Butcan we do as we please? We can certainly stipulate an arbitrary definition for “logic,” just as we could stipulate that henceforth we will use the word “giraffe” to designate cows. But stipulating a meaning for “logic” is not the same as saying what logic is.
Still, it may seem mysterious why an investigation of the nature of logic should require detailed attention to our traditional conceptions of logic. After all, one does not investigate the nature of gold by attending to our predecessors’ conceptions of gold. Our predecessors believed all kinds of false things about gold (e.g., that it could be obtained by alchemical transformation from many other substances). And they lacked any grasp of what we now regard as thedefining feature of gold: the number of protons in the nucleus of gold atoms. Studying our predecessors’ beliefs about the essence of gold is next to worthless if we want to know what distinguishes gold from other substances. If we really want to know what gold is, we need to studygold. If we want to know what logic is, then, shouldn’t we studylogic, rather than conceptions of logic? Isn’t there every reason to believe that our predecessors’ conceptions of logic are as unreliable as their conceptions of gold?
This apparently cogent line of argument slurs over an important disanalogy between
21For example, in thePrinciples of Mathematics, Russell claims to have refuted Kant’s view that
mathematical proof requires non-formal reasoning (1903:§4,§§433-4), but the issue has only been joined if Russell means the same thing as Kant did (or close enough) by “formal.”
the concept of gold and the concept of logic. Because gold is a natural kind concept, our grasp of it (even our best experts’ grasps) can be partial and confused. As long as we can identify some paradigm samples of gold, and as long as we conceive of gold as a single substance, all of whose instances behave similarly in the natural order, we have managed to attach our thought togold (and not to, say, the disjunctive kindgold-or-pyrite or the phenomenal kind golden-colored metal).22 Nature herself makes up for the slack in our collective understanding. That is why we can investigate the nature of gold without worrying too much about how our predecessors conceived of it. All we need to know is that we are experimenting on and theorizing about the very same stuff to which they applied the conceptgold.
Investigating the nature of logic, I suggest, is not like this. Logic is not a natural kind concept. It does not play a role in laws of nature, and so the natural order of the world cannot take up the slack between our ways of grabbing onto the concept and the concept itself. Thus we must attend more closely to the ways in which our predecessors marked out the subject if we want to ensure that we are talking about the same thing. This is not to say that there is no room to criticize our predecessors’ conceptions. We want to be able to say that our predecessors were wrong (at least in part) about both the scope and the essential characteristics of logic. But our model for such criticism cannot be our criticism of our predecessors’ conceptions of gold. The concept oflogic, I suggest, is more usefully compared withlegal concepts such asnegligence,property, orcontract. The correct application of these concepts requires much more sensitivity to past usage and past theory than does the correct application of gold. That is why judges must attend studiously to precedent. And that is why an investigation of the nature and bounds of logic must attend to the tradition of demarcating logic.
A philosopher with platonistic leanings in the philosophy of mathematics might object that it is chauvinistic to restrict natural kinds to kinds studied by the empirical sciences. Surely we can distinguish “natural” and “unnatural” kinds in mathematics, too. It is widely
held, for instance, that the convergent analyses of effective calculability given by G¨odel, Kleene, Church, and Turing amount to adiscovery of the nature of effective calculability, in much the same way as the atomic theory made possible the discovery of the nature of gold. Might not mathematical results also show us how to demarcate logic? Kneale and Kneale 1962 suggest that G¨odel’s incompleteness results show that “logic extends no further than [first-order] quantification theory”:
When Frege wrote, the scope of logic had not been delimited precisely, and his [logicist] thesis seemed plausible just because the reader could then make an easy transition in thought from quantification theory to the theory of sets and arithmetic. But G¨odel has revealed a profound difference between quantification theory, which is complete, and the theory of sets, which is not. In the interests of clarity it therefore seems best to reserve the name “logic” for the former, and this is in fact what most mathematicians do when they are engaged upon their ordinary concerns. (724; cf. 741)
Perhaps, then, it is the mathematical logician, not the philosopher of logic, who is best placed to tell us what logic is—just as it is thechemist, not the philosopher of science, who is best placed to tell us what gold is.
I do not want to deny that technical results like G¨odel’s can reveal natural conceptual joints. But one should not overplay the analogy between chemistry and mathematical logic. The concept of logic plays as important a role in philosophy as it does in mathematics (perhaps a more important role). So although mathematical results are relevant to the demarcation of logic, they cannot bear the whole burden. Even the Kneales appeal to the tradition (specifically the tradition of taking the logical enterprise to be that of “classifying and articulating the principles of formally valid inference,” 739) to support the demarcation they favor (741).
One might still resist my conclusion that an intelligent, principled demarcation of logic must be grounded in a thorough study of thehistory of conceptions of logic. But what are the alternatives? If we do not look to history, how do we know when we have gotten the right demarcation? I see only two possible replies, and neither, I will argue, is satisfactory. The first reply is to reject the question. On this approach, there is no such thing as the
“right” demarcation. One can construct any concept of logicality one wishes—simply by stipulation. None is any better or worse than any other, except in relation to a particular purpose. That we call them concepts of “logicality” has only psychological significance.
To see what is wrong with this reply, try substituting “negligence” for “logicality.” Of course we can stipulate various meanings for “negligence,” but if that’sall we can do, then we lose something very important: continuity of subject matter with our intellectual (and legal) predecessors. We want to be talking about the same thing they were talking about, so we can profit from or correct their reasoning, and no amount of stipulation can ensure that we are doing so. Continuity of subject matter is especially important in connection with logicality, because the reason we care about the concept of logicality is the role it plays in debates in philosophy and the foundations of mathematics—ongoing debates, with histories. As Sellars 1967 puts the point,23
The history of philosophy is thelingua franca which makes communication be- tween philosophers, at least of different points of view, possible. Philosophy without the history of philosophy, if not empty or blind, is at least dumb. (1)
The second reply is to invoke our intuitions about logicality as the standard against which we judge proposed demarcations. One can find such appeals to intuitions in much work on the demarcation of logic. For example, Sher 1991 writes:
The distinction between logical and extralogical terms is founded on our pre- theoretical intuitions that logical consequences are distinguished from material consequences in being necessary and formal. To reject this intuition is to drop the foundation of Tarski’s logic. To accept it is to provide a ground for the division of terms into logical and extralogical. (51, emphasis added)
Feferman objects that Sher’s criterion for logicality assimilates logic too much to mathe- matics, adding that the persuasiveness of his objection “. . . will evidently depend on one’s gut feelings about the nature of logic . . . ” (A:11). Are we reduced, in the end, to weighing one person’s intuitions against another’s gut feelings?
This methodology is ultimately not very satisfying. Our intuitions about logicality are not a kind of perception of an extramental reality: they are historical artifacts, a product
of our logical and philosophical educations. To the extent that there is intersubjective agreement about them,24 it should be attributed to a shared tradition, not access to a tradition-independent reality. As Stewart Shapiro and others have pointed out, the very idea of “pretheoretical logical intuitions” is dubious. Students beginning an introductory logic class typically have inferential intuitions, but they can be brought to distinguish logically valid inferences from materially valid ones only by instruction. All of our intuitions about logicality bear the stamp of theory. If we have the intuition that logic must be “formal,” this is not because of some kind of extrasensory perception of the essence of logic, but because we have encountered the idea so often in the course of our philosophical training.
This is not to say that we should ignore all of our intuitions about logicality. But before invoking these intuitions in justifying or criticizing a proposed demarcation of logic, we ought to seek their sources in the philosophical tradition. By studying the history of conceptions of logicality, we can seewhy philosophers have the intuitions they do. Knowing this, we can proceed to ask which intuitions we still have reasons to have, which go together, and which come from incompatible traditions. Historical reflection is a way to make our “brute intuitions” less brute.25
It is remarkable how far back one needs to go to achieve the kind of historical under- standing I am describing. Twentieth century logicians (e.g., Russell and Tarski) often invoke “formality” as a criterion of logicality without saying much about what it means or why it is an appropriate criterion to use in characterizing logic.26 Responsibility for these tasks is implicitly deferred to a prior (unspecified) tradition. In what follows, I will be arguing that we cannot get clear about the intuitions that guide contemporary debates about the demarcation of logic unless we go all the way back to Kant.
Adapting the Kantian slogan to yet another purpose, we might say: “intellectual history without conceptual analysis may be empty, but analysis without history is blind.”
24And perhaps there is not much: as Warmbr¯od 1999 points out, different philosophers have very
different intuitions about logicality (513).
25As Ian Hacking says (in another context): “The ‘fly-bottle’ was shaped by prehistory, and only
archaeology can display its shape” (1973:188).