to be emphasized in pragmatic demarcations. For example,Wagner (1987, 10–11) invokes Lindstrom’s theorem—that first-orderlogic is the only logic that is either complete or compact andsatisfies the Löwenheim-Skolem theorem—in arguing that logicshould be limited to first-order logic, and Kneale and Kneale (1962,724, 741) invoke Gödel’s incompleteness theorems to similareffect. Although nothing about the idea of an analytical demarcationexcludes appeal to properties of whole systems, analytical demarcationstend to appeal tolocal properties of particular expressionsrather than global systemic properties.

Finally, on a pragmatic demarcation, what counts as logic may dependon the current state of scientific and mathematical theory. If theadvance of science results in an increase or decrease in the resourcesneeded for deductive systematization of science (or whatever is thefavored task of logic), what counts as logic changes accordingly(Warmbrod 1999, 533). On an analytical demarcation, by contrast,whether particular resources are logical depends only on whether theyhave the favored property. If they do not, and if it turns out thatthey are needed for the deductive systematization of theories, then theproper conclusion to draw is that logic alone is not adequate for thistask.

8. Problem or pseudoproblem?

Now that we have gotten a sense for the tremendous variety ofapproaches to the problem of logical constants, let us step back andreflect on the problem itself and its motivation. We can distinguishfour general attitudes toward the problem of logical constants: thoseof the Demarcater, the Debunker, the Relativist, and the Deflater.

Demarcaters hold that the demarcation of logical constantsis a genuine and important problem, whose solution can be expected toilluminate the nature and special status of logic. On their view, thetask of logic is to study features that arguments possess in virtue oftheir logical forms or structures.^[25] Although there may be some sense inwhich the argument

is a good or “valid” argument, it is notformally valid. Onthe Demarcater’s view, logicians who investigate the (non-formal) kindof “validity” possessed by \(\refp{chicago-north}\) are straying fromthe proper provinceof logic into some neighboring domain (here, geography or lexicography;in other cases, mathematics or metaphysics). For the Demarcater, then,understanding the distinction between logical and nonlogical constantsis essential for understanding what logic is about. (For a forcefulstatement of the Demarcater’s point of view, see Kneale 1956.)

Debunkers, on the other hand, hold that the so-called“problem of logical constants” is a pseudoproblem (Bolzano 1929,§186; Lakoff 1970, 252–4; Coffa 1975; Etchemendy 1983, 1990,ch. 9; Barwise and Feferman 1985, 6; Read 1994). They do notdispute that logicians have traditionally concerned themselves withargument forms in which a limited number of expressions occuressentially. What they deny is that these expressions and argumentforms define thesubject matter of logic. On their view, logicis concerned with validitysimpliciter, not just validity thatholds in virtue of a limited set of “logical forms.” The logician’smethod for studying validity is to classify arguments by theirforms, but these forms (and the logical constants that in part definethem) are logic’stools, not its subject matter. The forms andconstants with which logicians are concerned at a particular point inthe development of logic are just a reflection of the logicians’progress (up to that point) in systematically classifying validinferences. Asking what is special about these forms and constants isthus a bit like asking what is special about the mountains that can beclimbed in a day: “The information so derived will be too closelydependent upon the skill of the climber to tell us much aboutgeography” (Coffa 1975, 114). What makes people logicians is not theirconcern with “and”, “or”, and “not”, but their concern with validity,consequence, consistency, and proof, and the distinctive methods theybring to their investigations.

A good way to see the practical difference between Debunkers andDemarcaters is by contrasting their views on the use of counterexamplesto show invalidity. Demarcaters typically hold that one can show anargument to be invalid by exhibiting another argument with the samelogical form that has true premises and a false conclusion. Of course,an argument will always instantiate multiple forms. For example, theargument

can be seen as an instance of the propositional logical form

as well as the more articulated form

As Massey (1975) reminds us, the fact that there are other argumentswith the form \(\refp{propform}\) that have true premises and a false conclusion doesnot show that \(\refp{firefighter}\) is invalid (or even that it is“formally” invalid).The Demarcater will insist that a genuine counterexample to the formalvalidity of \(\refp{firefighter}\) would have to exhibit thefulllogical structure of \(\refp{firefighter}\), which is not \(\refp{propform}\) but\(\refp{quantform}\). Thus the Demarcater’suse of counterexamples to demonstrate the formal invalidity ofarguments presupposes a principled way of discerning thefulllogical structure of an argument, and hence of distinguishing logicalconstants from nonlogical constants.^[26]

The Debunker, by contrast, rejects the idea that one of the manyargument forms \(\refp{firefighter}\) instantiates should be privileged asthe logical form of \(\refp{firefighter}\). On the Debunker’sview, counterexamples nevershow anything about a particular argument. All they show is that aform is invalid (that is, that it has invalid instances). Toshow that a particularargument is invalid, one sort of Debunkerholds, one needs to describe a possible situation in which the premiseswould be true and the conclusion false, and to give a formalcounterexample is not to dothat.

The Demarcater will object that the Debunker’s tolerant attitudeleaves us with no coherent distinction between logic and otherdisciplines. For surely it is the chemist, not the logician, who willbe called upon to tell us whether the following argument is a goodone:

Without a principled distinction between logical and nonlogicalconstants, it seems, logic would need to be a kind of universalscience: not just a canon for inference, but an encyclopedia. If logicis to be a distinctive discipline, the Demarcater will argue, it mustconcern itself not with all kinds of validity or goodness of arguments,but with a special, privileged kind:formal validity.

Against this, the Debunker might insist that deductive validity is afeature arguments have by virtue of the meanings of the terms containedin them, so that anyone who understands the premises and conclusion ofan argument must be in a position to determine, without recourse toempirical investigation, whether it is valid. On this conception, logicis the study ofanalytic truth, consequence, consistency, andvalidity. Because the relation between premise and conclusion in\(\refp{litmus}\)depends on empirical facts, not the meanings of terms, \(\refp{litmus}\) is notdeductively valid.^[27]

This response will not be available to those who have reservationsabout theanalytic/synthetic distinction. An important example is Tarski (1936a; 1936b; 1983;1987; 2002), who was much concerned to define logical truth andconsequence in purely mathematical terms, without appealing to suspectmodal or epistemic notions. On Tarski’s account, an argument is validjust in case there is no interpretation of its nonlogical constants onwhich the premises are true and the conclusion false. On this account,an argument containing no nonlogical constants is valid just in case itis materially truth-preserving (it is not the case that its premisesare true and its conclusion false). Thus, as Tarski notes, ifevery expression of a language counted as a logical constant,logical validity would reduce to material truth preservation (or, onlater versions of Tarski’s definition, to material truth preservationon every nonempty domain) (1983, 419). Someone who found this resultintolerable might take it to show either that theremust be aprincipled distinction between logical and nonlogical constants (theDemarcater’s conclusion), or that Tarski’s definition is misguided (theDebunker’s conclusion; see Etchemendy 1990, ch. 9).

Tarski’s own reaction was more cautious. After concluding that thedistinction is “certainly not quite arbitrary” (1983, 418), hewrites:

Perhaps it will be possible to find important objectivearguments which will enable us to justify the traditional boundarybetween logical and extra-logical expressions. But I also consider itto be quite possible that investigations will bring no positiveresults in this direction, so that we shall be compelled to regardsuch concepts as ‘logical consequence’, ‘analyticalstatement’, and ‘tautology’ as relative conceptswhich must, on each occasion, be related to a definite, although ingreater or less degree arbitrary, division of terms into logical andextra-logical. (420; see also Tarski 1987)

Here Tarski is describing a position distinct from both theDemarcater’s position and the Debunker’s. TheRelativistagrees with the Demarcater that logical consequence must be understoodasformal consequence, and so presupposes a distinctionbetween logical and nonlogical constants. But she agrees with theDebunker that we should not ask, “Which expressions are logicalconstants and which are not?” The way she reconciles these apparentlyconflicting positions is byrelativizing logical consequenceto a choice of logical constants. For each setC of logicalconstants, there will be a corresponding notion ofC-consequence. None of these notions is to be identified withconsequencesimpliciter; different ones are useful fordifferent purposes. In the limiting case, where every expression of thelanguage is taken to be a logical constant, we get materialconsequence, but this is no more (and no less)the consequencerelation than any of the others.

Like the Relativist, theDeflater seeks a moderate middleground between the Demarcater and the Debunker. The Deflater agreeswith the Demarcater that there is a real distinction between logicaland nonlogical constants, and between formally and materially validarguments. She rejects the Relativist’s position that logicalconsequence is a relative notion. But she also rejects the Demarcater’sproject of finding precise and illuminating necessary and sufficientconditions for logical constancy. “Logical constant”, she holds, is a“family resemblance” term, so we should not expect to uncover a hiddenessence that all logical constants share. As Wittgenstein said aboutthe concept of number: “the strength of the thread does not reside inthe fact that some one fibre runs through its whole length, but in theoverlapping of many fibres” (Wittgenstein 1958, §67). That doesnot mean that there is no distinction between logical and nonlogicalconstants, any more than our inability to give a precise definition of“game” means that there is no difference between games and otheractivities. Nor does it mean that the distinction does not matter. Whatit means is that we should not expect a principled criterion forlogical constancy that explains why logic has a privilegedepistemological or semantic status. (For a nice articulation of thiskind of view, see Gómez-Torrente 2002.)

The debate between these four positions cannot be resolved here,because to some extent “the proof is in the pudding.” A compelling andilluminating account of logical constants—one that vindicated adisciplinary segregation of \(\refp{chicago-north}\) from\(\refp{firefighter}\) by showing how these arguments are importantlydifferent—might give us reason to be Demarcaters. But it isimportant not to get so caught up in the debates between differentDemarcaters, or between Demarcaters and Debunkers, that one loses sightof the other positions one might take toward the problem of logical constants.

Further reading

Other recent general discussions of the problem of logical constantsinclude Peacocke 1976, McCarthy 1998, Warmbrod 1999, Sainsbury 2001,ch. 6, and Gómez-Torrente 2002. Tarski 1936b is essentialbackground to all of these.

For a discussion of grammatical criteria for logical terms, seeQuine 1980 and Føllesdal’s (1980) reply.

For a discussion of the Davidsonian approach, seeDavidson 1984, Evans 1976, Lycan 1989, Lepore and Ludwig 2002,and Edwards 2002.

Tarski 1986 is a brief and cogent exposition of thepermutation-invariance approach. For elaboration andcriticism, see McCarthy 1981, van Bentham 1989,Sher 1991, McGee 1996, Feferman 1999 and 2010, Bonnay 2008,and Dutilh Novaes 2014. Bonnay 2014 surveys recent workin this area.

Hacking 1979 and Peacocke 1987 are good representatives of the twoversions of the inferential characterization approach discussed above.Popper’s papers (1946–7, 1947) are still worth reading; seeSchroeder-Heister 1984 for critical discussion and Koslow 1999 for amodern approach reminiscent of Popper’s. See also Kneale 1956, Kremer1988, Prawitz 1985 and 2005, Tennant 1987, ch. 9, Dummett 1991, ch. 11, Došen 1994, Hodes2004, and Read 2010.

For examples of pragmatic demarcations, see Wagner 1987 andWarmbrod 1999. A different kind of pragmatic approach canbe found in Brandom (2000, ch. 1; 2008, ch. 2), who characterizeslogical vocabulary in terms of itsexpressive role.

For critiques of the whole project of demarcating the logicalconstants, see Coffa 1975, Etchemendy (1983; 1990, ch. 9), and Read1994.

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analytic/synthetic distinction |Davidson, Donald |definitions |logic: modal |logic: substructural |logic: temporal |logical consequence |logical form |semantics: proof-theoretic |Tarski, Alfred: truth definitions

Acknowledgments

I am grateful to Fabrizio Cariani, Kosta Došen, SolomonFeferman, Mario Gómez-Torrente, Graham Priest, Greg Restall,Gila Sher, and an anonymous reviewer for comments that helped improve this entry. Parts ofthe entry are derived from chapters 1, 2, 3, and 6 of my dissertation,“What Does It Mean to Say that Logic is Formal?”(University of Pittsburgh, 2000).