QUINE ON LOGIC
In ‘Two Dogmas of Empiricism’ (From a Logical Point of View 2nd Edition, Harper & Row 1961, pp. 2046), one of the dogmas Quine attacked was the view that sentences can be divided into two nonempty, mutually exclusive and exhaustive classes– the analytic and the synthetic. (This doctrine was utilised by the positivists in an attempt to explain why logic and mathematics are respectable and metaphysics not, even though the propositions of logic and mathematics seem as little open to empirical confirmation or refutation as those of metaphysics.) One type of supposed analytic truth, exemplified by ‘all bachelors are unmarried men’, was characterised as the category of truths resulting from logical truths (e.g. ‘all unmarried men are unmarried’) by substitution of synonyms for synonyms (‘bachelors’ for ‘unmarried men’). Quine denies the existence of such analyticities because of his doubts as to the scientific respectability of the concept of synonymy.fn1 We will return to this scepticism later but, even if it is justified, it leaves open the possibility that the logical truths themselves, such as ‘all unmarried men are unmarried’, are analytic.
Nonetheless Quine rejected this view also.fn2 He denies, firstly, that logical truths are conventional in a sense in which other truths are not: anything which could reasonably be described as a set of human conventions must be finite, he says, whilst there are infinitely many logical truths; hence logic is needed to derive the logical truths from the finite conventions, thus implicating the conventionalist approach in vicious circularity. Secondly, he considers the characterisation of logical truths as ‘true in virtue of the meaning of the logical words’ as correct, on the only reading of this phrase he can make sense of, but trivially so. For p to be true in virtue of circumstances C can only mean, for Quine, that Tc entails p, where Tc is some theory describing the circumstances. Now let M be some meaning theory and p be a logical truth like ‘If the table is brown then the table is brown’. Certainly M entails p. But so does O where O is any other theory one likes– horticultural, historical, true or false; for in classical logic, logical truths are entailed by anything. So there is nothing especially linguistic about the truth of logical truths. (See Quine’s Philosophy of Logic p. 96.)
Here Quine interprets ‘s is true by meaning alone’ to mean that the truth of sentence s must only be entailed by a meaning theory and not by any other type of (consistent) theory. But there is another, arguably more natural way, of construing ‘true by meaning alone’:– s is true by meaning alone iff a true meaning theory M entails, on its own, that s is true (or entails that p, where s is a name of p). By ‘on its own’ is meant: without need of augmentation by any other theory.fn3 On this account, it is irrelevant that a physical theory T entails ‘If the table is brown then the table is brown.’, the important point is that a true meaning theory also does so by itself. Similarly (if Quine’s scepticism about synonymy is unjustified) such a theory M will also entail, on its own, propositions such as bachelors are unmarried men, vixens are female foxes etc., propositions which are not entailed by theories in general. But it will not entail, by itself, that the table is brown; so this latter proposition is not analytic.
Relative, then, to agreement on the background logic for the metalanguage we have a sharp distinction between analytic and synthetic sentences. For any sentence s, s is analytic iff s is derivable from a true meaning theory M for the language of s. Two questions immediately present themselves here– what is a meaning theory and what is the correct background logic? For instance, unless we have ‘if p then p’ as a theorem of our background logic, we will not be able to derive the truth of the instance of the theorem above, i.e. with "the table is brown" instantiating p. How do we decide what is the correct logic? In ‘Two Dogmas of Empiricism’, Quine maintains that this is a pragmatic matter entirely on a par with (indeed part of) choosing the correct empirical theory of the world. The right logic is the one which figures in the best empirical theory. So even if he agreed that logical truths are analytic, in the above sense, he might claim it is an empirical matter which are the analytic truths! For Quine, even the laws of logic and mathematics are revisable in the light of experience, though because of their centrality in our belief scheme we are more reluctant to revise these sentences than nonlogical ones:
Any statement can be held true come what may … no statement is immune to revision
Two Dogmas of Empiricism op. cit. p. 43
But this ‘Two Dogmas’ view is arguably incoherent, at least if we generalise it to apply to logical inference rules as well as to logical theorems. This generalisation seems entirely in the spirit of Quine’s ‘logic empiricism’ and yields the Revisability Thesis:
any inference rule can be nontrivially revised so that one fails to accept it where one once accepted it, or vice versa.
A trivial revision occurs when there are terms in the rule which have a determinate meaning when the rule has its initial status and a distinct determinate meaning when its status is changed– when it is no longer accepted– or has become accepted where once it was not. The qualification that the revision in question must be nontrivial is necessary of the revisability thesis is not to be utterly uncontentious. (Everyone excepts that e.g. ‘0=1’ could come to be accepted if ‘=‘ comes to mean ‘less than’.)fn4
The Revisability Thesis, is, however, incoherent, or at least inconsistent with other very plausible assumptions which Quine himself holds. To see this, suppose T is a scientific theory one holds today and some sentences of T contain a symbol * which is a logical sign in the sense that one accepts various rules of inference involving *. Suppose A is an observational prediction one derives from T using the rules for * and that B cannot be derived by means of the rules one currently accepts. By altering the rules of inference (including the structural rules which lay down which ways of chaining inferences together are permissible) one will always be able to reverse this situation so that B is derivable using the new rules and A is not. If Quine is right, there will be a way of effecting such a change of rules which is nontrivial: which does not consist simply in associating new meanings with logical words.
Quine is faced with a problem. What are the empirical consequences of T? If any sentence derivable by some system we could nontrivially adopt is a consequence, then all sentences are consequences of T, i.e. both A and B will be derivable by some system of rules or other. If only sentences derivable by every system of rules we could nontrivially adopt are consequence then T has no consequences. Either way, no two theories have distinct empirical contents which is disastrous for Quine because in his view empirical content is the only content a theory can have.
So Quine has to discriminate among the various sets of inference rules and explain why some generate genuine consequences, others do not. But he faces a dilemma. He might, as a holist, say that we should consider theories T and possible logics L together, not in isolation, and class áT,Lñ as the correct pair of theory plus logic if the resulting combination of T plus its set of observational consequences O (as determined by logic L) makes best empirical sense of the totality of all true observation sentences. This fits in well with his holism but not with his empiricism about language understanding. For there is nothing to say that the logic or logics which come out right on this criterion will bear any relation to the logic we actually use today with respect to theory T. This extreme holist theory utterly divorces the notion of logical consequence from our actual linguistic practice with respect to logical words in a way which is unacceptable to anyway who thinks language understanding must have something to do with our verbal behaviour.
Alternatively, Quine might relativise logical consequence to time: he might say the inference rules we find actually use today give us the logical consequences_{today} of T. To be sure, T might have different logical consequences_{tomorrow}, but it has a determinate set of consequences today. On this view, as well as the dispositions which ground the rules we currently accept we also have dispositions to alter those dispositions: we might under certain circumstances acquire different dispositions. fn5
But now Quine is discriminating between dispositions to change dispositions. Some future dispositions one is disposed to acquire do form part of one's current understanding of certain terms (observation categoricals); but the future dispositions to adopt certain new rules involving the connectives do not. For if they did, the logical consequences_{tomorrow} would be every bit as much logical consequences (simpliciter) of T as the logical consequences_{today}. In other words, such changes to the new rules must be trivial changes, contrary to the revisability thesis that any system of rules could be nontrivially adopted or abandoned. Hence if Quine’s logic empiricism applies not only to purported theorems but also to the logically more fundamental notions of inference and consequence, then it is simply untenable and inconsistent with other assumptions he makes.
Quine’s later views seem to oscillate between the complete opposite position to the revisability one– namely that every change of logical rules brings a change of meaning– to a more moderate position in which only some aspects of our use of a logical word are essential to its meaning. This is his verdict function theory of the logical connectives first found in §13 of Word and Object. The meaning of the connectives is standardly given by the truth tables. But what is it for a speaker to understand a symbol as expressing a truthfunctional connective, as represented in the truth tables? As in the case of stimulus meaning, Quine tries to get at truth conditions via the more behaviourally tractable notion of believetrue conditions, or assent/dissent conditions. The basic idea is that if we find a community using a connective *, we attempt to find out its verdict table, the pattern of assents and dissents given to (comprehensible) complex * sentences along with their immediate constituents, when the complex sentence and its constituents are taken closely together. If assent matches T, dissent F, as they occur in the truth table for a connective C, then * expresses C in that language. But there is a problem:– there is a third possibility in addition to assent A and dissent D, namely being undecided U. And such indecision seems to destroy functionality unique output values for each possible set of input values. For instance, suppose we are undecided on P = ‘the baby will be a boy’, Q = ‘the baby will be a girl’ and R = ‘the baby will be a redhead’. Then we will be undecided on P&R and Q&R, and also on PÚR and QÚR. Thus in this case input <U,U> gets U as output. But P&Q gets dissent as its response– <U,U> to D– whilst PÚQ gets assent– <U,U> to A– so that functionality fails. There is no unique output for <U,U> as input.
Quine thinks this shows a certain indeterminacy in the meanings of the connectives, but it is arguable that if one considered, as determining the meaning of the operators, reactions not involving uncertainty though perhaps still including agnosticism due to vagueness or other types of indeterminacy (I) such as reference failure, no failure of functionality will result. This might be achieved by considering patterns to logical words and constituents drawn solely from the subjective observational sector of language i.e. the sector containing sentences such as ‘A rabbitlike thing is present’ rather than ‘A rabbit is present’ etc. One might then find the following verdict tables for & and Ú:



Q 




Q 


P&Q 
A 
I 
D 

P Ú Q 
A 
I 
D 

A 
A 
I 
D 

A 
A 
A 
A 
P 
I 
I 
I 
D 
P 
I 
A 
I 
I 

D 
D 
D 
D 

D 
A 
I 
D 
with I being the indeterminate value in the only three cases not fixed by the principle that conjunctions get assent when both conjuncts get assent and dissent when at least one gets dissent (disjunction likewise with assent interchanged with dissent; in each case, the above table is only one of the 27 possibilities compatible with that principle). For negation there are the three possibilities below.
P 
~P 
P 
— P 
P 
¬ P 
A 
D 
A 
D 
A 
D 
U 
U 
U 
U/A 
U 
U/D 
D 
A 
D 
A 
D 
A 
Now what happens when we apply these connectives to the language in general where, in contrast with the situation in the subjective sector, we can differentiate between agnosticism due to uncertainty as to the facts and agnosticism (I) due not to ignorance but due rather to the presence of an indeterminate or borderline case– a blob neither determinately red nor not red, for example. It can be shown that the AUD tables below, where U indicates absence of definite assent or dissent for whatever reason, are the only ones compatible with the AID verdict response tables above (cf. The Roots of Reference p. 77):–



Q 




Q 


P&Q 
A 
U 
D 

P Ú Q 
A 
U 
D 

A 
A 
U 
D 

A 
A 
A 
A 
P 
U 
U 
U/D 
D 
P 
U 
A 
U/A 
U 

D 
D 
D 
D 

D 
A 
U 
D 
In other words one is allowed no leeway in one’s response to a conjunction in the objective sector of language except when one is not prepared to be definite about either of the conjuncts. In that case one cannot assent, if one understands conjunction according to the AID tables, since we can see from these that assent to a conjunction implies assent to both conjuncts. But dissent is not ruled out (nor, obviously, agnosticism) since dissent to a conjunction does not determine, in the AID tables, the response to either conjunct.
But how might one use the tables to show that the analyticity of logic is not an empirical matter? One idea might be that if a sentence must be given verdict A, on pain of misunderstanding connectives according to the verdict tables, then it is analytic. But this will not do since it is easy to see that the verdict tables are compatible with giving U to all sentences, granted one gives U to all atomic sentences. Indeed the verdict tables allow one to give D to a logical truth such as (PÚ~P) & (PÚ~P), since one can give U to (PÚ~P), if one is indecisive on P, and so D to (PÚ~P) & (PÚ~P) since it is a conjunction both of whose conjuncts are given the response U. One could say that one is not allowed to give U to a conjunction whose conjuncts one is undecided on if the two conjuncts are incompatible, as the baby will be a boy and the baby will be a girl are. And P and ~P are incompatible. But one could not say this if one was attempting to explain the nature of logic and so of logical consequence and incompatibility as grounded in the verdict tables; for then the explanandum, the thing to be explained (e.g. logical relations such as incompatibility), would also be in the explanans, the thing doing the explanation.
This negative result does not mean, however, that the verdict table theory of meaning for the connectives lays no constraints down as to which logic is correct. For one can ask not which sentences must receive assent, and so be counted as laws (answer none, as we have seen) but which inference rules we must conform to. Such rules– analytic rules– we can count as sound in virtue of the meaning of the connectives. Suppose, for instance, that we said that a rule is analytic iff it is not possible to give a higher verdict (on the scale D < U < A) to the premiss than to the conclusion. Then one can check that the system of rules below is analytic.
P&Q 

P&Q 

P 

Q 

~ ~ P 

P 
P 

Q 

P Ú Q 

P Ú Q 

P 

~ ~ P 
The logic which results from this definition of analytic validity is, however, exceedingly weak. If, on the other hand, one lays down the following definition of what it is to abide by an inference rule (for a multiple conclusion system of logic):
a) Whenever one assents to all the premisses and dissents to all the conclusions but C then one assents to C.
b) Whenever one dissents from all the conclusions and assents to all premisses but P then one dissents from P.
then, on certain (nontrivial, and arguably not universally applicable) assumptions about the results of chaining together sequences of inference steps– on certain assumptions about what the correct structural rules are, that is– the result is classical logic. Hence there are grounds for thinking that logic is not an empirical matter, that certain operational rules are analytic, operational rules being nonstructural rules such as those in they system given above. Given the requisite structural rules (but here large problems arise as to how one decides which structural rules are right) use of these analytic operational rules in the background logic of a meaning theory enables one to show that certain truths are true in virtue of meaning alone.
fn1: In fact Quine accepts that ‘bachelor’ is stimulussynonymous with ‘unmarried man’ for each speaker; however even on the Word and Object account, ‘bachelor’ has no shared meaning common to all speakers, since it is not observational.
fn2: Particularly in earlier papers such as ‘Truth by Convention’ and ‘Carnap on Logical Truth’, collected in The Ways of Paradox.
fn3: An additional consideration is the following: the traditional account is of the form Meaning Theory M entails that "if the table is brown then the table is brown" is true; and the entailed sentence is not a logical truth, even though the sentence quoted in it is.
fn4: Most radically, Quine might hold that all change of logical rules is nontrivial because there is no such thing as the meaning of a logical word, so no such thing as a logical word expressing a different meaning today from yesterday. But the doctrine that any system of rules can be changed nontrivially to any other does not depend on such a radical scepticism about meaning which anyway Quine abandons in his verdict matrix theory, as we shall see.
fn5: Such higherorder dispositions to form lowerorder dispositions are the very stuff of Quine’s theory of perception and of observation categoricals; cf. Roots of Reference pp. 18, 6667.