Logical Knowledge

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LOGICAL KNOWLEDGE

"Logical knowledge" can be understood in two ways: as knowledge of the laws of logic and as knowledge derived by means of deductive reasoning. Most of the following is concerned with the first of these interpretations; the second will be treated briefly at the end. Furthermore, only deductive logic will be treated: As yet, there is no set of laws of inductive logic enjoying the kind of consensus acceptance accorded to deductive logic.

To begin with, we must specify what is a law of logic—not an entirely straightforward task. There are three, not all mutually exclusive, conceptions of logic laws. First, one could take them to be valid schemata (of statements), such as the familiar law of excluded middle, "p or not p ". A second conception is that they are valid rules of inference, such as the familiar modus ponens —that is, from "p →q " and p infer q. The third conception of logic law, due to Gottlob Frege and Bertrand Russell, takes them to be maximally general, true (not valid) second-order quantified statements (see Goldfarb, 1979). The following discussion is confined, by and large, to the second conception; but the philosophical problems canvassed arise with respect to the other conceptions as well.

In order to appreciate the problems involved in the analysis of knowledge of logical laws, note first that, however these laws are conceived, knowledge of them appears to be propositional. That is, to know a law of logic is to know that a rule of inference (or a schema) is valid (or a statement true). But, given the classical analysis of knowledge as justified true belief, it follows that knowledge of the validity of a rule of inference requires justification. There are two uncontroversially entrenched forms of justification: inductive and deductive justification. By the nature of inductive reasoning an inductive justification of validity shows, at best, that a rule of inference usually leads from true premises to a true conclusion (or that it is sufficiently highly likely to do so). This is too weak; a valid rule of inference, as noted above, necessarily leads from true premises to true conclusions. So it appears that the justification of validity must be deductive.

On the basis of this conclusion it can be shown that the justification of the validity of any rule of inference either is circular or involves an infinite regress. The argument has two parts. To begin with, there certainly are deductive justifications of rules of inference that raise no serious philosophical questions. Take the justification of the rule "existential specification" in Benson Mates's widely used Elementary Logic : "To justify this rule,… we observe that … we may … obtain the inference it permits [using certain basic rules] … Assuming … that the basic rules … are [valid], … the above description of how any [existential specification] inference can be made using only [those] rules … shows that [existential specification] is [valid], too" (Mates, 1972, p. 123). The rule is justified by explicitly assuming the validity of other rules, so the justification here is only relative. If all logical laws are justified in this way, then, plausibly, the justification of any given rule will be either circular, by explicitly assuming its own validity, or will involve an infinite regress.

One might conclude from this that there must be some set of rules that are not justified on the basis of the assumed validity of other rules. Let us call these rules fundamental. Unfortunately, there is a simple argument that the justification of fundamental rules will involve a similar circularity or infinite regress.

What counts as a deductive justification of a proposition depends on what forms of inference are taken to be valid. For, if any rule of inference used in an argument is invalid, then the argument could not constitute a deductive justification of anything. Let us formulate this point as: A deductive argument presupposes the validity of the rules of inference it employs. Given this formulation, we can state an intuitive principle: If an argument for the validity of a rule of inference presupposes the validity of that very rule, then the argument is circular. To distinguish this notion of circularity from the one used above, let us call this pragmatic circularity, and the former, direct circularity.

Suppose a fundamental rule of ρ is justified by an argument π. Now either π employs nonfundamental rules, or it does not. Suppose π employs a nonfundamental rule σ. By the first part of the argument, σ is justified by assuming the validity of fundamental rules. Again, either the justification of σ assumes the validity of ρ or it does not. Now assume further that if an argument employs a rule whose justification assumes the validity of another, then it presupposes the validity of the second. Thus, in the first case, the justification of ρ is pragmatically circular. In the second case, the justification of ρ presupposes the validity of a set of other fundamental rules.

Now suppose that π does not employ nonfundamental rules. Then, either it employs ρ or it does not. In the first case the justification is pragmatically circular. In the second, again, the justification of ρ presupposes the validity of a set of other fundamental rules. Hence, the justification of any fundamental rule either is pragmatically circular or involves an infinite regress. (See Goodman 1983, pp. 63–64; see also Bickenbach 1978, Dummett 1973, and Haack 1976.)

One might object to the notion of circularity of argument used in the second part of the argument. Unlike the more familiar variant of circularity, the conclusion in this case is not actually assumed as a premise but is presupposed by the inferential transitions. Thus, it is unclear that this sort of circular argument suffers from the principal difficulty afflicting the more familiar sort of circular argument, namely, that every conclusion is justifiable by its means.

This, however, is not a very strong objection. One might reply, to begin with, that pragmatically circular arguments are just as objectionable as directly circular ones in that both assume that the conclusion is not in question, by assuming its truth in the one case and by acting as if it were true in the other. Moreover, while it is unclear that every rule of inference is justifiable by a pragmatically circular argument, it is clear that such an argument can justify both rules that we take to be valid and rules that we take to be fallacies of reasoning. For example, the following is an argument demonstrating the validity of the fallacy of affirming the consequent (see Haack, 1976):

Suppose "p →q " is true.
Suppose q is true.
By the truth table for "→," if p is true and "p →q " is true, then q is true.
By (2) and (3), p is true and "p →q " is true.
Hence, p is true.

Second, one might accept that deductive justification is not appropriate for fundamental logical laws but conclude that there is another kind of justification, neither deductive nor inductive, for these laws. There have been two proposals about a third kind of justification.

One proposal, due to Herbert Feigl (1963), claims that fundamental logical laws require pragmatic, instrumental justification. An immediate difficulty is, What counts as a pragmatic justification of a logical law? Surely, if there is anything that a rule of inference is supposed to do for us, it is to enable us to derive true conclusions from true premises. So, it looks as if to justify a logical law pragmatically is to show that it is suited for this purpose. And that seems to require showing that it is valid. Feigl is aware of this problem and argues that, in the context of a pragmatic justification, circularity is not a problem, since all that such a justification is required to do is provide a recommendation in favor of doing things in some particular way, not a proof that this way necessarily works. It is not clear, however, that this constitutes a compelling response to the philosophical problem of justifying deduction, since, far from needing a letter of reference before employing deductive reasoning, its use is inescapable.

Another proposal for a third kind of justification is due to J. E. Bickenbach (1978), who argues that rules of inference are justified because they "fit with" specific instances of arguments that we accept as valid; for this reason he calls this kind of justification "instantial." The problem with this approach is that, in the case of rules of inference having some claim to being fundamental, such as modus ponens, it is plausible that we take the validity of the rule to be conceptually prior to the validity of any instance of it. For example, in the case of modus ponens, where there appear to be counterinstances to the rule, such as the sorites paradox, we take the problem to lie not in modus ponens but in vague concepts. Hence, whatever force "instantial" justification has, it seems incapable of conferring on fundamental rules of inference the kind of conceptual status we take them to have.

One might simply accept the conclusion of the argument, that fundamental logical laws cannot be justified, as indicating the philosophical status of these laws: They are simply constitutive rules of our practice of deductive justification. That is, there is no such thing as deductive justification that fails to conform to these rules, just as there is no such thing as the game of chess in which the queen is allowed to move in the same way as the knight. This third response leads to at least two philosophical questions: (1) How do we identify the fundamental laws of logic? (2) Is there such a thing as criticism or justification, as opposed to mere acceptance of a deductive practice?

A natural way to answer the first question is to take the fundamental rules to be determined by the meanings of the logical constants. This answer has been developed in some detail by Dag Prawitz (1977) and Michael Dummett (1991). Following Gerhard Gentzen (1969), they take the natural deduction introduction and elimination rules for a logical constant to be determined by the meaning of that constant. (More detail on the answer is provided in the final paragraph of this article.) Part of an answer to the second question has been provided by A. N. Prior (1967) and Nuel Belnap (1961), who showed that there exist sets of rules of inference that we can recognize as internally incoherent.

This third response has the consequence that our relation to the fundamental laws of logic is not one of knowledge classically construed and, hence, is different from our relation to other laws, such as the laws of physics, or of a country.

We turn now to the notion of knowledge derived from deductive reasoning. The question this notion raises, first studied by J. S. Mill (1950, bk. 2, chap. 3), is to explain how deductive reasoning could be simultaneously necessary and informative. It is undeniable that we can understand the premises and the conclusion of an argument without knowing that the former implies that latter; this is what makes it possible for us to gain information by means of deductive reasoning. This fact does not by itself conflict with the necessity of deductive implication, since there is no conflict between the existence of something and our lack of knowledge thereof. But, a problem can arise if the explanation of the necessity of deductive implication entails constraints on the notion of understanding. The following are two ways in which the problem of deduction arises.

First, consider Robert Stalnaker's (1987) analysis of the notions of proposition and of understanding. The proposition expressed by a statement is a set of the possible worlds, the set of those worlds in which the proposition is true. To understand a statement is to know the proposition it expresses; hence, to understand a statement is to know which possible worlds are those in which the proposition it expresses is true. These claims have two consequences: First, that all necessary statements, and hence all deductive valid statements, express the same proposition, namely, the set of all possible worlds; second, to understand any necessary statement is to know that the proposition it expresses is the set of all possible worlds. From these consequences it would seem to follow that in virtue of understanding any valid statement, one would know that it is necessarily true. It seems plausible that if one understands the premises and the conclusion of a valid argument, then one must also understand the conditional whose antecedent is the conjunction of the premises and whose consequent is the conclusion. But if the argument is valid, so is this conditional. Hence, if an argument is valid, then anyone who understood its premises and conclusion would know that this conditional expressed a necessary truth. It is now plausible to conclude that one can know whether an argument is valid merely on the basis of understanding its premises and conclusion by knowing whether the corresponding conditional expressed a necessary truth.

Next, consider Dummett's (1973, 1991) analysis of deductive implication. According to this analysis, deductive implication is based on the meanings of the logical constants. Thus, for example, the fact that p and q imply "p and q " is explained by the fact that the meaning of "and" is such that the truth condition of "p and q " is satisfied just in case those of p and of q are. Similarly, the meaning of the existential quantifier is such that if the truth condition of "a is F " is satisfied, then so must the truth condition of "There is an F ". Thus, corresponding to each logical constant, there is an account of the truth conditions of logically complex statements in which that constant occurs as the principal connective, in terms of the truth conditions of its substatements. This account explains the validity of rules of inference to those statements from their substatements and hence determines the set of fundamental rules, rules whose validity must be acknowledged by anyone who understands the meanings of the logical constants. But there are, as we have seen, cases in which we can understand the premises and the conclusion of an argument without knowing that the former implies the latter. So, how is deductive implication to be explained in those cases? This question is easy to answer if all the inferential transitions in these arguments are instances of fundamental rules determined by the senses of the constants. But the fact is otherwise; we acknowledge a number of rules of inference that are not reducible to fundamental rules. The problem is thus not an epistemological one; it arises because our conception of deductive implication includes rules whose necessity is not explainable on the basis of our understanding of the logical constants.