Logical Truth

First published Tue May 30, 2006; substantive revision Wed Sep 21, 2022

On standard views, logic has as one of its goals to characterize (andgive us practical means to tell apart) a peculiar set of truths, thelogical truths, of which the following English sentences are examplesstandardly taken as paradigmatic:

(1) If death is badonly if life is good, and death is bad, then life is good.
(2) If no desire isvoluntary and some beliefs are desires, then some beliefs are notvoluntary.
(3) If Drasha is a catand all cats are mysterious, then Drasha is mysterious.

As it turns out, it is very hard to think of universally acceptedideas about what the generic properties of logical truths are orshould be. A widespread, perhaps universally accepted idea is thatpart of what should distinguish logical truths from other kinds oftruths is that logical truths should have a yet to be fully understoodmodal force. It is typical to hold that, in some sense or senses of“could”, a logical truth could not be false or,alternatively, that in some sense or senses of “must”, alogical truth must be true. But there is little if any agreement abouthow the relevant modality should be understood.

Another widespread idea is that part of what should distinguishlogical truths is that they should be in some sense yet to be fullyunderstood “formal”. That a logical truth is formalimplies at the very least that all the sentences which are appropriatereplacement instances of its logical form are logical truths too. Inthis context, the logical form of a sentence \(S\) is supposed to be acertain schema determined uniquely by \(S\), a schema of which \(S\)is a replacement instance, and of which sentences with the same formas \(S\) are replacement instances too. A form has at the very leastthe property that the expressions in it which are not schematicletters (the “logical expressions”) are widely applicableacross different areas of discourse. Among people who accept the ideaof formality there would be wide agreement that the forms of (1), (2)and (3) would be something like \((1')\), \((2')\) and \((3')\)respectively:

\((1')\) If \(a\) is\(P\) only if \(b\) is \(Q\), and \(a\) is \(P\), then \(b\) is \(Q\).
\((2')\) If no \(Q\)is \(R\) and some \(P\)s are \(Q\)s, then some \(P\)s are not \(R\).
\((3')\) If \(a\) is a\(P\) and all \(P\)s are \(Q\), then \(a\) is \(Q\).

\((1')\), \((2')\) and \((3')\) do seem to give rise to sentences thatintuitively must be true for all appropriate replacements of theletters “\(a\)”, “\(b\)”, “\(P\)”,“\(Q\)”, and “\(R\)”. And expressions such as“if”, “and”, “some”,“all”, etc., which are paradigmatic logical expressions,do seem to be widely applicable across different areas of discourse.But the idea that logical truths are or should be formal is certainlynot universally accepted. And even among those who accept it, there islittle if any agreement about what generic criteria determine the formof an arbitrary sentence.^[1]

A remarkable fact about logical truth is that many have thought itplausible that the set of logical truths of certain rich formalizedlanguages is characterizable in terms of concepts of standardmathematics. In particular, on some views the set of logical truths ofa language of that kind is always the set of sentences of the languagederivable in a certain calculus. On other, more widespread views, theset of logical truths of a language of that kind can be identifiedwith the set of sentences that are valid across a certain range ofmathematical interpretations (where validity is something related tobut different from the condition that all the sentences that arereplacement instances of its form be true too; see below, section2.3). One main achievement of early mathematical logic was preciselyto show how to characterize notions of derivability and validity interms of concepts of standard mathematics. (Sections 2.2 and 2.3 givea basic description of the standard mathematically characterizednotions of derivability and validity for standard quantificationallanguages, with references to other entries.)

In part 1 of this entry we will describe in very broad outline themain existing views about how to understand the ideas of modality andformality relevant to logical truth. (A more detailed treatment ofthese views is available in other entries mentioned below, andespecially in the entries on theanalytic/synthetic distinction andlogical constants.) In part 2 we will describe, also in outline, a particular set ofphilosophical issues that arise when one considers the attemptedmathematical characterizations of logical truth. The question ofwhether or in what sense these characterizations are correct is boundwith the question of what is or should be our specific understandingof the ideas of modality and formality.^[2]

1. The Nature of Logical Truth

1.1 Modality

As we said above, it seems to be universally accepted that, if thereare any logical truths at all, a logical truth ought to be such thatit could not be false, or equivalently, it ought to be such that itmust be true. But as we also said, there is virtually no agreementabout the specific character of the pertinent modality.

Except among those who reject the notion of logical truth altogether,or those who, while accepting it, reject the notion of logical form,there is wide agreement that at least part of the modal force of alogical truth ought to be due to its being a particular case of a trueuniversal generalization over the possible values of the schematicletters in “formal” schemata like \((1')-(3')\). (Thesevalues may but need not be expressions.) On what is possibly theoldest way of understanding the logical modality, that modal force isentirely due to this property: thus, for example, on this view to saythat (1) must be true can only mean that (1) is a particular case ofthe true universal generalization “For all suitable \(P\),\(Q\), \(a\) and \(b\), if \(a\) is a \(P\) only if \(b\) is a \(Q\),and \(a\) is a \(P\), then \(b\) is a \(Q\)”. On one traditional(but not uncontroversial) interpretation, Aristotle's claim that theconclusion of asyllogismos must be true if the premises aretrue ought to be understood in this way. In a famous passage of thePrior Analytics, he says: “Asyllogismos isspeech (logos) in which, certain things being supposed,something other than the things supposed results of necessity (exanankes) because they are so” (24b18–20). Think of(2) as asyllogismos in which the “thingssupposed” are (2a) and (2b), and in which the thing that“results of necessity” is (2c):

(2a) No desire isvoluntary.
(2b) Some beliefs aredesires.
(2c) Some beliefs arenot voluntary.

On the interpretation we are describing, Aristotle's view is that tosay that (2c) results of necessity from (2a) and (2b) is to say that(2) is a particular case of the true universal generalization“For all suitable \(P\), \(Q\) and \(R\), if no \(Q\) is \(R\)and some \(P\)s are \(Q\)s, then some \(P\)s are not \(R\)”. Forthis interpretation see e.g. Alexander of Aphrodisias, 208.16 (quotedby Łukasiewicz 1957, §41), Bolzano (1837, §155) andŁukasiewicz (1957, §5).

In many other ancient and medieval logicians, “must”claims are understood as universal generalizations about actual items(even if they are not always understood as universal generalizationson “formal” schemata). Especially prominent is Diodorus'view that a proposition is necessary just in case it is true at alltimes (see Mates 1961, III, §3). Note that this makes sense ofthe idea that (2) must be true but, say, “People watch TV”could be false, for surely this sentence was not true in Diodorus'time. Diodorus' view appears to have been very common in the MiddleAges, when authors like William of Sherwood and Walter Burley seem tohave understood the perceived necessity of conditionals like (2) astruth at all times (see Knuuttila 1982, pp. 348–9). Anunderstanding of necessity as eternity is frequent also in laterauthors; see e.g., Kant,Critique of Pure Reason, B 184. Infavor of the mentioned interpretation of Aristotle and of theDiodorean view it might be pointed out that we often use modallocutions to stress the consequents of conditionals that follow frommere universal generalizations about the actual world, as in “Ifgas prices go up, necessarily the economy slows down”.

Many authors have thought that views of this sort do not account forthe full strength of the modal import of logical truths. A nowadaysvery common, but (apparently) late view in the history of philosophy,is that the necessity of a logical truth does not merely imply thatsome generalization about actual items holds, but also implies thatthe truth would have been true at a whole range of counterfactualcircumstances. Leibniz assigned this property to necessary truths suchas those of logic and geometry, and seems to have been one of thefirst to speak of the counterfactual circumstances as “possibleuniverses” or worlds (see the letter to Bourguet, pp.572–3, for a crisp statement of his views that contrasts themwith the views in the preceding paragraph; Knuuttila 1982, pp. 353 ff.detects the earliest transparent talk of counterfactual circumstancesand of necessity understood as at least implying truth in all of thesein Duns Scotus and Buridan; see also the entry onmedieval theories of modality). In contemporary writings the understanding of necessity as truth inall counterfactual circumstances, and the view that logical truths arenecessary in this sense, are widespread—although many, perhapsmost, authors adopt “reductivist” views of modality thatsee talk of counterfactual circumstances as no more than disguisedtalk about certain actualized (possibly abstract) items, such aslinguistic descriptions. Even Leibniz seems to have thought of his“possible universes” as ideas in the mind of God. (SeeLewis 1986 for an introduction to the contemporary polemics in thisarea.)

However, even after Leibniz and up to the present, many logicians seemto have avoided a commitment to a strong notion of necessity as truthin all (actual and) counterfactual circumstances. Thus Bolzano, inline with his interpretation of Aristotle mentioned above,characterizes necessary propositions as those whose negation isincompatible with purely general truths (see Bolzano 1837, §119).Frege says that “the apodictic judgment [i.e., roughly, thejudgment whose content begins with a ‘necessarily’governing the rest of the content] is distinguished from the assertoryin that it suggests the existence of universal judgments from whichthe proposition can be inferred, while in the case of the assertoryone such a suggestion is lacking” (Frege 1879, §4). Tarskiis even closer to the view traditionally attributed to Aristotle, forit is pretty clear that for him to say that e.g. (2c)“must” be true if (2a) and (2b) are true is to say that(2) is a particular case of the “formal” generalization“For all suitable \(P\), \(Q\) and \(R\), if no \(Q\) is \(R\)and some \(P\)s are \(Q\)s, then some \(P\)s are not \(R\)” (seeTarski 1936a, pp. 411, 415, 417, or the corresponding passages inTarski 1936b; see also Ray 1996). Quine is known for his explicitrejection of any modality that cannot be understood in terms ofuniversal generalizations about the actual world (see especially Quine1963). In some of these cases, this attitude is explained by adistrust of notions that are thought not to have reached a fullyrespectable scientific status, like the strong modal notions; it isfrequently accompanied in such authors, who are often practicinglogicians, by the proposal to characterize logical truth as a speciesof validity (in the sense of 2.3 below). (See Williamson 2013, ch. 3,and 2017, for a recent endorsement of the idea that logical truths areto be seen as instances of true universal generalizations over theirlogical forms, based on the scientific or “abductive”fruitfulness of the idea, though not on a disdain of modalnotions.)

On a recent view developed by Beall and Restall (2000, 2006), calledby them “logical pluralism”, the concept of logical truthcarries a commitment to the idea that a logical truth is true in allof a range of items or “cases”, and its necessity consistsin the truth of such a general claim (see Beall and Restall 2006, p.24). However, the concept of logical truth does not single out aunique range of “cases” as privileged in determining anextension for the concept, or, what amounts to the same thing, theuniversal generalization over a logical form can be interpreted viadifferent equally acceptable ranges of quantification; instead, thereare many such equally acceptable ranges and corresponding extensionsfor “logical truth”, which may be chosen as a function ofcontextual interests. This means that, for the logical pluralist, manysets have a right to be called “the set of logical truths”(and “the set of logical necessities”), each in theappropriate context.^[3] (See the entry onlogical pluralism.) On another recent understanding of logical necessity as a species ofgenerality, proposed by Rumfitt (2015), the necessity of a logicaltruth consists just in its being usable under all sets ofsubject-specific ways of drawing implications (provided these setssatisfy certain structural rules); or, more roughly, just in its beingapplicable no matter what sort of reasoning is at stake. On this view,a more substantive understanding of the modality at stake in logicaltruth is again not required. It may be noted that, although hepostulates a variety of subject-specific implication relations,Rumfitt rejects pluralism about logical truth in the sense of Bealland Restall (see his 2015, p. 56, n. 23.), and in fact thinks that theset of logical truths in a standard quantificational language ischaracterized by the standard classical logic.

Yet another sense in which it has been thought that truths like(1)-(3), and logical truths quite generally, “could” notbe false or “must” be true is epistemic. It is an oldobservation, going at least as far back as Plato, that some truthscount as intuitively known by us even in cases where we don't seem tohave any empirical grounds for them. Truths that are knowable onnon-empirical grounds are calleda priori (an expression thatbegins to be used with this meaning around the time of Leibniz; seee.g. his “Primæ Veritates”, p. 518). The axioms andtheorems of mathematics, the lexicographic and stipulativedefinitions, and also the paradigmatic logical truths, have been givenas examples. If it is accepted that logical truths areapriori, it is natural to think that they must be true or couldnot be false at least partly in the strong sense that their negationsare incompatible with what we are able to know non-empirically.

Assuming that sucha priori knowledge exists in some way orother, much recent philosophy has occupied itself with the issue ofhow it is possible. One traditional (“rationalist”) viewis that the mind is equipped with a special capacity to perceivetruths through the examination of the relations between pure ideas orconcepts, and that the truths reached through the correct operation ofthis capacity count as knowna priori. (See, e.g., Leibniz's“Discours de Métaphysique”, §§23 ff.;Russell 1912, p. 105; BonJour 1998 is a very recent example of a viewof this sort.) An opposing traditional (“empiricist”) viewis that there is no reason to postulate that capacity, or even thatthere are reasons not to postulate it, such as that it is“mysterious”. (See the entry onrationalism vs. empiricism.) Some philosophers, empiricists and otherwise, have attempted toexplaina priori knowledge as arising from some sort ofconvention or tacit agreement to assent to certain sentences (such as(1)) and use certain rules. Hobbes in his objections to Descartes'Meditations (“Third Objections”, IV, p. 608)proposes a wide-ranging conventionalist view. The later Wittgenstein(on one interpretation) and Carnap are distinguished proponents of“tacit agreement” and conventionalist views (see e.g.Wittgenstein 1978, I.9, I.142; Carnap 1939, §12, and 1963, p.916, for informal exposition of Carnap's views; see also Coffa 1991,chs. 14 and 17). Strictly speaking, Wittgenstein and Carnap think thatlogical truths do not express propositions at all, and are justvacuous sentences that for some reason or other we find useful tomanipulate; thus it is only in a somewhat diminished sense that we canspeak of (a priori) knowledge of them. However, in typicalrecent exponents of “tacit agreement” and conventionalistviews, such as Boghossian (1997), the claim that logical truths do notexpress propositions is rejected, and it is accepted that theexistence of the agreement provides full-blowna prioriknowledge of those propositions.

The “rational capacity” view and the“conventionalist” view agree that, in a broad sense, theepistemic ground of logical truths resides in our ability to analyzethe meanings of their expressions, be these understood as conventionsor as objective ideas. For this reason it can be said that theyexplain the apriority of logical truths in terms of their analyticity.(See the entry on theanalytic/synthetic distinction.) Kant's explanation of the apriority of logical truths has seemedharder to extricate.^[4] A long line of commentators of Kant has noted that, if Kant's view isthat all logical truths are analytic, this would seem to be in tensionwith his characterizations of analytic truths. Kant characterizesanalytic truths as those where the concept of the predicate iscontained in or identical with the concept of the subject, and, morefundamentally, as those whose denial is contradictory. But it hasappeared to those commentators that these characterizations, whileapplying to strict tautologies such as “Men are men” or“Bearded men are men”, would seem to leave out much ofwhat Kant himself counts as logically true, including syllogisms suchas (2) (see e.g. Mill 1843, bk. II, ch. vi, §5; Husserl 1901,vol. II, pt. 2, §66; Kneale and Kneale 1962, pp. 357–8;Parsons 1967; Maddy 1999). This and the apparent lack of clearpronouncements of Kant on the issue has led at least Maddy (1999) andHanna (2001) to consider (though not accept) the hypothesis that Kantviewed some logical truths as synthetica priori. On aninterpretation of this sort, the apriority of many logical truthswould be explained by the fact that they would be required by thecognitive structure of the transcendental subject, and specifically bythe forms of judgment.^[5] But the standard interpretation is to attribute to Kant the view thatall logical truths are analytic (see e.g. Capozzi and Roncaglia 2009).On an interpretation of this sort, Kant's forms of judgment may beidentified with logical concepts susceptible of analysis (see e.g.Allison 1983, pp. 126ff.). An extended defense of the interpretationthat Kant viewed all logical truths as analytic, including avindication of Kant against the objections of the line of commentatorsmentioned above, can be found in Hanna (2001), §3.1. Asubstantively Kantian contemporary theory of the epistemology of logicand its roots in cognition is developed in Hanna (2006); this theorydoes not seek to explain the apriority of logic in terms of itsanalyticity, and appeals instead to a specific kind of logicalintuition and a specific cognitive logic faculty. (Compare also theanti-aprioristic and anti-analytic but broadly Kantian view of Maddy2007, mentioned below.)

The early Wittgenstein shares with Kant the idea that the logicalexpressions do not express meanings in the way that non-logicalexpressions do (see 1921, 4.0312). Consistently with this view, heclaims that logical truths do not “say” anything (1921,6.11). But he seems to reject conventionalist and “tacitagreement” views (1921, 6.124, 6.1223). It is not that logicaltruths do not say anything because they are mere instruments for somesort of extrinsically useful manipulation; rather, they“show” the “logical properties” that the worldhas independently of our decisions (1921, 6.12, 6.13). It is unclearhow apriority is explainable in this framework. Wittgenstein calls thelogical truths analytic (1921, 6.11), and says that “one canrecognize in the symbol alone that they are true” (1921, 6.113).He seems to have in mind the fact that one can “see” thata logical truth of truth-functional logic must be valid by inspectionof a suitable representation of its truth-functional content (1921,6.1203, 6.122). But the extension of the idea to quantificationallogic is problematic, despite Wittgenstein's efforts to reducequantificational logic to truth-functional logic; as we now know,there is no algorithm for deciding if a quantificational sentence isvalid. What is perhaps more important, Wittgenstein gives nodiscernible explanation of why in principle all the “logicalproperties” of the world should be susceptible of beingreflected in an adequate notation.

Against the “rational capacity”,“conventionalist”, Kantian and early Wittgensteinianviews, other philosophers, especially radical empiricists andnaturalists (not to speak of epistemological skeptics), have rejectedthe claim thata priori knowledge exists (hence byimplication also the claim that analytic propositions exist), and theyhave proposed instead that there is only an illusion of apriority.Often this rejection has been accompanied by criticism of the otherviews. J.S. Mill thought that propositions like (2) seemapriori merely because they are particular cases of early and veryfamiliar generalizations that we derive from experience, like“For all suitable \(P\), \(Q\) and \(R\), if no \(Q\) is \(R\)and some \(P\)s are \(Q\)s, then some \(P\)s are not \(R\)” (seeMill 1843, bk. II, ch. viii). Bolzano held a similar view (see Bolzano1837, §315). Quine (1936, §III) famously criticized theHobbesian view noting that since the logical truths are infinite innumber, our ground for them must not lie just in a finite number ofexplicit conventions, for logical rules are presumably needed toderive an infinite number of logical truths from a finite number ofconventions (an argument derived from Carroll 1895; see Soames 2018,ch. 10, for exposition and Gómez-Torrente 2019 for criticism ofthe argument). Later Quine (especially 1954) criticized Carnap'sconventionalist view, largely on the grounds that there seems to be nonon-vague distinction between conventional truths and truths that aretacitly left open for refutation, and that to the extent that sometruths are the product of convention or “tacit agreement”,such agreement is characteristic of many scientific hypotheses andother postulations that seem paradigmatically non-analytic. (See Griceand Strawson 1956 and Carnap 1963 for reactions to these criticisms.)Quine (especially 1951) also argued that accepted sentences ingeneral, including paradigmatic logical truths, can be best seen assomething like hypotheses that are used to deal with experience, anyof which can be rejected if this helps make sense of the empiricalworld (see Putnam 1968 for a similar view and a purported example). Onthis view there cannot be strictlya priori grounds for anytruth. Three recent subtle anti-aprioristic positions are Maddy's(2002, 2007), Azzouni's (2006, 2008), and Sher's (2013). For Maddy,logical truths area posteriori, but they cannot bedisconfirmed merely by observation and experiment, since they formpart of very basic ways of thinking of ours, deeply embedded in ourconceptual machinery (a conceptual machinery that is structurallysimilar to Kant's postulated transcendental organization of theunderstanding). Similarly, for Azzouni logical truths are equallya posteriori, though our sense that they must be true comesfrom their being psychologically deeply ingrained; unlike Maddy,however, Azzouni thinks that the logical rules by which we reason areopaque to introspection. Sher provides an attempt at combining aQuinean epistemology of logic with a commitment to a metaphysicallyrealist view of the modal ground of logical truth.

One way in whicha priori knowledge of a logical truth suchas (1) would be possible would be ifa priori knowledge ofthe fact that (1) is a logical truth, or of the universalgeneralization “For all suitable \(a\), \(P\), \(b\) and \(Q\),if \(a\) is \(P\) only if \(b\) is \(Q\), and \(a\) is \(P\), then\(b\) is \(Q\)” were possible. One especially noteworthy kind ofskeptical consideration in the epistemology of logic is that thepossibility of inferentiala priori knowledge of these factsseems to face a problem of circularity or of infinite regress. If weare to obtain inferentiala priori knowledge of those facts,then we will presumably follow logical rules at some point, includingpossibly the rule ofmodus ponens whose very correctnessmight well depend in part on the fact that (1) is a logical truth oron the truth of the universal generalization “For all suitable\(a\), \(P\), \(b\) and \(Q\), if \(a\) is \(P\) only if \(b\) is\(Q\), and \(a\) is \(P\), then \(b\) is \(Q\)”. In any case, itseems clear that not all claims of this latter kind, expressing that acertain truth is a logical truth or that a certain logical schema istruth-preserving, could be given ana priori inferentialjustification without the use of some of the same logical rules whosecorrectness they might be thought to codify. The point can again bereasonably derived from Carroll (1895). Some of the recent literatureon this consideration, and on anti-skeptical rejoinders, includesDummett (1973, 1991) and Boghossian (2000).

1.2 Formality

On most views, even if it were true that logical truths are true inall counterfactual circumstances,a priori, and analytic,this would not give sufficient conditions for a truth to be a logicaltruth. On most views, a logical truth also has to be in some sense“formal”, and this implies at least that all truths thatare replacement instances of its form are logical truths too (andhence, on the assumption of the preceding sentence, true in allcounterfactual circumstances,a priori, and analytic). To usea slight modification of an example of Albert of Saxony (quoted byBocheński 1956, §30.07), “If a widow runs, then afemale runs” should be true in all counterfactual circumstances,a priori, and analytic if any truth is. However, “If awidow runs, then a log runs” is a replacement instance of itsform, and in fact it even has the same form on any view of logicalform (something like “If a \(P\) \(Q\)s, then an \(R\)\(Q\)s”), but it is not even truesimpliciter. So onmost views, “If a widow runs, then a female runs” is not alogical truth.

For philosophers who accept the idea of formality, as we said above,the logical form of a sentence is a certain schema in which theexpressions that are not schematic letters are widely applicableacross different areas of discourse.^[6] If the schema is the form of a logical truth, all of its replacementinstances are logical truths. The idea that logic is especiallyconcerned with (replacement instances of) schemata is of courseevident beginning with Aristotle and the Stoics, in all of whom theword usually translated by “figure” is preciselyschema. In Aristotle a figure is actually an even moreabstract form of a group of what we would now call“schemata”, such as (2′). Our schemata are closer towhat in the Aristotelian syllogistic are the moods; but there seems tobe no word for “mood” in Aristotle (except possiblyptoseon in 42b30 ortropon in 43a10; see Smith 1989,pp. 148–9), and thus no general reflection on the notion offormal schemata. There is explicit reflection on the contrast betweenthe formal schemata or moods and the matter (hyle) ofsyllogismoi in Alexander of Aphrodisias (53.28ff., quoted byBocheński 1956, §24.06), and there has been ever since. Thematter are the values of the schematic letters.

The idea that the non-schematic expressions in logical forms, i.e. thelogical expressions, are widely applicable across different areas ofdiscourse is also present from the beginning of logic, and recurs inall the great logicians. It appears indirectly in many passages fromAristotle, such as the following: “All the sciences are relatedthrough the common things (I call common those which they use in orderto demonstrate from them, but not those that are demonstrated in themor those about which something is demonstrated); and logic is relatedto them all, as it is a science that attempts to demonstrateuniversally the common things” (Posterior Analytics,77a26–9); “we don't need to take hold of the things of allrefutations, but only of those that are characteristic of logic; forthese are common to every technique and ability”(Sophistical Refutations, 170a34–5). (In these texts“logic” is an appropriate translation ofdialektike; see Kneale and Kneale 1962, I, §3, whoinform us thatlogike is used for the first time with itscurrent meaning in Alexander of Aphrodisias.) Frege says that“the firmest proof is obviously the purely logical, which,prescinding from the particularity of things, is based solely on thelaws on which all knowledge rests” (1879, p. 48; see also 1885,where the universal applicability of the arithmetical concepts istaken as a sign of their logicality). The same idea is conspicuous aswell in Tarski (1941, ch. II, §6).

That logical expressions include paradigmatic cases like“if”, “and”, “some”,“all”, etc., and that they must be widely applicableacross different areas of discourse is what we might call “theminimal thesis” about logical expressions. But beyond this thereis little if any agreement about what generic feature makes anexpression logical, and hence about what determines the logical formof a sentence. Most authors sympathetic to the idea that logic isformal have tried to go beyond the minimal thesis. It would begenerally agreed that being widely applicable across different areasof discourse is only a necessary, not sufficient property of logicalexpressions; for example, presumably most prepositions are widelyapplicable, but they are not logical expressions on any implicitgeneric notion of a logical expression. Attempts to enrich the notionof a logical expression have typically sought to provide furtherproperties that collectively amount to necessary and sufficientconditions for an expression to be logical.

One idea that has been used in such characterizations, and that isalso present in Aristotle, is that logical expressions do not,strictly speaking, signify anything; or, that they do not signifyanything in the way that substantives, adjectives and verbs signifysomething. “Logic [dialektike] is not a science ofdetermined things, or of any one genus” (PosteriorAnalytics, 77a32–3). We saw that the idea was still presentin Kant and the early Wittgenstein. It reemerged in the Middle Ages.The main sense of the word “syncategorematic” as appliedto expressions was roughly this semantic sense (see Kretzmann 1982,pp. 212 ff.). Buridan and other late medieval logicians proposed thatcategorematic expressions constitute the “matter” ofsentences while the syncategorematic expressions constitute their“form” (see the text quoted by Bocheński 1956,§26.11). (In a somewhat different, earlier, grammatical sense ofthe word, syncategorematic expressions were said to be those thatcannot be used as subjects or predicates in categorical propositions;see Kretzmann 1982, pp. 211–2.) The idea of syncategorematicityis somewhat imprecise, but there are serious doubts that it can serveto characterize the idea of a logical expression, whatever this maybe. Most prepositions and adverbs are presumably syncategorematic, butthey are also presumably non-logical expressions. Conversely,predicates such as “are identical”, “is identicalwith itself”, “is both identical and not identical withitself”, etc., which are resolutely treated as logical in recentlogic, are presumably categorematic. (They are of course categorematicin the grammatical sense, in which prepositions and adverbs areequally clearly syncategorematic.)

Most other proposals have tried to delineate in some other way theAristotelian idea that the logical expressions have some kind of“insubstantial” meaning, so as to use it as a necessaryand sufficient condition for logicality. One recent suggestion is thatlogical expressions are those that do not allow us to distinguishdifferent individuals. One way in which this has been made precise isthrough the characterization of logical expressions as those whoseextension or denotation over any particular domain of individuals isinvariant under permutations of that domain. (See Tarski and Givant1987, p. 57, and Tarski 1966; for related proposals see also McCarthy1981, Sher 1991, ch. 3, McGee 1996, Feferman 1999, Bonnay 2008, Woods2016 and Griffiths and Paseau 2022, among others.) A permutation of adomain is a one-to-one correspondence between the domain and itself.For example, if \(D\) is the domain {Aristotle, Caesar, Napoleon,Kripke}, one permutation is the correspondence that assigns each manto himself; another is the correspondence \(P\) that assigns Caesar toAristotle (in mathematical notation,\(P(\text{Aristotle})=\text{Caesar}\)), Napoleon to Caesar, Kripke toNapoleon, and Aristotle to Kripke. That the extension of an expressionover a domain is invariant under a permutation of that domain meansthat the induced image of that extension under the permutation is theextension itself (the “induced image” of an extensionunder a permutation \(Q\) is what the extension becomes when in placeof each object \(o\) one puts the object \(Q(o)\)). The extension of“philosopher” over \(D\) is not invariant under thepermutation \(P\) above, for that extension is \(\{\text{Aristotle},\text{Kripke}\}\), whose induced image under \(P\) is\(\{\text{Caesar}, \text{Aristotle}\}\). This is favorable to theproposal, for “philosopher” is certainly not widelyapplicable, and so non-logical on most views. On the other hand, thepredicate “are identical” has as its extension over \(D\)the set of pairs

\(\{ \langle \text{Aristotle, Aristotle} \rangle, \langle\text{Caesar, Caesar} \rangle, \langle \text{Napoleon, Napoleon}\rangle, \langle \text{Kripke, Kripke} \rangle\};\)

its induced image under \(P\), and under any other permutation of\(D\), is that very same set of pairs (as the reader may check); soagain this is favorable to the proposal. (Other paradigmatic logicalexpressions receive more complicated extensions over domains, but theextensions they receive are invariant under permutations. For example,on one usual way of understanding the extension of “and”over a domain, this is the function that assigns, to each pair\(\langle S_1, S_2 \rangle\), where \(S_1\) and \(S_2\) are sets ofinfinite sequences of objects drawn from \(D\), the intersection of\(S_1\) and \(S_2\); and this function is permutation invariant.) Oneproblem with the proposal is that many expressions that seem clearlynon-logical, because they are not widely applicable, are neverthelessinvariant under permutations, and thus unable to distinguish differentindividuals. The simplest examples are perhaps non-logical predicatesthat have an empty extension over any domain, and hence have emptyinduced images as well. “Male widow” is one example;versions of it can be used as counterexamples to the differentversions of the idea of logicality as permutation invariance (seeGómez-Torrente 2002), and it's unclear that the proponent ofthe idea can avoid the problem in any non ad hoc way.

Another popular recent way of delineating the Aristotelian intuitionof the semantic “insubstantiality” of logical expressionsappeals to the concept of “pure inferentiality”. The ideais that logical expressions are those whose meaning, in some sense, isgiven by “purely inferential” rules. (See Kneale 1956,Hacking 1979, Peacocke 1987, Hodes 2004, among others.) A necessaryproperty of purely inferential rules is that they regulate onlyinferential transitions between verbal items, not between extra-verbalassertibility conditions and verbal items, or between verbal items andactions licensed by those items. A certain inferential rule licensesyou to say “It rains” when it rains, but it's not“purely inferential”. A rule that licenses you to say“A is a female whose husband died before her” when someonesays “A is a widow”, however, is not immediatelydisqualified as purely inferential. Now, presumably in some sense themeaning of “widow” is given by this last rule togetherperhaps with the converse rule, that licenses you to say “A is awidow” when someone says “A is a female whose husband diedbefore her”. But “widow” is not a logicalexpression, since it's not widely applicable; so one needs topostulate more necessary properties that “purelyinferential” rules ought to satisfy. A number of such conditionsare postulated in the relevant literature (see e.g. Belnap 1962 (areply to Prior 1960), Hacking 1979 and Hodes 2004). However, even whenthe notion of pure inferentiality is strengthened in these ways,problems remain. Most often the proposal is that an expression islogical just in case certain purely inferential rules give its wholemeaning, including its sense, or the set of aspects of its use thatneed to be mastered in order to understand it (as in Kneale 1956,Peacocke 1987 and Hodes 2004). However, it seems clear that someparadigmatic logical expressions have extra sense attached to themthat is not codifiable purely inferentially. For example, inductivereasoning involving “all” seems to be part of the sense ofthis expression, but it's hard to see how it could be codified bypurely inferential rules (as noted by Sainsbury 1991, pp. 316–7;see also Dummett 1991, ch. 12). A different version of the proposalconsists in saying that an expression is logical just in case certainpurely inferential rules that are part of its sense suffice todetermine its extension (as in Hacking 1979). But it seems clear thatif the extension of, say, “are identical” is determined bya certain set of purely inferential rules that are part of its sense,then the extension of “are identical and are not malewidows” is equally determined by the same rules, which arguablyform part of its sense; yet “are identical and are not malewidows” is not a logical expression (see Gómez-Torrente2002).

In view of problems of these and other sorts, some philosophers haveproposed that the concept of a logical expression is not associatedwith necessary and sufficient conditions, but only with some necessarycondition related to the condition of wide applicability, such as thecondition of “being very relevant for the systematization ofscientific reasoning” (see Warmbrōd 1999 for a position ofthis type). Others (Gómez-Torrente 2021) have proposed thatthere may be a set of necessary and sufficient conditions, if theseare not much related to the idea of semantic“insubstantiality” and are instead pragmatic and suitablyvague; for example, many expressions are excluded directly by thecondition of wide applicability; and prepositions are presumablyexcluded by some such implicit condition as “a logicalexpression must be one whose study is useful for the resolution ofsignificant problems and fallacies in reasoning”. To be sure,these proposals give up on the extended intuition of semantic“insubstantiality”, and may be somewhat unsatisfactory forthat reason.

Some philosophers have reacted even more radically to the problems ofusual characterizations, claiming that the distinction between logicaland non-logical expressions must be vacuous, and thus rejecting thenotion of logical form altogether. (See e.g. Orayen 1989, ch. 4,§2.2; Etchemendy 1990, ch. 9; Read 1994; Priest 2001.) Thesephilosophers typically think of logical truth as a notion roughlyequivalent to that of analytic truthsimpliciter. But theyare even more liable to the charge of giving up on extended intuitionsthan the proposals of the previous paragraph.

For more thorough treatments of the ideas of formality and of alogical expression see the entrylogical constants, and MacFarlane 2000.

2. The Mathematical Characterization of Logical Truth

2.1 Formalization

One important reason for the successes of modern logic is its use ofwhat has been called “formalization”. This term is usuallyemployed to cover several distinct (though related) phenomena, all ofthem present in Frege (1879). One of these is the use of a completelyspecified set of artificial symbols to which the logician assignsunambiguous meanings, related to the meanings of corresponding naturallanguage expressions, but much more clearly delimited and strippedfrom the notes that in those natural language expressions seem (to thelogician) irrelevant to truth-conditional content; this is especiallytrue of symbols meant to represent the logical expressions of naturallanguage. Another phenomenon is the stipulation of a completelyprecise grammar for the formulae construed out of the artificialsymbols, formulae that will be “stripped” versions ofcorrelate sentences in natural language; this grammar amounts to analgorithm for producing formulae starting from the basic symbols. Athird phenomenon is the postulation of a deductive calculus with avery clear specification of axioms and rules of inference for theartificial formulae (see the next section); such a calculus isintended to represent in some way deductive reasoning with thecorrelates of the formulae, but unlike ordinary deductions,derivations in the calculus contain no steps that are not definiteapplications of the specified rules of inference.

Instead of attempting to characterize the logical truths of a naturallanguage like English, the Fregean logician attempts to characterizethe artificial formulae that are “stripped” correlates ofthose logical truths in a Fregean formalized language. In first-orderFregean formalized languages, among these formulae one findsartificial correlates of (1), (2) and (3), things like

\(((\text{Bad}(\textit{death}) \rightarrow \text{Good}(\textit{life}))\ \& \ \text{Bad}(\textit{death})) \rightarrow\text{Good}(\textit{life}).\)

\((\forallx(\text{Desire}(x) \rightarrow \neg \text{Voluntary}(x)) \ \&\\exists x(\text{Belief}(x) \ \&\ \text{Desire}(x)))\)
\(\rightarrow \exists x(\text{Belief}(x) \ \&\ \neg\text{Voluntary}(x)).\)

\((\text{Cat}(\textit{drasha}) \ \&\ \forall x(\text{Cat}(x)\rightarrow \text{Mysterious}(x)))\)
\(\rightarrow \text{Mysterious}(\textit{drasha}).\)

(See the entry onlogic, classical.) Fregean formalized languages include also classical higher-orderlanguages. (See the entry onlogic, second-order and higher-order.) The logical expressions in these languages are standardly taken to bethe symbols for the truth-functions, the quantifiers, identity andother symbols definable in terms of those (but there are dissentingviews on the status of the higher-order quantifiers; see 2.4.3below).

The restriction to artificial formulae raises a number of questionsabout the exact value of the Fregean enterprise for the demarcation oflogical truths in natural language; much of this value depends on howmany and how important are perceived to be the notes stripped from thenatural language expressions that are correlates of the standardlogical expressions of formalized languages. But whatever one's viewof the exact value of formalization, there is little doubt that it hasbeen very illuminating for logical purposes. One reason is that it'ssometimes relatively clear that the stripped notes are irrelevant totruth-conditional content (this is especially true of the use ofnatural language logical expressions for doing mathematics). Anotherof the reasons is that the fact that the grammar and meaning of theartificial formulae is so well delimited has permitted the developmentof proposed characterizations of logical truth that use only conceptsof standard mathematics. This in turn has allowed the study of thecharacterized notions by means of standard mathematical techniques.The next two sections describe the two main approaches tocharacterization in broad outline.^[7]

2.2 Derivability

We just noted that the Fregean logician's formalized grammar amountsto an algorithm for producing formulae from the basic artificialsymbols. This is meant very literally. As was clear to mathematicallogicians from very early on, the basic symbols can be seen as (orcodified by) natural numbers, and the formation rules in theartificial grammar can be seen as (or codified by) simple computablearithmetical operations. The grammatical formulae can then be seen as(or codified by) the numbers obtainable from the basic numbers aftersome finite series of applications of the operations, and thus theirset is characterizable in terms of concepts of arithmetic and settheory (in fact arithmetic suffices, with the help of sometricks).

Exactly the same is true of the set of formulae that are derivable ina formalized deductive calculus. A formula \(F\) is derivable in aFregean calculus \(C\) just in case \(F\) is obtainable from theaxioms of \(C\) after some finite series of applications of the rulesof inference of \(C\). But the axioms are certain formulae built bythe process of grammatical formation, so they can be seen as (orcodified by) certain numbers; and the rules of inference can again beseen as (or codified by) certain computable arithmetical operations.So the derivable formulae can be seen as (or codified by) the numbersobtainable from the axiom numbers after some finite series ofapplications of the inference operations, and thus their set is againcharacterizable in terms of concepts of standard mathematics (againarithmetic suffices).

In the time following Frege's revolution, there appears to have been awidespread belief that the set of logical truths of any Fregeanlanguage could be characterized as the set of formulae derivable insome suitably chosen calculus (hence, essentially, as the set ofnumbers obtainable by certain arithmetical operations). Frege himselfsays, speaking of the higher-order language in his“Begriffsschrift”, that through formalization (in thethird sense above) “we arrive at a small number of laws inwhich, if we add those contained in the rules, the content of all thelaws is included, albeit in an undeveloped state” (Frege 1879,§13). The idea follows straightforwardly from Russell'sconception of mathematics and logic as identical (see Russell 1903,ch. I, §10; Russell 1920, pp. 194–5) and his thesis that“by the help of ten principles of deduction and ten otherpremises of a general logical nature (…), all mathematics canbe strictly and formally deduced” (Russell 1903, ch. I,§4). See also Bernays (1930, p. 239): “[throughformalization] it becomes evident that all logical inference(…) can be reduced to a limited number of logical elementaryprocesses that can be exactly and completely enumerated”. In theopening paragraphs of his paper on logical consequence, Tarski (1936a,1936b) says that the belief was prevalent before the appearance ofGödel's incompleteness theorems (see subsection 2.4.3 below forthe bearing of these theorems on this issue). In recent times,apparently due to the influence of Tarskian arguments such as the onementioned towards the end of subsection 2.4.3, the belief in theadequacy of derivability characterizations seems to have waned (seee.g. Prawitz 1985 for a similar appraisal).

2.3 Model-theoretic Validity

Even on the most cautious way of understanding the modality present inlogical truths, a sentence is a logical truth only if no sentencewhich is a replacement instance of its logical form is false. (Thisidea is only rejected by those who reject the notion of logical form.)It is a common observation that this property, even if it isnecessary, is not clearly sufficient for a sentence to be a logicaltruth. Perhaps there is a sentence that has this property but is notreally logically true, because one could assign someunexpressedmeanings to the variables and the schematic letters in itslogical form, and under those meanings the form would be a false sentence.^[8] On the other hand, it is not clearly incorrect to think that asentence is a logical truth if no collective assignment of meanings tothe variables and the schematic letters in its logical form would turnthis form into a false sentence. Say that a sentence isuniversally valid when it has this property. A standardapproach to the mathematical characterization of logical truth,alternative to the derivability approach, uses always some version ofthe property of universal validity, proposing it in each case as bothnecessary and sufficient for logical truth. Note that if a sentence isuniversally valid then, even if it's not logically true, it will betrue. So all universally valid sentences are correct at leastin this sense.

Apparently, the first to use a version of universal validity andexplicitly propose it as both necessary and sufficient for logicaltruth was Bolzano (see Bolzano 1837, §148; and Coffa 1991, pp.33–4 for the claim of priority). The idea is also present inother mathematicians of the nineteenth century (see e.g. Jané2006), and was common in Hilbert's school. Tarski (1936a, 1936b) wasthe first to indicate in a fully explicit way how the version ofuniversal validity used by the mathematicians could itself be given acharacterization in terms of concepts of standard mathematics, in thecase of Fregean formalized languages with an algorithmic grammar.Essentially Tarski's characterization is widely used today in the formof what is known as themodel-theoretic notion of validity,and it seems fair to say that it is usually accepted that this notiongives a reasonably good delineation of the set of logical truths forFregean languages.

The notion of model-theoretic validity mimics the notion of universalvalidity, but is defined just with the help of the set-theoreticapparatus developed by Tarski (1935) for the characterization ofsemantic concepts such as satisfaction, definability, and truth. (Seethe entry onTarski's truth definitions.) Given a Fregean language, astructure for the language is aset-theoretical object composed of a set-domain taken together with anassignment of extensions drawn from that domain to its non-logicalconstants. A structure is meant by most logicians to represent anassignment of meanings: its domain gives the range or“meaning” of the first-order variables (and induces rangesof the higher-order variables), and the extensions that the structureassigns to the non-logical constants are “meanings” thatthese expressions could take. Using the Tarskian apparatus, onedefines for the formulae of the Fregean language the notion of truthin (or satisfaction by) a set-theoretic structure (with respect to aninfinite sequence assigning an object of the domain to each variable).And finally, one defines a formula to be model-theoretically validjust in case it is true in all structures for its language (withrespect to all infinite sequences). Let's abbreviate “\(F\) istrue in all structures” as “MTValid\((F)\)”. Themodel-theoretic characterization makes it clear that“MTValid\((F)\)” is definable purely in terms of conceptsof set theory. (The notion of model-theoretic validity for Fregeanlanguages is explained in thorough detail in the entries onclassical logic andsecond-order and higher-order logic; see also the entry onmodel theory.)^[9]

(If \(F\) is a formula of a first-order language without identity,then if no replacement instance of the form of \(F\) is false, this isa sufficient condition for \(F\)'s being model-theoretically valid. Asit turns out, if \(F\) is not model-theoretically valid, then somereplacement instance of its form whose variables range over thenatural numbers and whose non-logical constants are arithmeticalexpressions will be false. This can be justified by means of arefinement of the Löwenheim-Skolem theorem. See the entry onlogic, classical, and Quine 1970, ch. 4, for discussion and references. No similarresults hold for higher-order languages.)

The “MT” in “MTValid\((F)\)” stresses the factthat model-theoretic validity is different from universal validity.The notion of a meaning assignment which appears in the description ofuniversal validity is a very imprecise and intuitive notion, while thenotion of a structure appearing in a characterization ofmodel-theoretic validity is a fairly precise and technical one. Itseems clear that the notion of a structure for Fregean formalizedlanguages is minimally reasonable, in the sense that a structuremodels the power of one or several meaning assignments to make false(the logical form of) some sentence. As we will mention later, theconverse property, that each meaning assignment's validity-refutingpower is modeled by some structure, is also a natural but moredemanding requirement on a notion of structure.

2.4 The Problem of Adequacy

The fact that the notions of derivability and model-theoretic validityare definable in standard mathematics seems to have been a veryattractive feature of them among practicing logicians. But thisattractive feature of course does not justify by itself taking eithernotion as an adequate characterization of logical truth. On mostviews, with a mathematical characterization of logical truth weattempt to delineate a set of formulae possessing a number ofnon-mathematical properties. Which properties these are variesdepending on our pretheoretic conception of, for example, the featuresof modality and formality. (By “pretheoretic” it's notmeant “previous to any theoretical activity”; there couldhardly be a “pretheoretic” conception of logical truth inthis sense. In this context what's meant is “previous to thetheoretical activity of mathematical characterization”.) But onany such conception there will be external, non-mathematical criteriathat can be applied to evaluate the question whether a mathematicalcharacterization is adequate. In this last section we will outlinesome of the basic issues and results on the question whetherderivability and model-theoretic validity are adequate in thissense.

2.4.1 Analysis and Modality

One frequent objection to the adequacy of model-theoretic validity isthat it does not provide a conceptual analysis of the notion oflogical truth, even for sentences of Fregean formalized languages (seee.g. Pap 1958, p. 159; Kneale and Kneale 1962, p. 642; Field 1989, pp.33–4; Etchemendy 1990, ch. 7). This complaint is especiallycommon among authors who feel inclined to identify logical truth andanalyticitysimpliciter (see e.g. Kneale and Kneale,ibid., Etchemendy 1990, p. 126). If one thinks of the conceptof logical truth simply as the concept of analytic truth, it isespecially reasonable to accept that the concept of logical truth doesnot have much to do with the concept of model-theoretic validity, forpresumably this concept does not have much to do with the concept ofanalyticity. To say that a formula is model-theoretically valid meansthat there are no set-theoretic structures in which it is false;hence, to say that a formula is not model-theoretically valid meansthat there are set-theoretic structures in which it is false. But tosay that a sentence is or is not analytic presumably does not meananything about the existence or non-existence of set-theoreticstructures. Note that we could object to derivability on the samegrounds, for to say that a sentence is or is not analytic presumablydoes not mean anything about its being or not being the product of acertain algorithm (compare Etchemendy 1990, p. 3). (One furtherpeculiar, much debated claim in Etchemendy 1990 is that true claims ofthe form “\(F\) is logically true” or “\(F\) is notlogically true” should themselves be logical truths (while thecorresponding claims “MTValid\((F)\)” and “NotMTValid\((F)\)” are not logical truths). Etchemendy's claim isperhaps defensible under a conception of logical truth as analyticitysimpliciter, but certainly doubtful on more traditionalconceptions of logical truth, on which the predicate “is alogical truth” is not even a logical expression. SeeGómez-Torrente 1998/9 and Soames 1999, ch. 4 fordiscussion.)

Analogous “no conceptual analysis” objections can be madeif we accept that the concept of logical truth has some other strongmodal notes unrelated to analyticity; for example, if we accept thatit is part of the concept of logical truth that logical truths aretrue in all counterfactual circumstances, or necessary in some otherstrong sense. Sher (1996) accepts something like the requirement thata good characterization of logical truth should be given in terms of amodally rich concept. However, she argues that the notion ofmodel-theoretic validity is strongly modal, and so the “noconceptual analysis” objection is actually wrong: to say that aformula is or is not model-theoretically valid is to make amathematical existence or non-existence claim, and according to Sherthese claims are best read as claims about the possibility andnecessity of structures. (Shalkowski 2004 argues that Sher's defenseof model-theoretic validity is insufficient, on the basis of a certainmetaphysical conception of logical necessity. Etchemendy 2008relatedly argues that Sher's defense is based on inadequaterestrictions on the modality relevant to logical truth. See also thecritical discussion of Sher in Hanson 1997.) García-Carpintero(1993) offers a view related to Sher's: model-theoretic validityprovides a (correct) conceptual analysis of logical truth for Fregeanlanguages, because the notion of a set-theoretical structure is infact a subtle refinement of the modal notion of a possible meaningassignment. Azzouni (2006), ch. 9, also defends the view thatmodel-theoretic validity provides a correct conceptual analysis oflogical truth (though restricted to first-order languages), on thebasis of a certain deflationist conception of the (strong) modalityinvolved in logical truth.

The standard view of set-theoretic claims, however, does not see themas strong modal claims—at best, some of them are modal in theminimal sense that they are universal generalizations or particularcases of these. But it is at any rate unclear that this is the basisfor a powerful objection to model-theoretic validity or toderivability, for, even if we accept that the concept of logical truthis strongly modal, it is unclear that a good characterization oflogical truth ought to be a conceptual analysis. An analogy mighthelp. It is widely agreed that the characterizations of the notion ofcomputability in standard mathematics, e.g. as recursiveness, are insome sense good characterizations. Note that the concept ofcomputability is modal, in a moderately strong sense; itseems to be about what beings like us could do with certain symbols ifthey were free from certain limitations—not about, say, whatexisting beings have done or will do. However, to say that a certainfunction is recursive is not to make a modal claim about it, but acertain purely arithmetical claim. So recursiveness is widely agreedto provide a good characterization of computability, but it clearlydoes not provide a conceptual analysis. Perhaps it could be arguedthat the situation with model-theoretic validity, or derivability, orboth, is the same.

A number of philosophers explicitly reject the requirement that a goodcharacterization of logical truth should provide a conceptualanalysis, and (at least for the sake of argument) do not question theusual view of set-theoretic claims as non-modal, but have argued thatthe universe of set-theoretic structures somehow models the universeof possible structures (or at least the universe of possibleset-theoretic structures; see McGee 1992, Shapiro 1998, Sagi 2014). Inthis indirect sense, the characterization in terms of model-theoreticvalidity would grasp part of the strong modal force that logicaltruths are often perceived to possess. McGee (1992) gives an elegantargument for this idea: it is reasonable to think that given anyset-theoretic structure, even one construed out of non-mathematicalindividuals, actualized or not, there is a set-theoretic structureisomorphic to it but construed exclusively out of pure sets; but anysuch pure set-theoretic structure is, on the usual view, an actualizedexistent; so every possible set-theoretic structure is modeled by aset-theoretic structure, as desired. (The significance of this relieson the fact that in Fregean languages a formula is true in a structureif and only if it is true in all the structures isomorphic to it.)

But model-theoretic validity (or derivability) might be theoreticallyadequate in some way even if some possible meaning-assignments are notmodeled straightforwardly by (actual) set-theoretic structures. Formodel-theoretic validity to be theoretically adequate, it might beheld, it is enough if we have other reasons to think that it isextensionally adequate, i.e. that it coincides in extension with ourpreferred pretheoretic notion of logical truth. In subsections 2.4.2and 2.4.3 we will examine some existing arguments for and against theplain extensional adequacy of derivability and model-theoreticvalidity for Fregean languages.

2.4.2 Extensional Adequacy: A General Argument

If one builds one's deductive calculus with care, one will be able toconvince oneself that all the formulae derivable in the calculus arelogical truths. The reason is that one can have used one's intuitionvery systematically to obtain that conviction: one can have includedin one's calculus only axioms of which one is convinced that they arelogical truths; and one can have included as rules of inference rulesof which one is convinced that they produce logical truths whenapplied to logical truths. Using another terminology, this means that,if one builds one's calculus with care, one will be convinced that thederivability characterization of logical truth for formulae of theformalized language will besound with respect to logicaltruth.

It is equally obvious that if one has at hand a notion ofmodel-theoretic validity for a formalized language which is based on aminimally reasonable notion of structure, then all logical truths (ofthat language) will be model-theoretically valid. The reason issimple: if a formula is not model-theoretically valid then there is astructure in which it is false; but this structure must then model ameaning assignment (or assignments) on which the formula (or itslogical form) is false; so it will be possible to construct a formulawith the same logical form, whose non-logical expressions have, bystipulation, the particular meanings drawn from that collectivemeaning assignment, and which is therefore false. But then the idea offormality and the weakest conception of the modal force of logicaltruths uncontroversially imply that the original formula is notlogically true. Using another terminology, we can conclude thatmodel-theoretic validity iscomplete with respect to logicaltruth.

Let's abbreviate “\(F\) is derivable in calculus \(C\)” by“DC\((F)\)” and “\(F\) is a logical truth (in ourpreferred pretheoretical sense)” by “LT\((F)\)”.Then, if \(C\) is a calculus built to suit our pretheoretic conceptionof logical truth, the situation can be summarized thus:

(4) \(\text{DC}(F)\Rightarrow \text{LT}(F) \Rightarrow \text{MTValid}(F).\)

The first implication is the soundness of derivability; the second isthe completeness of model-theoretic validity.

In order to convince ourselves that the characterizations of logicaltruth in terms of DC\((F)\) and MTValid\((F)\) are extensionallyadequate we should convince ourselves that the converse implicationshold too:

(5)\(\text{MTValid}(F) \Rightarrow \text{LT}(F) \Rightarrow\text{DC}(F).\)

Obtaining this conviction, or the conviction that these implicationsdon't in fact hold, turns out to be difficult in general. But a remarkof Kreisel (1967) establishes that a conviction that they hold can beobtained sometimes. In some cases it is possible to give amathematical proof that derivability (in some specified calculus\(C\)) is complete with respect to model-theoretic validity, i.e. aproof of

(6)\(\text{MTValid}(F) \Rightarrow \text{DC}(F).\)

Kreisel called attention to the fact that (6) together with (4)implies that model-theoretic validity is sound with respect to logicaltruth, i.e., that the first implication of (5) holds. (Strictlyspeaking, this is a strong generalization of Kreisel's remark, whichin place of “\(\text{LT}(F)\)” had something like“\(F\) is true in allclass structures”(structures with a class, possibly proper, as domain of the individualvariables).) This means that when (6) holds the notion ofmodel-theoretic validity offers an extensionally correctcharacterization of logical truth. (See Etchemendy 1990, ch. 11,Hanson 1997, Gómez-Torrente 1998/9, and Field 2008, ch. 2, forversions of this observation, and Smith 2011 and Griffiths 2014 forobjections.) Also, (6), together with (4), implies that the notion ofderivability is complete with respect to logical truth (the secondimplication in (5)) and hence offers an extensionally correctcharacterization of this notion. Note that this reasoning is verygeneral and independent of what our particular pretheoretic conceptionof logical truth is.

An especially significant case in which this reasoning can be appliedis the case of standard Fregean first-order quantificationallanguages, under a wide array of pretheoretic conceptions of logicaltruth. It is typical to accept that all formulae derivable in atypical first-order calculus are universally valid, true in allcounterfactual circumstances,a priori and analytic if anyformula is.^[10] So (4) holds under a wide array of pretheoretic conceptions in thiscase. (6) holds too for the typical calculi in question, in virtue ofGödel's completeness theorem, so (5) holds. This means that onecan convince oneself that both derivability and model-theoreticvalidity are extensionally correct characterizations of our favoritepretheoretic notion of logical truth for first-order languages, if ourpretheoretic conception is not too eccentric. The situation is not soclear in other languages of special importance for the Fregeantradition, the higher-order quantificational languages.

2.4.3 Extensional Adequacy: Higher-order Languages

It follows from Gödel's first incompleteness theorem that alreadyfor a second-order language there is no calculus \(C\) wherederivability is sound with respect to model-theoretic validity andwhich makes true (6) (for the notion of model-theoretic validity asusually defined for such a language). We may call this resulttheincompleteness of second-order calculi with respect to model-theoreticvalidity. Said another way: for every second-order calculus \(C\)sound with respect to model-theoretic validity there will be a formula\(F\) such that \(\text{MTValid}(F)\) but it is not the case that\(\text{DC}(F)\).

In this situation it's not possible to apply Kreisel's argument for(5). In fact, the incompleteness of second-order calculi shows that,given any calculus \(C\) satisfying (4), one of the implications of(5) is false (orboth are):either derivability in\(C\) is incomplete with respect to logical truthormodel-theoretic validity is unsound with respect to logical truth.

Different authors have extracted opposed lessons from incompleteness.A common reaction is to think that model-theoretic validity must beunsound with respect to logical truth. This is especially frequent inphilosophers on whose conception logical truths must beapriori or analytic. One idea is that the results ofapriori reasoning or of analytic thinking ought to be codifiablein a calculus. (See e.g. Wagner 1987, p. 8.) But even if we grant thisidea, it's doubtful that the desired conclusion follows. Suppose that(i) everya priori or analytic reasoning must be reproduciblein a calculus. We accept also, of course, that (ii)for everycalculus \(C\) sound with respect to model-theoretic validitythere is a model-theoretically valid formula that is notderivable in \(C\). From all this it doesn't follow that (iii)there is a model-theoretically valid formula \(F\) such thatfor every calculus \(C\) sound for model-theoretic validity\(F\) is not derivable inC. From (iii) and (i) it follows ofcourse that there are model-theoretically valid formulae that are notobtainable bya priori or analytic reasoning. But the stepfrom (ii) to (iii) is a typical quantificational fallacy. From (i) and(ii) it doesn't follow that there is any model-theoretically validformula which is not obtainable bya priori or analyticreasoning. The only thing that follows (from (ii) alone under theassumptions that model-theoretic validity is sound with respect tological truth and that logical truths area priori andanalytic) is that no calculus sound with respect to model-theoreticvalidity can by itself modelall thea priori oranalytic reasonings that people are able to make. But it's notsufficiently clear that this should be intrinsically problematic.After all,a priori and analytic reasonings must start frombasic axioms and rules, and for all we know a reflective mind may havean inexhaustible ability to find new truths and truth-preserving rulesbya priori or analytic consideration of even a meager stockof concepts. The claim that all analytic truths ought to be derivablein one single calculus is perhaps plausible on the view thatanalyticity is to be explained by conventions or “tacitagreements”, for these agreements are presumably finite innumber, and their implications are presumably at most effectivelyenumerable. But this view is just one problematic idea about howapriority and analyticity should be explicated. (See also Etchemendy1990, chs. 8, 9, for an argument for the unsoundness of higher-ordermodel-theoretic validity based on the conception of logical truth asanalyticitysimpliciter, and Gómez-Torrente 1998/9,Soames 1999, ch. 4, Paseau 2014, Florio and Incurvati 2019, 2021, andGriffiths and Paseau 2022, ch. 9, for critical reactions.)

Another type of unsoundness arguments attempt to show that there issome higher-order formula that is model-theoretically valid but isintuitively false in a structure whose domain is a proper class. (The“intended interpretation” of set theory, if it exists atall, might be one such structure, for it is certainly not a set; seethe entry onset theory.) These arguments thus question the claim that each meaningassignment's validity-refuting power is modeled by some set-theoreticstructure, a claim which is surely a corollary of the firstimplication in (5). (In McGee 1992 there is a good example; there iscritical discussion in Gómez-Torrente 1998/9.) The mostwidespread view among set theorists seems to be that there are noformulae with that property in Fregean languages, but it's certainlynot an absolutely firm belief of theirs. Note that these argumentsoffer a challenge only to the idea that universal validity (as definedin section 2.3) is adequately modeled by set-theoretic validity, notto the soundness of a characterization of logical truth in terms ofuniversal validity itself, or in terms of a species of validity basedon some notion of “meaning assignment” different from theusual notion of a set-theoretic structure. (The arguments we mentionedin the preceding paragraph and in 2.4.1 would have deeper implicationsif correct, for they easily imply challenges to all characterizationsin terms of species of validity as well). In fact, worries of thiskind have prompted the proposal of a different kind of notions ofvalidity (for Fregean languages), in which set-theoretic structuresare replaced with suitable values of higher-order variables in ahigher-order language for set theory, e.g. with “pluralinterpretations” (see Boolos 1985, Rayo and Uzquiano 1999,Williamson 2003, Florio and Incurvati 2021; see also Florio andLinnebo 2021 and the entry onplural quantification). Both set-theoretic and proper class structures are modeled by suchvalues, so these particular worries of unsoundness do not affect thiskind of proposals.

In general, there are no fully satisfactory philosophical argumentsfor the thesis that model-theoretic validity is unsound with respectto logical truth in higher-order languages. Are there then any goodreasons to think that derivability (in any calculus sound formodel-theoretic validity) must be incomplete with respect to logicaltruth? There don't seem to be any absolutely convincing reasons forthis view either. The main argument (the first version of which wasperhaps first made explicit in Tarski 1936a, 1936b) seems to be this.As noted above, Gödel's first incompleteness theorem implies thatfor any calculus for a higher-order language there will be amodel-theoretically valid formula that will not be derivable in thecalculus. As it turns out, the formula obtained by the Gödelconstruction is also always intuitively true in all domains(set-theoretical or not), and it's reasonable to think of it asuniversally valid. (It's certainly not a formula false in a properclass structure.) The argument concludes that for any calculus thereare logically true formulae that are not derivable in it.

From this it has been concluded that derivability (in any calculus)must be incomplete with respect to logical truth. But a fundamentalproblem is that this conclusion is based on two assumptions that willnot necessarily be granted by the champion of derivability: first, theassumption that the expressions typically cataloged as logical inhigher-order languages, and in particular the quantifiers inquantifications of the form \(\forall X\) (where \(X\) is ahigher-order variable), are in fact logical expressions; and second,the assumption that being universally valid is a sufficient conditionfor logical truth. On these assumptions it is certainly veryreasonable to think that derivability, in any calculus satisfying (4),must be incomplete with respect to logical truth. But in the absenceof additional considerations, a critic may question the assumptions,and deny relevance to the argument. The second assumption wouldprobably be questioned e.g. from the point of view that logical truthsmust be analytic, for there is no conclusive reason to think thatuniversally valid formulae must be analytic. The first assumptionactually underlies any conviction one may have that (4) holds for anyone particular higher-order calculus. (Note that if we denied that thehigher-order quantifiers are logical expressions we could equally denythat the arguments presented above against the soundness ofmodel-theoretic validity with respect tological truth arerelevant at all.) That the higher-order quantifiers are logical hasoften been denied on the grounds that they are semantically too“substantive”. It is often pointed out in this connectionthat higher-order quantifications can be used to define sophisticatedset-theoretic properties that one cannot define just with the help offirst-order quantifiers. (Defenders of the logical status ofhigher-order quantifications, on the other hand, point to the wideapplicability of the higher-order quantifiers, to the fact that theyare analogous to the first-order quantifiers, to the fact that theyare typically needed to provide categorical axiomatizations ofmathematical structures, etc. See Quine (1970), ch. 5, for thestandard exponent of the restrictive view, and Boolos (1975) andShapiro (1991) for standard exponents of the liberal view.)

Bibliography

Alexander of Aphrodisias,In Aristotelis Analyticorum PriorumLibrum I Commentarium, M. Wallies (ed.), Berlin: Reimer,1883.
Allison, H., 1983,Kant's Transcendental Idealism. AnInterpretation and Defense, New Haven: Yale UniversityPress.
Aristotle,Analytica Priora et Posteriora, W.D. Ross(ed.), Oxford: Clarendon Press, 1964.
–––,Topica et Sophistici Elenchi, W.D.Ross (ed.), Oxford: Clarendon Press, 1974.
Azzouni, J., 2006,Tracking Reason: Proof, Consequence andTruth. Oxford: Oxford University Press.
–––, 2008, “The Compulsion to Believe:Logical Inference and Normativity”,Protosociology, 25:69–88.
Beall, Jc and G. Restall, 2000, “Logical Pluralism”,Australasian Journal of Philosophy, 78: 475–93.
–––, 2006,Logical Pluralism, Oxford:Clarendon Press.
Belnap, N.D., 1962, “Tonk, Plonk and Plink”,Analysis, 22: 130–4.
Bernays, P., 1930, “The Philosophy of Mathematics andHilbert's Proof Theory”, translated by P. Mancosu, in Mancosu(ed.),From Brouwer to Hilbert, Oxford: Oxford UniversityPress, 1998.
Bocheński, I.M., 1956,Formale Logik, Munich:Alber.
Boghossian, P., 1997, “Analyticity”, in B. Hale and C.Wright (eds.),A Companion to the Philosophy of Language,Oxford: Blackwell, pp. 331–68.
–––, 2000, “Knowledge of Logic”, inP. Boghossian and C. Peacocke (eds.),New Essays on the APriori, Oxford: Clarendon Press, pp. 229–54.
Bolzano, B., 1837,Theory of Science, partial translationby R. George, Oxford: Blackwell, 1972.
BonJour, L., 1998,In Defense of Pure Reason, New York:Cambridge University Press.
Bonnay, D., 2008, “Logicality and Invariance”,Bulletin of Symbolic Logic, 14: 29–68.
Boolos, G., 1975, “On Second-Order Logic”,Journalof Philosophy, 72: pp. 509–27.
–––, 1985, “Nominalist Platonism”,in hisLogic, Logic, and Logic, Cambridge, MA: HarvardUniversity Press, pp. 73–87.
Capozzi, M. and G. Roncaglia, 2009, “Logic and Philosophy ofLogic from Humanism to Kant”, in L. Haaparanta (ed.),TheDevelopment of Modern Logic, Oxford: Oxford University Press, pp.78–158.
Carnap, R., 1939,Foundations of Logic and Mathematics(International Encyclopaedia of Unified Science, Vols. I-II), Chicago:University of Chicago Press.
–––, 1963, “Replies and SystematicExpositions”, in P. A. Schilpp (ed.),The Philosophy ofRudolf Carnap, La Salle, IL: Open Court, pp. 859–1013.
Carroll, L., 1895, “What the Tortoise Said toAchilles”,Mind, 4: 278–80.
Chihara, C., 1998, “Tarski's Thesis and the Ontology ofMathematics”, in M. Schirn (ed.),The Philosophy ofMathematics Today, Oxford: Oxford University Press, pp.157–72.
Coffa, J.A., 1991,The Semantic Tradition from Kant toCarnap, Cambridge: Cambridge University Press.
Dogramaci, S., 2017, “Why Is a Valid Inference a GoodInference?”,Philosophy and Phenomenological Research,94: 61–96.
Dummett, M., 1973, “The Justification of Deduction”,in hisTruth and Other Enigmas, Cambridge, MA: HarvardUniversity Press, 1978, pp. 290–318.
–––, 1981,Frege. Philosophy ofLanguage, Cambridge, MA: Harvard University Press.
–––, 1991,The Logical Basis ofMetaphysics, Cambridge, MA: Harvard University Press.
Estrada-González, L., 2012, “Models of Possibilismand Trivialism”,Logic and Logical Philosophy, 21:175–205.
Etchemendy, J., 1990,The Concept of Logical Consequence,Cambridge, MA: Harvard University Press.
–––, 2008, “Reflections onConsequence”, in D. Patterson (ed.),New Essays on Tarskiand Philosophy, Oxford: Oxford University Press, pp.263–99.
Feferman, S., 1999, “Logic, Logics and Logicism”,Notre Dame Journal of Formal Logic, 40: 31–54.
Field, H., 1989,Realism, Mathematics and Modality,Oxford: Blackwell.
–––, 2008,Saving Truth from Paradox,Oxford: Oxford University Press.
–––, 2015, “What Is LogicalValidity?”, in C. R. Caret and O. T. Hjortland (eds.),Foundations of Logical Consequence, Oxford: Oxford UniversityPress, pp. 33–70.
Florio, S. and L. Incurvati, 2019, “Metalogic and theOvergeneration Argument”,Mind, 128: 761–93.
–––, 2021, “Overgeneration in the HigherInfinite”, in G. Sagi and J. Woods (eds.),The SemanticConception of Logic. Essays on Consequence, Invariance, andMeaning, Oxford: Oxford University Press, pp. 142–59.
Florio, S. and O. Linnebo, 2021,The Many and the One. APhilosophical Study of Plural Logic, Oxford: Oxford UniversityPress.
Franks, C., 2014, “Logical Nihilism”, in P. Rush(ed.),The Metaphysics of Logic, Cambridge: CambridgeUniversity Press, pp. 109–27.
Frege, G., 1879, “Begriffsschrift, a Formula Language,Modeled upon that of Arithmetic, for Pure Thought”, translatedby S. Bauer-Mengelberg, in J. van Heijenoort (ed.),From Frege toGödel, Cambridge, MA: Harvard University Press, 1967, pp.1–82.
–––, 1885, “On Formal Theories ofArithmetic”, in hisCollected Papers on Mathematics, Logicand Philosophy, B. McGuinness (ed.), Oxford: Blackwell, 1984, pp.112–21.
García-Carpintero, M., 1993, “The Grounds for theModel-Theoretic Account of the Logical Properties”,NotreDame Journal of Formal Logic, 34: 107–31.
Glanzberg, M., 2015, “Logical Consequence and NaturalLanguage”, in C. R. Caret and O. T. Hjortland (eds.),Foundations of Logical Consequence, Oxford: Oxford UniversityPress, pp. 71–120.
Gómez-Torrente, M., 1998/9, “Logical Truth andTarskian Logical Truth”,Synthese, 117:375–408.
–––, 2002, “The Problem of LogicalConstants”,Bulletin of Symbolic Logic, 8:1–37.
–––, 2008, “Are There Model-TheoreticLogical Truths that Are not Logically True?”, in D. Patterson(ed.),New Essays on Tarski and Philosophy, Oxford: OxfordUniversity Press, pp. 340–68.
–––, 2019, “Soames on the LogicalEmpiricists on Truth, Meaning, Convention, and Logical Truth”,Philosophical Studies, 176: 1357–65.
–––, 2021, “The Problem of LogicalConstants and the Semantic Tradition: From Invariantist Views to aPragmatic Account”, in G. Sagi and J. Woods (eds.),TheSemantic Conception of Logic. Essays on Consequence, Invariance, andMeaning, Cambridge: Cambridge University Press, pp.35–54.
Grice, P. and P.F. Strawson, 1956, “In Defense of aDogma”, in Grice,Studies in the Way of Words,Cambridge, MA: Harvard University Press, 1989, pp. 196–212.
Griffiths, O., 2014, “Formal and InformalConsequence”,Thought, 3: 9–20.
Griffiths, O. and A. C. Paseau, 2022,One True Logic: A MonistManifesto, Oxford: Oxford University Press.
Hacking, I., 1979, “What Is Logic?”,Journal ofPhilosophy, 76: 285–319.
Hanna, R., 2001,Kant and the Foundations of AnalyticPhilosophy, Oxford: Clarendon Press.
–––, 2006,Rationality and Logic,Cambridge, MA: MIT Press.
Hanson, W., 1997, “The Concept of LogicalConsequence”,Philosophical Review, 106:365–409.
–––, 2006, “Actuality, Necessity, andLogical Truth”,Philosophical Studies, 130:437–59.
–––, 2014, “Logical Truth in ModalLanguages: Reply to Nelson and Zalta”,PhilosophicalStudies, 167: 327–39.
Hobbes, T., “Troisièmes Objections”, inDescartes,Œuvres Philosophiques, vol. II, F.Alquié (ed.), Paris: Garnier, 1967, pp. 599–631.
Hodes, H., 2004, “On the Sense and Reference of a LogicalConstant”,Philosophical Quarterly, 54:134–65.
Husserl, E., 1901,Logical Investigations, Vol. II,London: Routledge, 2001.
Iacona, A., 2018,Logical Form. Between Logic and NaturalLanguage, Cham: Springer.
Jané, I., 2006, “What Is Tarski'sCommonConcept of Consequence?”,Bulletin of Symbolic Logic,12: 1–42.
Kant, I.,Critique of Pure Reason, translation by P.Guyer and A. W. Wood, Cambridge: Cambridge University Press,1998.
Kneale, W., 1956, “The Province of Logic”, in H. D.Lewis (ed.),Contemporary British Philosophy (3rd Series),London: Allen & Unwin.
Kneale, W. and M. Kneale, 1962,The Development of Logic,Oxford: Clarendon Press.
Knuuttila, S., 1982, “Modal Logic”, in N. Kretzmann,A. Kenny and J. Pinborg (eds.),The Cambridge History of LaterMedieval Philosophy, Cambridge: Cambridge University Press, pp.342–57.
Kreisel, G., 1967, “Informal Rigour and CompletenessProofs”, in I. Lakatos (ed.),Problems in the Philosophy ofMathematics, Amsterdam: North-Holland, pp. 138–71.
Kretzmann, N., 1982, “Syncategoremata, Sophismata,Exponibilia”, in N. Kretzmann, A. Kenny and J. Pinborg (eds.),The Cambridge History of Later Medieval Philosophy,Cambridge: Cambridge University Press, pp. 211–45.
Leibniz, G.W., Letter to Bourguet (XII), in C.I. Gerhardt (ed.),Die philosophische Schriften von G.W. Leibniz, Hildesheim:Olms, 1965, vol. III, pp. 572–6.
–––, “Primæ Veritates”, in L.Couturat (ed.),Opuscules et Fragments Inédits deLeibniz, Hildesheim: Olms, 1961, pp. 518–23.
–––, “Discours deMétaphysique”, in C.I. Gerhardt (ed.),Diephilosophische Schriften von G.W. Leibniz, Hildesheim: Olms,1965, vol. IV, pp. 427–63.
–––, “Analysis Linguarum”, in L.Couturat (ed.),Opuscules et Fragments Inédits deLeibniz, Hildesheim: Olms, 1961, pp. 351–4.
Lewis, D.K., 1986,On the Plurality of Worlds, Oxford:Blackwell.
Łukasiewicz, J., 1957,Aristotle's Syllogistic from theStandpoint of Modern Formal Logic, second edition, Oxford:Clarendon Press.
McCarthy, T., 1981, “The Idea of a Logical Constant”,Journal of Philosophy, 78: 499–523.
MacFarlane, J., 2000,What does It Mean to Say that Logic IsFormal?, Ph.D. thesis, University of Pittsburgh, PhilosophyDepartment.
–––, 2002, “Frege, Kant, and the Logic inLogicism”,Philosophical Review, 111: 25–65.
McGee, V., 1992, “Two Problems with Tarski's Theory ofConsequence”,Proceedings of the Aristotelian Society(new series), 92: 273–92.
–––, 1996, “Logical Operations”,Journal of Philosophical Logic, 25: 567–80.
Maddy, P., 1999, “Logic and the Discursive Intellect”,Notre Dame Journal of Formal Logic, 40: 94–115.
–––, 2002, “A Naturalistic Look atLogic”,Proceedings and Addresses of the AmericanPhilosophical Association, 76 (2): 61–90.
–––, 2007,Second Philosophy. A NaturalisticMethod, Oxford: Oxford University Press.
Mates, B., 1961,Stoic Logic, Berkeley: University ofCalifornia Press.
Mill, J.S., 1843,A System of Logic, in hisCollectedWorks, vol. 7, Toronto: University of Toronto Press, 1973.
Mortensen, C., 1989, “Anything Is Possible”,Erkenntnis, 30: 319–37.
Nelson, M. and E. N. Zalta, 2012, “A Defense of ContingentLogical Truths”,Philosophical Studies, 157:153–62.
Orayen, R., 1989,Lógica, Significado yOntología, Mexico City: UNAM.
Pap, A., 1958,Semantics and Necessary Truth, New Haven:Yale University Press.
Parsons, C., 1969, “Kant's Philosophy of Arithmetic”,in hisMathematics in Philosophy, Ithaca: Cornell UniversityPress, 1983, pp. 110–49.
Paseau, A. C., 2014, “The Overgeneration Argument(s): ASuccinct Refutation”,Analysis, 74: 40–7.
Peacocke, C., 1987, “Understanding Logical Constants: ARealist's Account”,Proceedings of the British Academy,73: 153–200.
Prawitz, D., 1985, “Remarks on Some Approaches to theConcept of Logical Consequence”,Synthese, 62:153–71.
Priest, G., 2001, “Logic: One or Many?”, in J. Woodsand B. Brown (eds.),Logical Consequence: Rival Approaches,Oxford: Hermes Science Publishing, pp. 23–38.
Prior, A.N., 1960, “The Runabout Inference-Ticket”,Analysis, 21: 38–9.
Putnam, H., 1968, “The Logic of Quantum Mechanics”, inhisMathematics, Matter and Method. Philosophical Papers, Volume1, Cambridge: Cambridge University Press, 1975, pp.174–197.
Quine, W.V., 1936, “Truth by Convention”, in hisThe Ways of Paradox and Other Essays, revised edition,Cambridge, MA: Harvard University Press, 1976, pp. 77–106.
–––, 1951, “Two Dogmas ofEmpiricism”, in hisFrom a Logical Point of View,second edition revised, Cambridge, MA: Harvard University Press, 1980,pp. 20–46.
–––, 1954, “Carnap and LogicalTruth”, in hisThe Ways of Paradox and Other Essays,revised edition, Cambridge, MA: Harvard University Press, 1976, pp.107–32.
–––, 1963, “Necessary Truth”, in hisThe Ways of Paradox and Other Essays, revised edition,Cambridge, MA: Harvard University Press, 1976, pp. 68–76.
–––, 1970,Philosophy of Logic,Englewood Cliffs, NJ: Prentice-Hall.
Ray, G., 1996, “Logical Consequence: a Defense ofTarski”,Journal of Philosophical Logic, 25:617–77.
Rayo, A. and G. Uzquiano, 1999, “Toward a Theory ofSecond-Order Consequence”,Notre Dame Journal of FormalLogic, 40: 315–25.
Read, S., 1994, “Formal and Material Consequence”,Journal of Philosophical Logic, 23: 247–65.
Rumfitt, I., 2015,The Boundary Stones of Thought. An Essay inthe Philosophy of Logic, Oxford: Clarendon Press.
Russell, B., 1903,The Principles of Mathematics, NewYork: Norton, 1938.
–––, 1912,The Problems of Philosophy,Oxford: Oxford University Press, 1976.
–––, 1920,Introduction to MathematicalPhilosophy, second edition. New York: Macmillan.
Russell, G., 2017, “An Introduction to LogicalNihilism”, in H. Leitgeb, I. Niiniluoto, P. Seppäläand E. Sober (eds.),Logic, Methodology and Philosophy of Science.Proceedings of the Fifteenth International Congress, London:College Publications, pp. 125–34.
–––, 2018, “Logical Nihilism: Could ThereBe No Logic?”, in E. Sosa and E. Villanueva (eds.),Philosophical Issues 28: Philosophy of Logic and InferentialReasoning, Oxford: Wiley Blackwell, pp. 308–24.
Sagi, G., 2014, “Models and Logical Consequence”,Journal of Philosophical Logic, 43: 943–964.
Sainsbury, M., 1991,Logical Forms, Oxford:Blackwell.
Shalkowski, S., 2004, “Logic and Absolute Necessity”,Journal of Philosophy, 101: 55–82.
Shapiro, S., 1991,Foundations without Foundationalism: a Casefor Second-Order Logic, Oxford: Clarendon Press.
–––, 1998, “Logical Consequence: Modelsand Modality”, in M. Schirn (ed.),The Philosophy ofMathematics Today, Oxford: Oxford University Press, pp.131–56.
Sher, G., 1991,The Bounds of Logic, Cambridge, MA: MITPress.
–––, 1996, “Did Tarski Commit‘Tarski's Fallacy’?”,Journal of SymbolicLogic, 61: 653–86.
–––, 2013, “The Foundational Problem ofLogic”,Bulletin of Symbolic Logic, 19:145–98.
Smith, P., 2011, “Squeezing Arguments”,Analysis, 71: 22–30.
Smith, R., 1989, “Notes to Book A”, in Aristotle,Prior Analytics, R. Smith (ed.), Indianapolis, IN: Hackett,pp. 105–81.
Soames, S., 1999,Understanding Truth, New York: OxfordUniversity Press.
–––, 2018,The Analytic Tradition inPhilosophy,Volume 2: A New Vision, Princeton, NJ:Princeton University Press.
Tarski, A., 1935, “The Concept of Truth in FormalizedLanguages”, translated by J.H. Woodger in A. Tarski,Logic,Semantics, Metamathematics, second edition, J. Corcoran (ed.),Indianapolis, IN: Hackett, 1983, pp. 152–278.
–––, 1936a, “On the Concept of LogicalConsequence”, translated by J.H. Woodger in A. Tarski,Logic, Semantics, Metamathematics, second edition, J.Corcoran (ed.), Indianapolis, IN: Hackett, 1983, pp.409–20.
–––, 1936b, “On the Concept of FollowingLogically”, translated by M. Stroińska and D. Hitchcock,History and Philosophy of Logic, 23 (2002):155–96.
–––, 1941,Introduction to Logic and to theMethodology of Deductive Science, translated by O. Helmer, NewYork: Oxford University Press.
–––, 1966, “What Are LogicalNotions?”, ed. by J. Corcoran,History and Philosophy ofLogic, 7 (1986): 143–54.
Tarski, A. and S. Givant, 1987,A Formalization of Set Theorywithout Variables, Providence, RI: American MathematicalSociety.
Wagner, S.J., 1987, “The Rationalist Conception ofLogic”,Notre Dame Journal of Formal Logic, 28:3–35.
Warmbrōd, K., 1999, “Logical Constants”,Mind, 108: 503–38.
Williamson, T., 2003, “Everything”, in D. Zimmermanand J. Hawthorne (eds.),Philosophical Perspectives 17: Languageand Philosophical Linguistics, Oxford: Blackwell, pp.415–65.
–––, 2013,Modal Logic as Metaphysics,Oxford: Oxford University Press.
–––, 2017, “Semantic Paradoxes andAbductive Methodology”, in B. Armour-Garb (ed.),Reflectionson the Liar, Oxford: Oxford University Press, pp.325–46.
Wittgenstein, L., 1921,Tractatus Logico-Philosophicus,translated by C.K. Ogden, London: Routledge, 1990.
–––, 1978,Remarks on the Foundations ofMathematics, revised edition, G.H. Von Wright, R. Rhees andG.E.M. Anscombe (eds.), Cambridge, MA: MIT Press.
Woods, J., 2016, “Characterizing Invariance”,Ergo, 3: 778–807.
Zalta, E., 1988, “Logical and Analytic Truths that Are NotNecessary”,Journal of Philosophy, 85:57–74.

Academic Tools

How to cite this entry.
Preview the PDF version of this entry at theFriends of the SEP Society.
Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

Logical Consequence and Entailment, PhilPapers category edited by Salvatore Florio.
Foundations of Logical Consequence Project at Arché, Philosophical Research Centre for Logic, Language,Metaphysics and Epistemology, University of Saint Andrews.

Acknowledgments

I thank Axel Barceló, Bill Hanson, Ignacio Jané, JohnMacFarlane and Edward N. Zalta for very helpful comments on an earlierversion of this entry.

Open access to the SEP is made possible by a world-wide funding initiative.
The Encyclopedia Now Needs Your Support
Please Read How You Can Help Keep the Encyclopedia Free

Browse

About

Support SEP

Mirror Sites

View this site from another server:

USA (Main Site)Philosophy, Stanford University

Info about mirror sites

Library of Congress Catalog Data: ISSN 1095-5054

Movatterモバイル変換