Toward the Pragmatic A Priori
In “The Structure of Logic and Its Relation to Other Systems” (1921), Lewis finally frees himself of Royce’s re-instatement through denial conception of logic. The main ground of Lewis’s rejection of this view is the claim that re-instatement through denial fails to distinguish between “good” and “bad” systems of logic. Lewis justifies this claim with an example of
a bad logic in which, reasoning badly according to our bad principles, we always get consistently bad results. And if we deny one of these principles, still by sticking to our bad method of reasoning, we can reaffirm the bad principle in conclusion.
(1921,507-508)
The example in fact is a
family of . . . systems . . . determined by the presence in the system of the proposition
(t) q<[p<(p<
where p, q, etc., are propositions, and p < q represents “p implies q,” or “if p is asserted, q may be asserted.” . . . Some of the systems in this family might be regarded as “good” logic, but most of them are “bad.” Such a “bad” logic may be developed logistically from the following formal postulates:
|
A. |
-(-p) = P |
|
B. |
-(p < -p) |
|
C. |
(p |
|
D. |
Ip < (q < r) < [q < (p < r)] |
|
E. |
(<7 < r) < [(P < q) < (P < r)] |
|
F. |
(P < q) < (-P < -q) |
(Def. of -p, the denial of p)
[Contraposition]
[Permuting Antecedents]
[Transitivity]
[Denying Antecedents]
Postulate F is obviously false as a general law of implication. It is interesting that postulate B seems to exclude the possibility that any proposition should lead to its own denial as a consequence, yet if P be any principle of the system, we can prove that -P < (-P < P). Hence the assertion of -P leads to the assertion (-P < P) (1921, 508n3; names of axioms in square brackets mine).
It’s not clear what exactly Lewis has in mind. Here’s one way of filling in the details. Assume that the rules of inference of this system, call it E, are modus ponens and substitution, as in the systems developed by Lewis through Survey. Then a substitution instance of Denying Antecedents is
(7) (-q < -p) <(--q<- -p)
Contraposition, Transitivity, and (7) then yield
(8) (p < q) < (--q<--p)
Given A and the definition of negation, (8) leads to the principle that implication is symmetric.
(9) (p < q) < (q < p)
Permuting the Antecedents of (9) yields
(10) <7 < ((p < q)
Now applying the symmetry of implication to the consequent (10) gives (:£).
Once we have (£) we get reaffirmation through denial of all the axioms. Let A be an axiom. An instance of (^) is
(11) A < (-A < (-A < A))
Since A is an axiom, from (11) we derive
(12) -A < (-A < A)
Now, from (12), any assertion or assumption of-A allows the derivation, by modus ponens twice, of A. This Lewis takes to be the re-instatement of A from its denial.
One might naturally raise a question of what exactly being a “bad” system of logic amounts to. It’s not clear whether Lewis can provide an answer in pragmatic terms or show that Z is a “bad” logic on the basis of that answer. But this is not critical for Lewis’s criticism to be effective. The crucial point against Royce is that more than one system exhibits reinstatement through denial of its axioms. This result cuts against Royce’s claim that the re-instatement criterion establishes the “absolute” truth of a unique set of logical principles.
However, when we look at Lewis’s derivation, it becomes less than clear that he has demonstrated the re-instatement through denial of these “bad” axioms. To begin with, note that “axiom” B is not used at all in these derivations, so Lewis’s re-instatement derivation doesn’t depend on it.18 It seems at first glance that what makes the derivation possible is the “axiom” F, Denying Antecedents, which evidently expresses a principle of inference we intuitively regard as fallacious. A second glance, however, puts this in question. Let’s consider Z as a system of material implication. Then the principle (|), which Lewis singles out as the main lemma of the re-instatement derivation, is truth-functionally equivalent to the valid schema “-p V -q V q." Hence its instance, (11), is equivalent to “-A V A.” Moreover, the principle immediately leading to re-instatement, (12), is equivalent to A. Thus, the derivation of the re-instatement principle (12) from (11) and A’s being an axiom is equivalent to the inference of A from -A V A and —A. This suggests that, from the perspective of material implication, the re-instatement derivation of A in Z has in fact nothing to do with the assumption of -A: the final step of the derivation is equivalent to an inference of A from A and the assumption -A, which would go through even in the absence of -A. Put otherwise, an axiom of a system is derivable in that system, no matter what else, if anything, is assumed. No denial need be involved, so no re-instatement is achieved.
How do things stand, then, for Lewis’s rejection of Royce’s criterion of logical truth? It seems to me to be something of an impasse. Royce would naturally claim that the re-instatement criterion is meant to be the ground for picking out the axioms of logic, while what Lewis has shown is no more than the fact that any (consistent?) arbitrarily selected set of propositions may be treated as the axioms of a system, and so any member of that set is derivable in that system even if its negation is assumed. But in response Lewis would surely ask, if it’s not a matter of being derivable from its negation, what exactly does re-instatement through denial amount to?
I cannot here discuss the prospects of advancing beyond this aporia, but will leave the reader with a suggestion. The way of understanding the re-instatement of the difference between affirmation and denial outlined at the end of the last section might provide Royce with an answer to the question I just now posed on Lewis’s behalf. That answer would not be decisive, but it points the way to advancing a conversation between Royce and Lewis on the limits of pragmatism for the a priori.
