## 1. The Nature of Logical Truth


### 1.1 Modality



As we said above, it seems to be universally accepted that, if there
are any logical truths at all, a logical truth ought to be such that
it could not be false, or equivalently, it ought to be such that it
must be true. But as we also said, there is virtually no agreement
about 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 a
logical truth ought to be due to its being a particular case of a true
universal generalization over the possible values of the schematic
letters in "formal" schemata like \((1')-(3')\). (These
values may but need not be expressions.) On what is possibly the
oldest way of understanding the logical modality, that modal force is
entirely due to this property: thus, for example, on this view to say
that (1) must be true can only mean that (1) is a particular case of
the 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 the
conclusion of a *syllogismos* must be true if the premises are
true ought to be understood in this way. In a famous passage of the
*Prior Analytics*, he says: "A *syllogismos* is
speech (*logos*) in which, certain things being supposed,
something other than the things supposed results of necessity (*ex
anankes*) because they are so" (24b18-20). Think of
(2) as a *syllogismos* in which the "things
supposed" are (2a) and (2b), and in which the thing that
"results of necessity" is (2c):


* (2a)  No desire is
voluntary.
* (2b)  Some beliefs are
desires.
* (2c)  Some beliefs are
not voluntary.



On the interpretation we are describing, Aristotle's view is that to
say 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 \(R\)". For
this interpretation see e.g. Alexander of Aphrodisias, 208.16 (quoted
by Lukasiewicz 1957, SS41), Bolzano (1837, SS155) and
Lukasiewicz (1957, SS5).



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 generalizations
on "formal" schemata). Especially prominent is Diodorus'
view that a proposition is necessary just in case it is true at all
times (see Mates 1961, III, SS3). Note that this makes sense of
the 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 Middle
Ages, when authors like William of Sherwood and Walter Burley seem to
have understood the perceived necessity of conditionals like (2) as
truth at all times (see Knuuttila 1982, pp. 348-9). An
understanding of necessity as eternity is frequent also in later
authors; see e.g., Kant, *Critique of Pure Reason*, B 184. In
favor of the mentioned interpretation of Aristotle and of the
Diodorean view it might be pointed out that we often use modal
locutions to stress the consequents of conditionals that follow from
mere universal generalizations about the actual world, as in "If
gas prices go up, necessarily the economy slows down".



Many authors have thought that views of this sort do not account for
the full strength of the modal import of logical truths. A nowadays
very common, but (apparently) late view in the history of philosophy,
is that the necessity of a logical truth does not merely imply that
some generalization about actual items holds, but also implies that
the truth would have been true at a whole range of counterfactual
circumstances. Leibniz assigned this property to necessary truths such
as those of logic and geometry, and seems to have been one of the
first to speak of the counterfactual circumstances as "possible
universes" or worlds (see the letter to Bourguet, pp.
572-3, for a crisp statement of his views that contrasts them
with the views in the preceding paragraph; Knuuttila 1982, pp. 353 ff.
detects the earliest transparent talk of counterfactual circumstances
and of necessity understood as at least implying truth in all of these
in Duns Scotus and Buridan; see also the entry on
 medieval theories of modality).
 In contemporary writings the understanding of necessity as truth in
all counterfactual circumstances, and the view that logical truths are
necessary in this sense, are widespread--although many, perhaps
most, authors adopt "reductivist" views of modality that
see talk of counterfactual circumstances as no more than disguised
talk about certain actualized (possibly abstract) items, such as
linguistic descriptions. Even Leibniz seems to have thought of his
"possible universes" as ideas in the mind of God. (See
Lewis 1986 for an introduction to the contemporary polemics in this
area.)



However, even after Leibniz and up to the present, many logicians seem
to have avoided a commitment to a strong notion of necessity as truth
in all (actual and) counterfactual circumstances. Thus Bolzano, in
line with his interpretation of Aristotle mentioned above,
characterizes necessary propositions as those whose negation is
incompatible with purely general truths (see Bolzano 1837, SS119).
Frege says that "the apodictic judgment [i.e., roughly, the
judgment whose content begins with a 'necessarily'
governing the rest of the content] is distinguished from the assertory
in that it suggests the existence of universal judgments from which
the proposition can be inferred, while in the case of the assertory
one such a suggestion is lacking" (Frege 1879, SS4). Tarski
is even closer to the view traditionally attributed to Aristotle, for
it 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 \(R\)" (see
Tarski 1936a, pp. 411, 415, 417, or the corresponding passages in
Tarski 1936b; see also Ray 1996). Quine is known for his explicit
rejection of any modality that cannot be understood in terms of
universal generalizations about the actual world (see especially Quine
1963). In some of these cases, this attitude is explained by a
distrust of notions that are thought not to have reached a fully
respectable scientific status, like the strong modal notions; it is
frequently accompanied in such authors, who are often practicing
logicians, by the proposal to characterize logical truth as a species
of 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 are
to be seen as instances of true universal generalizations over their
logical forms, based on the scientific or "abductive"
fruitfulness of the idea, though not on a disdain of modal
notions.)



On a recent view developed by Beall and Restall (2000, 2006), called
by them "logical pluralism", the concept of logical truth
carries a commitment to the idea that a logical truth is true in all
of a range of items or "cases", and its necessity consists
in the truth of such a general claim (see Beall and Restall 2006, p.
24). However, the concept of logical truth does not single out a
unique range of "cases" as privileged in determining an
extension for the concept, or, what amounts to the same thing, the
universal generalization over a logical form can be interpreted via
different equally acceptable ranges of quantification; instead, there
are many such equally acceptable ranges and corresponding extensions
for "logical truth", which may be chosen as a function of
contextual interests. This means that, for the logical pluralist, many
sets have a right to be called "the set of logical truths"
(and "the set of logical necessities"), each in the
appropriate
 context.[3]
 (See the entry on
 logical pluralism.)
 On another recent understanding of logical necessity as a species of
generality, proposed by Rumfitt (2015), the necessity of a logical
truth consists just in its being usable under all sets of
subject-specific ways of drawing implications (provided these sets
satisfy certain structural rules); or, more roughly, just in its being
applicable no matter what sort of reasoning is at stake. On this view,
a more substantive understanding of the modality at stake in logical
truth is again not required. It may be noted that, although he
postulates a variety of subject-specific implication relations,
Rumfitt rejects pluralism about logical truth in the sense of Beall
and Restall (see his 2015, p. 56, n. 23.), and in fact thinks that the
set of logical truths in a standard quantificational language is
characterized 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" not
be false or "must" be true is epistemic. It is an old
observation, going at least as far back as Plato, that some truths
count as intuitively known by us even in cases where we don't seem to
have any empirical grounds for them. Truths that are knowable on
non-empirical grounds are called *a priori* (an expression that
begins to be used with this meaning around the time of Leibniz; see
e.g. his "Primae Veritates", p. 518). The axioms and
theorems of mathematics, the lexicographic and stipulative
definitions, and also the paradigmatic logical truths, have been given
as examples. If it is accepted that logical truths are *a
priori*, it is natural to think that they must be true or could
not be false at least partly in the strong sense that their negations
are incompatible with what we are able to know non-empirically.



Assuming that such *a priori* knowledge exists in some way or
other, much recent philosophy has occupied itself with the issue of
how it is possible. One traditional ("rationalist") view
is that the mind is equipped with a special capacity to perceive
truths through the examination of the relations between pure ideas or
concepts, and that the truths reached through the correct operation of
this capacity count as known *a priori*. (See, e.g., Leibniz's
"Discours de Metaphysique", SSSS23 ff.;
Russell 1912, p. 105; BonJour 1998 is a very recent example of a view
of this sort.) An opposing traditional ("empiricist") view
is that there is no reason to postulate that capacity, or even that
there are reasons not to postulate it, such as that it is
"mysterious". (See the entry on
 rationalism vs. empiricism.)
 Some philosophers, empiricists and otherwise, have attempted to
explain *a priori* knowledge as arising from some sort of
convention 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, SS12, 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 that
logical truths do not express propositions at all, and are just
vacuous sentences that for some reason or other we find useful to
manipulate; thus it is only in a somewhat diminished sense that we can
speak of (*a priori*) knowledge of them. However, in typical
recent exponents of "tacit agreement" and conventionalist
views, such as Boghossian (1997), the claim that logical truths do not
express propositions is rejected, and it is accepted that the
existence of the agreement provides full-blown *a priori*
knowledge of those propositions.



The "rational capacity" view and the
"conventionalist" view agree that, in a broad sense, the
epistemic ground of logical truths resides in our ability to analyze
the meanings of their expressions, be these understood as conventions
or as objective ideas. For this reason it can be said that they
explain the apriority of logical truths in terms of their analyticity.
(See the entry on the
 analytic/synthetic distinction.)
 Kant's explanation of the apriority of logical truths has seemed
harder to
 extricate.[4]
 A long line of commentators of Kant has noted that, if Kant's view is
that all logical truths are analytic, this would seem to be in tension
with his characterizations of analytic truths. Kant characterizes
analytic truths as those where the concept of the predicate is
contained in or identical with the concept of the subject, and, more
fundamentally, as those whose denial is contradictory. But it has
appeared to those commentators that these characterizations, while
applying to strict tautologies such as "Men are men" or
"Bearded men are men", would seem to leave out much of
what Kant himself counts as logically true, including syllogisms such
as (2) (see e.g. Mill 1843, bk. II, ch. vi, SS5; Husserl 1901,
vol. II, pt. 2, SS66; Kneale and Kneale 1962, pp. 357-8;
Parsons 1967; Maddy 1999). This and the apparent lack of clear
pronouncements of Kant on the issue has led at least Maddy (1999) and
Hanna (2001) to consider (though not accept) the hypothesis that Kant
viewed some logical truths as synthetic *a priori*. On an
interpretation of this sort, the apriority of many logical truths
would be explained by the fact that they would be required by the
cognitive structure of the transcendental subject, and specifically by
the forms of
 judgment.[5]
 But the standard interpretation is to attribute to Kant the view that
all logical truths are analytic (see e.g. Capozzi and Roncaglia 2009).
On an interpretation of this sort, Kant's forms of judgment may be
identified with logical concepts susceptible of analysis (see e.g.
Allison 1983, pp. 126ff.). An extended defense of the interpretation
that Kant viewed all logical truths as analytic, including a
vindication of Kant against the objections of the line of commentators
mentioned above, can be found in Hanna (2001), SS3.1. A
substantively Kantian contemporary theory of the epistemology of logic
and its roots in cognition is developed in Hanna (2006); this theory
does not seek to explain the apriority of logic in terms of its
analyticity, and appeals instead to a specific kind of logical
intuition and a specific cognitive logic faculty. (Compare also the
anti-aprioristic and anti-analytic but broadly Kantian view of Maddy
2007, mentioned below.)



The early Wittgenstein shares with Kant the idea that the logical
expressions do not express meanings in the way that non-logical
expressions do (see 1921, 4.0312). Consistently with this view, he
claims that logical truths do not "say" anything (1921,
6.11). But he seems to reject conventionalist and "tacit
agreement" views (1921, 6.124, 6.1223). It is not that logical
truths do not say anything because they are mere instruments for some
sort of extrinsically useful manipulation; rather, they
"show" the "logical properties" that the world
has independently of our decisions (1921, 6.12, 6.13). It is unclear
how apriority is explainable in this framework. Wittgenstein calls the
logical truths analytic (1921, 6.11), and says that "one can
recognize in the symbol alone that they are true" (1921, 6.113).
He seems to have in mind the fact that one can "see" that
a logical truth of truth-functional logic must be valid by inspection
of a suitable representation of its truth-functional content (1921,
6.1203, 6.122). But the extension of the idea to quantificational
logic is problematic, despite Wittgenstein's efforts to reduce
quantificational logic to truth-functional logic; as we now know,
there is no algorithm for deciding if a quantificational sentence is
valid. What is perhaps more important, Wittgenstein gives no
discernible explanation of why in principle all the "logical
properties" of the world should be susceptible of being
reflected in an adequate notation.



Against the "rational capacity",
"conventionalist", Kantian and early Wittgensteinian
views, other philosophers, especially radical empiricists and
naturalists (not to speak of epistemological skeptics), have rejected
the claim that *a priori* knowledge exists (hence by
implication also the claim that analytic propositions exist), and they
have proposed instead that there is only an illusion of apriority.
Often this rejection has been accompanied by criticism of the other
views. J.S. Mill thought that propositions like (2) seem *a
priori* merely because they are particular cases of early and very
familiar 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 \(R\)" (see
Mill 1843, bk. II, ch. viii). Bolzano held a similar view (see Bolzano
1837, SS315). Quine (1936, SSIII) famously criticized the
Hobbesian view noting that since the logical truths are infinite in
number, our ground for them must not lie just in a finite number of
explicit conventions, for logical rules are presumably needed to
derive an infinite number of logical truths from a finite number of
conventions (an argument derived from Carroll 1895; see Soames 2018,
ch. 10, for exposition and Gomez-Torrente 2019 for criticism of
the argument). Later Quine (especially 1954) criticized Carnap's
conventionalist view, largely on the grounds that there seems to be no
non-vague distinction between conventional truths and truths that are
tacitly left open for refutation, and that to the extent that some
truths are the product of convention or "tacit agreement",
such agreement is characteristic of many scientific hypotheses and
other postulations that seem paradigmatically non-analytic. (See Grice
and Strawson 1956 and Carnap 1963 for reactions to these criticisms.)
Quine (especially 1951) also argued that accepted sentences in
general, including paradigmatic logical truths, can be best seen as
something like hypotheses that are used to deal with experience, any
of which can be rejected if this helps make sense of the empirical
world (see Putnam 1968 for a similar view and a purported example). On
this view there cannot be strictly *a priori* grounds for any
truth. Three recent subtle anti-aprioristic positions are Maddy's
(2002, 2007), Azzouni's (2006, 2008), and Sher's (2013). For Maddy,
logical truths are *a posteriori*, but they cannot be
disconfirmed merely by observation and experiment, since they form
part of very basic ways of thinking of ours, deeply embedded in our
conceptual machinery (a conceptual machinery that is structurally
similar to Kant's postulated transcendental organization of the
understanding). Similarly, for Azzouni logical truths are equally
*a posteriori*, though our sense that they must be true comes
from their being psychologically deeply ingrained; unlike Maddy,
however, Azzouni thinks that the logical rules by which we reason are
opaque to introspection. Sher provides an attempt at combining a
Quinean epistemology of logic with a commitment to a metaphysically
realist view of the modal ground of logical truth.



One way in which *a priori* knowledge of a logical truth such
as (1) would be possible would be if *a priori* knowledge of
the fact that (1) is a logical truth, or 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 \(Q\)" were possible. One especially noteworthy kind of
skeptical consideration in the epistemology of logic is that the
possibility of inferential *a priori* knowledge of these facts
seems to face a problem of circularity or of infinite regress. If we
are to obtain inferential *a priori* knowledge of those facts,
then we will presumably follow logical rules at some point, including
possibly the rule of *modus ponens* whose very correctness
might well depend in part on the fact that (1) is a logical truth or
on 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, it
seems clear that not all claims of this latter kind, expressing that a
certain truth is a logical truth or that a certain logical schema is
truth-preserving, could be given an *a priori* inferential
justification without the use of some of the same logical rules whose
correctness they might be thought to codify. The point can again be
reasonably derived from Carroll (1895). Some of the recent literature
on this consideration, and on anti-skeptical rejoinders, includes
Dummett (1973, 1991) and Boghossian (2000).


### 1.2 Formality



On most views, even if it were true that logical truths are true in
all counterfactual circumstances, *a priori*, and analytic,
this would not give sufficient conditions for a truth to be a logical
truth. On most views, a logical truth also has to be in some sense
"formal", and this implies at least that all truths that
are replacement instances of its form are logical truths too (and
hence, on the assumption of the preceding sentence, true in all
counterfactual circumstances, *a priori*, and analytic). To use
a slight modification of an example of Albert of Saxony (quoted by
Bochenski 1956, SS30.07), "If a widow runs, then a
female runs" should be true in all counterfactual circumstances,
*a priori*, and analytic if any truth is. However, "If a
widow runs, then a log runs" is a replacement instance of its
form, and in fact it even has the same form on any view of logical
form (something like "If a \(P\) \(Q\)s, then an \(R\)
\(Q\)s"), but it is not even true *simpliciter*. So on
most views, "If a widow runs, then a female runs" is not a
logical 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 the
expressions that are not schematic letters are widely applicable
across different areas of
 discourse.[6]
 If the schema is the form of a logical truth, all of its replacement
instances are logical truths. The idea that logic is especially
concerned with (replacement instances of) schemata is of course
evident beginning with Aristotle and the Stoics, in all of whom the
word usually translated by "figure" is precisely
*schema*. In Aristotle a figure is actually an even more
abstract form of a group of what we would now call
"schemata", such as (2'). Our schemata are closer to
what in the Aristotelian syllogistic are the moods; but there seems to
be no word for "mood" in Aristotle (except possibly
*ptoseon* in 42b30 or *tropon* in 43a10; see Smith 1989,
pp. 148-9), and thus no general reflection on the notion of
formal schemata. There is explicit reflection on the contrast between
the formal schemata or moods and the matter (*hyle*) of
*syllogismoi* in Alexander of Aphrodisias (53.28ff., quoted by
Bochenski 1956, SS24.06), and there has been ever since. The
matter are the values of the schematic letters.



The idea that the non-schematic expressions in logical forms, i.e. the
logical expressions, are widely applicable across different areas of
discourse is also present from the beginning of logic, and recurs in
all the great logicians. It appears indirectly in many passages from
Aristotle, such as the following: "All the sciences are related
through the common things (I call common those which they use in order
to demonstrate from them, but not those that are demonstrated in them
or those about which something is demonstrated); and logic is related
to them all, as it is a science that attempts to demonstrate
universally the common things" (*Posterior Analytics*,
77a26-9); "we don't need to take hold of the things of all
refutations, but only of those that are characteristic of logic; for
these are common to every technique and ability"
(*Sophistical Refutations*, 170a34-5). (In these texts
"logic" is an appropriate translation of
*dialektike*; see Kneale and Kneale 1962, I, SS3, who
inform us that *logike* is used for the first time with its
current 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 the
laws on which all knowledge rests" (1879, p. 48; see also 1885,
where the universal applicability of the arithmetical concepts is
taken as a sign of their logicality). The same idea is conspicuous as
well in Tarski (1941, ch. II, SS6).



That logical expressions include paradigmatic cases like
"if", "and", "some",
"all", etc., and that they must be widely applicable
across different areas of discourse is what we might call "the
minimal thesis" about logical expressions. But beyond this there
is little if any agreement about what generic feature makes an
expression logical, and hence about what determines the logical form
of a sentence. Most authors sympathetic to the idea that logic is
formal have tried to go beyond the minimal thesis. It would be
generally agreed that being widely applicable across different areas
of discourse is only a necessary, not sufficient property of logical
expressions; for example, presumably most prepositions are widely
applicable, but they are not logical expressions on any implicit
generic notion of a logical expression. Attempts to enrich the notion
of a logical expression have typically sought to provide further
properties that collectively amount to necessary and sufficient
conditions for an expression to be logical.



One idea that has been used in such characterizations, and that is
also present in Aristotle, is that logical expressions do not,
strictly speaking, signify anything; or, that they do not signify
anything in the way that substantives, adjectives and verbs signify
something. "Logic [*dialektike*] is not a science of
determined things, or of any one genus" (*Posterior
Analytics*, 77a32-3). We saw that the idea was still present
in Kant and the early Wittgenstein. It reemerged in the Middle Ages.
The main sense of the word "syncategorematic" as applied
to expressions was roughly this semantic sense (see Kretzmann 1982,
pp. 212 ff.). Buridan and other late medieval logicians proposed that
categorematic expressions constitute the "matter" of
sentences while the syncategorematic expressions constitute their
"form" (see the text quoted by Bochenski 1956,
SS26.11). (In a somewhat different, earlier, grammatical sense of
the word, syncategorematic expressions were said to be those that
cannot be used as subjects or predicates in categorical propositions;
see Kretzmann 1982, pp. 211-2.) The idea of syncategorematicity
is somewhat imprecise, but there are serious doubts that it can serve
to characterize the idea of a logical expression, whatever this may
be. Most prepositions and adverbs are presumably syncategorematic, but
they are also presumably non-logical expressions. Conversely,
predicates such as "are identical", "is identical
with itself", "is both identical and not identical with
itself", etc., which are resolutely treated as logical in recent
logic, are presumably categorematic. (They are of course categorematic
in the grammatical sense, in which prepositions and adverbs are
equally clearly syncategorematic.)



Most other proposals have tried to delineate in some other way the
Aristotelian idea that the logical expressions have some kind of
"insubstantial" meaning, so as to use it as a necessary
and sufficient condition for logicality. One recent suggestion is that
logical expressions are those that do not allow us to distinguish
different individuals. One way in which this has been made precise is
through the characterization of logical expressions as those whose
extension or denotation over any particular domain of individuals is
invariant under permutations of that domain. (See Tarski and Givant
1987, p. 57, and Tarski 1966; for related proposals see also McCarthy
1981, Sher 1991, ch. 3, McGee 1996, Feferman 1999, Bonnay 2008, Woods
2016 and Griffiths and Paseau 2022, among others.) A permutation of a
domain 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 man
to himself; another is the correspondence \(P\) that assigns Caesar to
Aristotle (in mathematical notation,
\(P(\text{Aristotle})=\text{Caesar}\)), Napoleon to Caesar, Kripke to
Napoleon, and Aristotle to Kripke. That the extension of an expression
over a domain is invariant under a permutation of that domain means
that the induced image of that extension under the permutation is the
extension itself (the "induced image" of an extension
under a permutation \(Q\) is what the extension becomes when in place
of each object \(o\) one puts the object \(Q(o)\)). The extension of
"philosopher" over \(D\) is not invariant under the
permutation \(P\) above, for that extension is \(\{\text{Aristotle},
\text{Kripke}\}\), whose induced image under \(P\) is
\(\{\text{Caesar}, \text{Aristotle}\}\). This is favorable to the
proposal, for "philosopher" is certainly not widely
applicable, and so non-logical on most views. On the other hand, the
predicate "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); so
again this is favorable to the proposal. (Other paradigmatic logical
expressions receive more complicated extensions over domains, but the
extensions 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 of
infinite sequences of objects drawn from \(D\), the intersection of
\(S\_1\) and \(S\_2\); and this function is permutation invariant.) One
problem with the proposal is that many expressions that seem clearly
non-logical, because they are not widely applicable, are nevertheless
invariant under permutations, and thus unable to distinguish different
individuals. The simplest examples are perhaps non-logical predicates
that have an empty extension over any domain, and hence have empty
induced images as well. "Male widow" is one example;
versions of it can be used as counterexamples to the different
versions of the idea of logicality as permutation invariance (see
Gomez-Torrente 2002), and it's unclear that the proponent of
the idea can avoid the problem in any non ad hoc way.



Another popular recent way of delineating the Aristotelian intuition
of the semantic "insubstantiality" of logical expressions
appeals to the concept of "pure inferentiality". The idea
is that logical expressions are those whose meaning, in some sense, is
given by "purely inferential" rules. (See Kneale 1956,
Hacking 1979, Peacocke 1987, Hodes 2004, among others.) A necessary
property of purely inferential rules is that they regulate only
inferential transitions between verbal items, not between extra-verbal
assertibility conditions and verbal items, or between verbal items and
actions licensed by those items. A certain inferential rule licenses
you 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 someone
says "A is a widow", however, is not immediately
disqualified as purely inferential. Now, presumably in some sense the
meaning of "widow" is given by this last rule together
perhaps with the converse rule, that licenses you to say "A is a
widow" when someone says "A is a female whose husband died
before her". But "widow" is not a logical
expression, since it's not widely applicable; so one needs to
postulate more necessary properties that "purely
inferential" rules ought to satisfy. A number of such conditions
are postulated in the relevant literature (see e.g. Belnap 1962 (a
reply to Prior 1960), Hacking 1979 and Hodes 2004). However, even when
the notion of pure inferentiality is strengthened in these ways,
problems remain. Most often the proposal is that an expression is
logical just in case certain purely inferential rules give its whole
meaning, including its sense, or the set of aspects of its use that
need to be mastered in order to understand it (as in Kneale 1956,
Peacocke 1987 and Hodes 2004). However, it seems clear that some
paradigmatic logical expressions have extra sense attached to them
that is not codifiable purely inferentially. For example, inductive
reasoning involving "all" seems to be part of the sense of
this expression, but it's hard to see how it could be codified by
purely inferential rules (as noted by Sainsbury 1991, pp. 316-7;
see also Dummett 1991, ch. 12). A different version of the proposal
consists in saying that an expression is logical just in case certain
purely inferential rules that are part of its sense suffice to
determine its extension (as in Hacking 1979). But it seems clear that
if the extension of, say, "are identical" is determined by
a certain set of purely inferential rules that are part of its sense,
then the extension of "are identical and are not male
widows" is equally determined by the same rules, which arguably
form part of its sense; yet "are identical and are not male
widows" is not a logical expression (see Gomez-Torrente
2002).



In view of problems of these and other sorts, some philosophers have
proposed that the concept of a logical expression is not associated
with necessary and sufficient conditions, but only with some necessary
condition related to the condition of wide applicability, such as the
condition of "being very relevant for the systematization of
scientific reasoning" (see Warmbrod 1999 for a position of
this type). Others (Gomez-Torrente 2021) have proposed that
there may be a set of necessary and sufficient conditions, if these
are not much related to the idea of semantic
"insubstantiality" and are instead pragmatic and suitably
vague; for example, many expressions are excluded directly by the
condition of wide applicability; and prepositions are presumably
excluded by some such implicit condition as "a logical
expression must be one whose study is useful for the resolution of
significant problems and fallacies in reasoning". To be sure,
these proposals give up on the extended intuition of semantic
"insubstantiality", and may be somewhat unsatisfactory for
that reason.



Some philosophers have reacted even more radically to the problems of
usual characterizations, claiming that the distinction between logical
and non-logical expressions must be vacuous, and thus rejecting the
notion of logical form altogether. (See e.g. Orayen 1989, ch. 4,
SS2.2; Etchemendy 1990, ch. 9; Read 1994; Priest 2001.) These
philosophers typically think of logical truth as a notion roughly
equivalent to that of analytic truth *simpliciter*. But they
are even more liable to the charge of giving up on extended intuitions
than the proposals of the previous paragraph.



For more thorough treatments of the ideas of formality and of a
logical expression see the entry
 logical 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 of
what has been called "formalization". This term is usually
employed to cover several distinct (though related) phenomena, all of
them present in Frege (1879). One of these is the use of a completely
specified set of artificial symbols to which the logician assigns
unambiguous meanings, related to the meanings of corresponding natural
language expressions, but much more clearly delimited and stripped
from the notes that in those natural language expressions seem (to the
logician) irrelevant to truth-conditional content; this is especially
true of symbols meant to represent the logical expressions of natural
language. Another phenomenon is the stipulation of a completely
precise grammar for the formulae construed out of the artificial
symbols, formulae that will be "stripped" versions of
correlate sentences in natural language; this grammar amounts to an
algorithm for producing formulae starting from the basic symbols. A
third phenomenon is the postulation of a deductive calculus with a
very clear specification of axioms and rules of inference for the
artificial formulae (see the next section); such a calculus is
intended to represent in some way deductive reasoning with the
correlates of the formulae, but unlike ordinary deductions,
derivations in the calculus contain no steps that are not definite
applications of the specified rules of inference.



Instead of attempting to characterize the logical truths of a natural
language like English, the Fregean logician attempts to characterize
the artificial formulae that are "stripped" correlates of
those logical truths in a Fregean formalized language. In first-order
Fregean formalized languages, among these formulae one finds
artificial correlates of (1), (2) and (3), things like


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


* \((\forall
x(\text{Desire}(x) \rightarrow \neg \text{Voluntary}(x)) \ \&\
\exists x(\text{Belief}(x) \ \&\ \text{Desire}(x)))\)
  

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


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

  \(\rightarrow \text{Mysterious}(\textit{drasha}).\)



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



The restriction to artificial formulae raises a number of questions
about the exact value of the Fregean enterprise for the demarcation of
logical truths in natural language; much of this value depends on how
many and how important are perceived to be the notes stripped from the
natural language expressions that are correlates of the standard
logical expressions of formalized languages. But whatever one's view
of the exact value of formalization, there is little doubt that it has
been very illuminating for logical purposes. One reason is that it's
sometimes relatively clear that the stripped notes are irrelevant to
truth-conditional content (this is especially true of the use of
natural language logical expressions for doing mathematics). Another
of the reasons is that the fact that the grammar and meaning of the
artificial formulae is so well delimited has permitted the development
of proposed characterizations of logical truth that use only concepts
of standard mathematics. This in turn has allowed the study of the
characterized notions by means of standard mathematical techniques.
The next two sections describe the two main approaches to
characterization in broad
 outline.[7]


### 2.2 Derivability



We just noted that the Fregean logician's formalized grammar amounts
to an algorithm for producing formulae from the basic artificial
symbols. This is meant very literally. As was clear to mathematical
logicians from very early on, the basic symbols can be seen as (or
codified by) natural numbers, and the formation rules in the
artificial grammar can be seen as (or codified by) simple computable
arithmetical operations. The grammatical formulae can then be seen as
(or codified by) the numbers obtainable from the basic numbers after
some finite series of applications of the operations, and thus their
set is characterizable in terms of concepts of arithmetic and set
theory (in fact arithmetic suffices, with the help of some
tricks).



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



In the time following Frege's revolution, there appears to have been a
widespread belief that the set of logical truths of any Fregean
language could be characterized as the set of formulae derivable in
some suitably chosen calculus (hence, essentially, as the set of
numbers obtainable by certain arithmetical operations). Frege himself
says, speaking of the higher-order language in his
"Begriffsschrift", that through formalization (in the
third sense above) "we arrive at a small number of laws in
which, if we add those contained in the rules, the content of all the
laws is included, albeit in an undeveloped state" (Frege 1879,
SS13). The idea follows straightforwardly from Russell's
conception of mathematics and logic as identical (see Russell 1903,
ch. I, SS10; Russell 1920, pp. 194-5) and his thesis that
"by the help of ten principles of deduction and ten other
premises of a general logical nature (...), all mathematics can
be strictly and formally deduced" (Russell 1903, ch. I,
SS4). See also Bernays (1930, p. 239): "[through
formalization] it becomes evident that all logical inference
(...) can be reduced to a limited number of logical elementary
processes that can be exactly and completely enumerated". In the
opening paragraphs of his paper on logical consequence, Tarski (1936a,
1936b) says that the belief was prevalent before the appearance of
Godel's incompleteness theorems (see subsection 2.4.3 below for
the bearing of these theorems on this issue). In recent times,
apparently due to the influence of Tarskian arguments such as the one
mentioned towards the end of subsection 2.4.3, the belief in the
adequacy of derivability characterizations seems to have waned (see
e.g. Prawitz 1985 for a similar appraisal).


### 2.3 Model-theoretic Validity



Even on the most cautious way of understanding the modality present in
logical truths, a sentence is a logical truth only if no sentence
which is a replacement instance of its logical form is false. (This
idea is only rejected by those who reject the notion of logical form.)
It is a common observation that this property, even if it is
necessary, is not clearly sufficient for a sentence to be a logical
truth. Perhaps there is a sentence that has this property but is not
really logically true, because one could assign some *unexpressed
meanings* to the variables and the schematic letters in its
logical 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 a
sentence is a logical truth if no collective assignment of meanings to
the variables and the schematic letters in its logical form would turn
this form into a false sentence. Say that a sentence is
*universally valid* when it has this property. A standard
approach to the mathematical characterization of logical truth,
alternative to the derivability approach, uses always some version of
the property of universal validity, proposing it in each case as both
necessary and sufficient for logical truth. Note that if a sentence is
universally valid then, even if it's not logically true, it will be
*true*. So all universally valid sentences are correct at least
in this sense.



Apparently, the first to use a version of universal validity and
explicitly propose it as both necessary and sufficient for logical
truth was Bolzano (see Bolzano 1837, SS148; and Coffa 1991, pp.
33-4 for the claim of priority). The idea is also present in
other mathematicians of the nineteenth century (see e.g. Jane
2006), and was common in Hilbert's school. Tarski (1936a, 1936b) was
the first to indicate in a fully explicit way how the version of
universal validity used by the mathematicians could itself be given a
characterization in terms of concepts of standard mathematics, in the
case of Fregean formalized languages with an algorithmic grammar.
Essentially Tarski's characterization is widely used today in the form
of what is known as the *model-theoretic notion of validity*,
and it seems fair to say that it is usually accepted that this notion
gives a reasonably good delineation of the set of logical truths for
Fregean languages.



The notion of model-theoretic validity mimics the notion of universal
validity, but is defined just with the help of the set-theoretic
apparatus developed by Tarski (1935) for the characterization of
semantic concepts such as satisfaction, definability, and truth. (See
the entry on
 Tarski's truth definitions.)
 Given a Fregean language, a *structure* for the language is a
set-theoretical object composed of a set-domain taken together with an
assignment of extensions drawn from that domain to its non-logical
constants. A structure is meant by most logicians to represent an
assignment of meanings: its domain gives the range or
"meaning" of the first-order variables (and induces ranges
of the higher-order variables), and the extensions that the structure
assigns to the non-logical constants are "meanings" that
these expressions could take. Using the Tarskian apparatus, one
defines for the formulae of the Fregean language the notion of truth
in (or satisfaction by) a set-theoretic structure (with respect to an
infinite sequence assigning an object of the domain to each variable).
And finally, one defines a formula to be model-theoretically valid
just in case it is true in all structures for its language (with
respect to all infinite sequences). Let's abbreviate "\(F\) is
true in all structures" as "MTValid\((F)\)". The
model-theoretic characterization makes it clear that
"MTValid\((F)\)" is definable purely in terms of concepts
of set theory. (The notion of model-theoretic validity for Fregean
languages is explained in thorough detail in the entries on
 classical logic
 and
 second-order and higher-order logic;
 see also the entry on
 model 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 is
a sufficient condition for \(F\)'s being model-theoretically valid. As
it turns out, if \(F\) is not model-theoretically valid, then some
replacement instance of its form whose variables range over the
natural numbers and whose non-logical constants are arithmetical
expressions will be false. This can be justified by means of a
refinement of the Lowenheim-Skolem theorem. See the entry on
 logic, classical,
 and Quine 1970, ch. 4, for discussion and references. No similar
results hold for higher-order languages.)



The "MT" in "MTValid\((F)\)" stresses the fact
that model-theoretic validity is different from universal validity.
The notion of a meaning assignment which appears in the description of
universal validity is a very imprecise and intuitive notion, while the
notion of a structure appearing in a characterization of
model-theoretic validity is a fairly precise and technical one. It
seems clear that the notion of a structure for Fregean formalized
languages is minimally reasonable, in the sense that a structure
models the power of one or several meaning assignments to make false
(the logical form of) some sentence. As we will mention later, the
converse property, that each meaning assignment's validity-refuting
power is modeled by some structure, is also a natural but more
demanding requirement on a notion of structure.


### 2.4 The Problem of Adequacy



The fact that the notions of derivability and model-theoretic validity
are definable in standard mathematics seems to have been a very
attractive feature of them among practicing logicians. But this
attractive feature of course does not justify by itself taking either
notion as an adequate characterization of logical truth. On most
views, with a mathematical characterization of logical truth we
attempt to delineate a set of formulae possessing a number of
non-mathematical properties. Which properties these are varies
depending on our pretheoretic conception of, for example, the features
of modality and formality. (By "pretheoretic" it's not
meant "previous to any theoretical activity"; there could
hardly be a "pretheoretic" conception of logical truth in
this sense. In this context what's meant is "previous to the
theoretical activity of mathematical characterization".) But on
any such conception there will be external, non-mathematical criteria
that can be applied to evaluate the question whether a mathematical
characterization is adequate. In this last section we will outline
some of the basic issues and results on the question whether
derivability and model-theoretic validity are adequate in this
sense.


#### 2.4.1 Analysis and Modality



One frequent objection to the adequacy of model-theoretic validity is
that it does not provide a conceptual analysis of the notion of
logical truth, even for sentences of Fregean formalized languages (see
e.g. Pap 1958, p. 159; Kneale and Kneale 1962, p. 642; Field 1989, pp.
33-4; Etchemendy 1990, ch. 7). This complaint is especially
common among authors who feel inclined to identify logical truth and
analyticity *simpliciter* (see e.g. Kneale and Kneale,
*ibid*., Etchemendy 1990, p. 126). If one thinks of the concept
of logical truth simply as the concept of analytic truth, it is
especially reasonable to accept that the concept of logical truth does
not have much to do with the concept of model-theoretic validity, for
presumably this concept does not have much to do with the concept of
analyticity. To say that a formula is model-theoretically valid means
that there are no set-theoretic structures in which it is false;
hence, to say that a formula is not model-theoretically valid means
that there are set-theoretic structures in which it is false. But to
say that a sentence is or is not analytic presumably does not mean
anything about the existence or non-existence of set-theoretic
structures. Note that we could object to derivability on the same
grounds, for to say that a sentence is or is not analytic presumably
does not mean anything about its being or not being the product of a
certain algorithm (compare Etchemendy 1990, p. 3). (One further
peculiar, much debated claim in Etchemendy 1990 is that true claims of
the form "\(F\) is logically true" or "\(F\) is not
logically true" should themselves be logical truths (while the
corresponding claims "MTValid\((F)\)" and "Not
MTValid\((F)\)" are not logical truths). Etchemendy's claim is
perhaps defensible under a conception of logical truth as analyticity
*simpliciter*, but certainly doubtful on more traditional
conceptions of logical truth, on which the predicate "is a
logical truth" is not even a logical expression. See
Gomez-Torrente 1998/9 and Soames 1999, ch. 4 for
discussion.)



Analogous "no conceptual analysis" objections can be made
if we accept that the concept of logical truth has some other strong
modal notes unrelated to analyticity; for example, if we accept that
it is part of the concept of logical truth that logical truths are
true in all counterfactual circumstances, or necessary in some other
strong sense. Sher (1996) accepts something like the requirement that
a good characterization of logical truth should be given in terms of a
modally rich concept. However, she argues that the notion of
model-theoretic validity is strongly modal, and so the "no
conceptual analysis" objection is actually wrong: to say that a
formula is or is not model-theoretically valid is to make a
mathematical existence or non-existence claim, and according to Sher
these claims are best read as claims about the possibility and
necessity of structures. (Shalkowski 2004 argues that Sher's defense
of model-theoretic validity is insufficient, on the basis of a certain
metaphysical conception of logical necessity. Etchemendy 2008
relatedly argues that Sher's defense is based on inadequate
restrictions on the modality relevant to logical truth. See also the
critical discussion of Sher in Hanson 1997.) Garcia-Carpintero
(1993) offers a view related to Sher's: model-theoretic validity
provides a (correct) conceptual analysis of logical truth for Fregean
languages, because the notion of a set-theoretical structure is in
fact a subtle refinement of the modal notion of a possible meaning
assignment. Azzouni (2006), ch. 9, also defends the view that
model-theoretic validity provides a correct conceptual analysis of
logical truth (though restricted to first-order languages), on the
basis of a certain deflationist conception of the (strong) modality
involved in logical truth.



The standard view of set-theoretic claims, however, does not see them
as strong modal claims--at best, some of them are modal in the
minimal sense that they are universal generalizations or particular
cases of these. But it is at any rate unclear that this is the basis
for a powerful objection to model-theoretic validity or to
derivability, for, even if we accept that the concept of logical truth
is strongly modal, it is unclear that a good characterization of
logical truth ought to be a conceptual analysis. An analogy might
help. It is widely agreed that the characterizations of the notion of
computability in standard mathematics, e.g. as recursiveness, are in
some sense good characterizations. Note that the concept of
comput*ability* is modal, in a moderately strong sense; it
seems to be about what beings like us could do with certain symbols if
they were free from certain limitations--not about, say, what
existing beings have done or will do. However, to say that a certain
function is recursive is not to make a modal claim about it, but a
certain purely arithmetical claim. So recursiveness is widely agreed
to provide a good characterization of computability, but it clearly
does not provide a conceptual analysis. Perhaps it could be argued
that the situation with model-theoretic validity, or derivability, or
both, is the same.



A number of philosophers explicitly reject the requirement that a good
characterization of logical truth should provide a conceptual
analysis, and (at least for the sake of argument) do not question the
usual view of set-theoretic claims as non-modal, but have argued that
the universe of set-theoretic structures somehow models the universe
of possible structures (or at least the universe of possible
set-theoretic structures; see McGee 1992, Shapiro 1998, Sagi 2014). In
this indirect sense, the characterization in terms of model-theoretic
validity would grasp part of the strong modal force that logical
truths are often perceived to possess. McGee (1992) gives an elegant
argument for this idea: it is reasonable to think that given any
set-theoretic structure, even one construed out of non-mathematical
individuals, actualized or not, there is a set-theoretic structure
isomorphic to it but construed exclusively out of pure sets; but any
such pure set-theoretic structure is, on the usual view, an actualized
existent; so every possible set-theoretic structure is modeled by a
set-theoretic structure, as desired. (The significance of this relies
on the fact that in Fregean languages a formula is true in a structure
if and only if it is true in all the structures isomorphic to it.)



But model-theoretic validity (or derivability) might be theoretically
adequate in some way even if some possible meaning-assignments are not
modeled straightforwardly by (actual) set-theoretic structures. For
model-theoretic validity to be theoretically adequate, it might be
held, it is enough if we have other reasons to think that it is
extensionally adequate, i.e. that it coincides in extension with our
preferred pretheoretic notion of logical truth. In subsections 2.4.2
and 2.4.3 we will examine some existing arguments for and against the
plain extensional adequacy of derivability and model-theoretic
validity for Fregean languages.


#### 2.4.2 Extensional Adequacy: A General Argument



If one builds one's deductive calculus with care, one will be able to
convince oneself that all the formulae derivable in the calculus are
logical truths. The reason is that one can have used one's intuition
very systematically to obtain that conviction: one can have included
in one's calculus only axioms of which one is convinced that they are
logical truths; and one can have included as rules of inference rules
of which one is convinced that they produce logical truths when
applied to logical truths. Using another terminology, this means that,
if one builds one's calculus with care, one will be convinced that the
derivability characterization of logical truth for formulae of the
formalized language will be *sound* with respect to logical
truth.



It is equally obvious that if one has at hand a notion of
model-theoretic validity for a formalized language which is based on a
minimally reasonable notion of structure, then all logical truths (of
that language) will be model-theoretically valid. The reason is
simple: if a formula is not model-theoretically valid then there is a
structure in which it is false; but this structure must then model a
meaning assignment (or assignments) on which the formula (or its
logical form) is false; so it will be possible to construct a formula
with the same logical form, whose non-logical expressions have, by
stipulation, the particular meanings drawn from that collective
meaning assignment, and which is therefore false. But then the idea of
formality and the weakest conception of the modal force of logical
truths uncontroversially imply that the original formula is not
logically true. Using another terminology, we can conclude that
model-theoretic validity is *complete* with respect to logical
truth.



Let's abbreviate "\(F\) is derivable in calculus \(C\)" by
"DC\((F)\)" and "\(F\) is a logical truth (in our
preferred pretheoretical sense)" by "LT\((F)\)".
Then, if \(C\) is a calculus built to suit our pretheoretic conception
of 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 is
the completeness of model-theoretic validity.



In order to convince ourselves that the characterizations of logical
truth in terms of DC\((F)\) and MTValid\((F)\) are extensionally
adequate we should convince ourselves that the converse implications
hold too:


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



Obtaining this conviction, or the conviction that these implications
don't in fact hold, turns out to be difficult in general. But a remark
of Kreisel (1967) establishes that a conviction that they hold can be
obtained sometimes. In some cases it is possible to give a
mathematical proof that derivability (in some specified calculus
\(C\)) is complete with respect to model-theoretic validity, i.e. a
proof 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 logical
truth, i.e., that the first implication of (5) holds. (Strictly
speaking, this is a strong generalization of Kreisel's remark, which
in place of "\(\text{LT}(F)\)" had something like
"\(F\) is true in all *class* structures"
(structures with a class, possibly proper, as domain of the individual
variables).) This means that when (6) holds the notion of
model-theoretic validity offers an extensionally correct
characterization of logical truth. (See Etchemendy 1990, ch. 11,
Hanson 1997, Gomez-Torrente 1998/9, and Field 2008, ch. 2, for
versions of this observation, and Smith 2011 and Griffiths 2014 for
objections.) Also, (6), together with (4), implies that the notion of
derivability is complete with respect to logical truth (the second
implication in (5)) and hence offers an extensionally correct
characterization of this notion. Note that this reasoning is very
general and independent of what our particular pretheoretic conception
of logical truth is.



An especially significant case in which this reasoning can be applied
is the case of standard Fregean first-order quantificational
languages, under a wide array of pretheoretic conceptions of logical
truth. It is typical to accept that all formulae derivable in a
typical first-order calculus are universally valid, true in all
counterfactual circumstances, *a priori* and analytic if any
formula
 is.[10]
 So (4) holds under a wide array of pretheoretic conceptions in this
case. (6) holds too for the typical calculi in question, in virtue of
Godel's completeness theorem, so (5) holds. This means that one
can convince oneself that both derivability and model-theoretic
validity are extensionally correct characterizations of our favorite
pretheoretic notion of logical truth for first-order languages, if our
pretheoretic conception is not too eccentric. The situation is not so
clear in other languages of special importance for the Fregean
tradition, the higher-order quantificational languages.


#### 2.4.3 Extensional Adequacy: Higher-order Languages



It follows from Godel's first incompleteness theorem that already
for a second-order language there is no calculus \(C\) where
derivability is sound with respect to model-theoretic validity and
which makes true (6) (for the notion of model-theoretic validity as
usually defined for such a language). We may call this result *the
incompleteness of second-order calculi with respect to model-theoretic
validity*. 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 (or *both* are): *either* derivability in
\(C\) is incomplete with respect to logical truth *or*
model-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 be
unsound with respect to logical truth. This is especially frequent in
philosophers on whose conception logical truths must be *a
priori* or analytic. One idea is that the results of *a
priori* reasoning or of analytic thinking ought to be codifiable
in a calculus. (See e.g. Wagner 1987, p. 8.) But even if we grant this
idea, it's doubtful that the desired conclusion follows. Suppose that
(i) every *a priori* or analytic reasoning must be reproducible
in a calculus. We accept also, of course, that (ii) *for every*
calculus \(C\) sound with respect to model-theoretic validity
*there is* a model-theoretically valid formula that is not
derivable in \(C\). From all this it doesn't follow that (iii)
*there is* a model-theoretically valid formula \(F\) such that
*for every* calculus \(C\) sound for model-theoretic validity
\(F\) is not derivable in *C*. From (iii) and (i) it follows of
course that there are model-theoretically valid formulae that are not
obtainable by *a priori* or analytic reasoning. But the step
from (ii) to (iii) is a typical quantificational fallacy. From (i) and
(ii) it doesn't follow that there is any model-theoretically valid
formula which is not obtainable by *a priori* or analytic
reasoning. The only thing that follows (from (ii) alone under the
assumptions that model-theoretic validity is sound with respect to
logical truth and that logical truths are *a priori* and
analytic) is that no calculus sound with respect to model-theoretic
validity can by itself model *all* the *a priori* or
analytic reasonings that people are able to make. But it's not
sufficiently clear that this should be intrinsically problematic.
After all, *a priori* and analytic reasonings must start from
basic axioms and rules, and for all we know a reflective mind may have
an inexhaustible ability to find new truths and truth-preserving rules
by *a priori* or analytic consideration of even a meager stock
of concepts. The claim that all analytic truths ought to be derivable
in one single calculus is perhaps plausible on the view that
analyticity is to be explained by conventions or "tacit
agreements", for these agreements are presumably finite in
number, and their implications are presumably at most effectively
enumerable. But this view is just one problematic idea about how
apriority and analyticity should be explicated. (See also Etchemendy
1990, chs. 8, 9, for an argument for the unsoundness of higher-order
model-theoretic validity based on the conception of logical truth as
analyticity *simpliciter*, and Gomez-Torrente 1998/9,
Soames 1999, ch. 4, Paseau 2014, Florio and Incurvati 2019, 2021, and
Griffiths and Paseau 2022, ch. 9, for critical reactions.)



Another type of unsoundness arguments attempt to show that there is
some higher-order formula that is model-theoretically valid but is
intuitively false in a structure whose domain is a proper class. (The
"intended interpretation" of set theory, if it exists at
all, might be one such structure, for it is certainly not a set; see
the entry on
 set theory.)
 These arguments thus question the claim that each meaning
assignment's validity-refuting power is modeled by some set-theoretic
structure, a claim which is surely a corollary of the first
implication in (5). (In McGee 1992 there is a good example; there is
critical discussion in Gomez-Torrente 1998/9.) The most
widespread view among set theorists seems to be that there are no
formulae with that property in Fregean languages, but it's certainly
not an absolutely firm belief of theirs. Note that these arguments
offer a challenge only to the idea that universal validity (as defined
in section 2.3) is adequately modeled by set-theoretic validity, not
to the soundness of a characterization of logical truth in terms of
universal validity itself, or in terms of a species of validity based
on some notion of "meaning assignment" different from the
usual notion of a set-theoretic structure. (The arguments we mentioned
in the preceding paragraph and in 2.4.1 would have deeper implications
if correct, for they easily imply challenges to all characterizations
in terms of species of validity as well). In fact, worries of this
kind have prompted the proposal of a different kind of notions of
validity (for Fregean languages), in which set-theoretic structures
are replaced with suitable values of higher-order variables in a
higher-order language for set theory, e.g. with "plural
interpretations" (see Boolos 1985, Rayo and Uzquiano 1999,
Williamson 2003, Florio and Incurvati 2021; see also Florio and
Linnebo 2021 and the entry on
 plural quantification).
 Both set-theoretic and proper class structures are modeled by such
values, so these particular worries of unsoundness do not affect this
kind of proposals.



In general, there are no fully satisfactory philosophical arguments
for the thesis that model-theoretic validity is unsound with respect
to logical truth in higher-order languages. Are there then any good
reasons to think that derivability (in any calculus sound for
model-theoretic validity) must be incomplete with respect to logical
truth? There don't seem to be any absolutely convincing reasons for
this view either. The main argument (the first version of which was
perhaps first made explicit in Tarski 1936a, 1936b) seems to be this.
As noted above, Godel's first incompleteness theorem implies that
for any calculus for a higher-order language there will be a
model-theoretically valid formula that will not be derivable in the
calculus. As it turns out, the formula obtained by the Godel
construction is also always intuitively true in all domains
(set-theoretical or not), and it's reasonable to think of it as
universally valid. (It's certainly not a formula false in a proper
class structure.) The argument concludes that for any calculus there
are 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 fundamental
problem is that this conclusion is based on two assumptions that will
not necessarily be granted by the champion of derivability: first, the
assumption that the expressions typically cataloged as logical in
higher-order languages, and in particular the quantifiers in
quantifications of the form \(\forall X\) (where \(X\) is a
higher-order variable), are in fact logical expressions; and second,
the assumption that being universally valid is a sufficient condition
for logical truth. On these assumptions it is certainly very
reasonable to think that derivability, in any calculus satisfying (4),
must be incomplete with respect to logical truth. But in the absence
of additional considerations, a critic may question the assumptions,
and deny relevance to the argument. The second assumption would
probably be questioned e.g. from the point of view that logical truths
must be analytic, for there is no conclusive reason to think that
universally valid formulae must be analytic. The first assumption
actually underlies any conviction one may have that (4) holds for any
one particular higher-order calculus. (Note that if we denied that the
higher-order quantifiers are logical expressions we could equally deny
that the arguments presented above against the soundness of
model-theoretic validity with respect to *logical truth* are
relevant at all.) That the higher-order quantifiers are logical has
often been denied on the grounds that they are semantically too
"substantive". It is often pointed out in this connection
that higher-order quantifications can be used to define sophisticated
set-theoretic properties that one cannot define just with the help of
first-order quantifiers. (Defenders of the logical status of
higher-order quantifications, on the other hand, point to the wide
applicability of the higher-order quantifiers, to the fact that they
are analogous to the first-order quantifiers, to the fact that they
are typically needed to provide categorical axiomatizations of
mathematical structures, etc. See Quine (1970), ch. 5, for the
standard exponent of the restrictive view, and Boolos (1975) and
Shapiro (1991) for standard exponents of the liberal view.)