The Problem of Deduction
The problem of deduction can be stated as follows: in virtue of what does one proposition logically necessitate another proposition? Propositions have logical powers of entailment, but what is the nature of those powers? Compare Hume’s question: what is the nature of causal power? Events necessitate other events, but it is question how they do that—what does causal power consist in? Entailment is a relation between distinct propositions, a relation that holds necessarily, and we want to know what constitutes it. There are two sides to the problem: metaphysical and epistemological. What is the objective nature of the relation of logical consequence, and what kind of knowledge do we have of that relation? And it is a problem because there seems to be nothing for logical necessity to be. Like causal necessity it is undetectable by the senses—we have no “impression” of it (internal or external). We make logical inferences, as we make causal inferences, but we are at a loss to see what grounds them. Is this just a matter of habit with no rational foundation? Just as there is a problem of induction, so there is a problem of deduction; and in both cases necessity is the problematic notion. It seems impossible to locate the necessity that is required for deduction. Logical powers are as elusive as causal powers.
This difficulty gives rise to skeptical scenarios. What if we encounter aliens who reason very differently from us, taking as logical consequences what we deem to be a palpable non-sequiturs? What if they accept modus ponens as we do except when the subject matter of the propositions concerns (say) psychological states? What if they deduce as we do except after time t at which point they do something completely different (as we would describe it)? How can we demonstrate that they are reasoning incorrectly? What can we point to in their mind that shows that they are not reasoning as they used to? And how can we justify our own belief that we are reasoning correctly? What is it about our grasp of propositions that explains and justifies how we deductively reason? What if someone just doesn’t find the normal inferences self-evident and questions our habitual certainty? What fact about a proposition establishes that it has certain logical consequences and not others?  How do we even get the idea of logical inference (compare Hume on the origin of the idea of causation)? What does the necessitating that we talk so glibly about? Has anyone ever seen it? Examine a proposition on all sides and you will not find logical necessity lurking in it. A proposition is a string of concepts (or possibly objects), but where is the relation of logical necessitation there? How does a proposition reach outside of itself to another proposition that allegedly follows from it? This seems like a mysterious joining of distinct existences.
Nothing about the linguistic form of a proposition can explain the logical powers of the proposition: phonetics and syntax are not the basis of entailment. Meaning is what must play the role of logical necessitation, but what is it about meaning that plays this role? The problem goes beyond questions of indeterminacy: even if meaning is determinate we still don’t know how it manages to generate entailments. We lamely say “truth in virtue of meaning”: but what does that mean? What is it about meaning (whatever it is) that explains how one proposition can entail another? Complex structured entities don’t generally have entailments, so what is it about complex structured meanings that suits them to give rise to entailments? Meanings have constituent structure, but that doesn’t by itself explain how they generate inter-propositional relations of logical consequence. Lots of things have constituent structure. It is quite possible to do linguistics and not worry about entailment at all. Nor does psychology help: nothing in the mind or behavior can add up to logical entailment—not contents of consciousness or dispositions to infer. We think we can “see” that one proposition entails another, but we can’t say what we are seeing. Is it that we somehow see the shadow of the entailed proposition in the entailing proposition? Do entailments somehow lurk in the crevices of a proposition? What if you were entailment-blind and simply couldn’t see what a proposition entails—would you thereby be blind to the proposition? How can what is inside a proposition determine what is outside it? Can we picture the entailments of a proposition—see them with our mind’s eye? It is not that the given proposition literally contains the propositions it entails, so that in seeing it you see them; the idea is rather that your grasp of a proposition leads you to grasp its logical consequences. But how does a proposition lead you thus? Does it whisper those propositions to you? The entailments are forced on you, we feel, but the force that forces you is elusive. Propositions are complete in themselves (to paraphrase Hume on cause and effect), so how can they make reference to other propositions that lie outside of them? Yet they do in the sense that they have logical entailments; we just can’t see what enables them to. Our deductive practices thus seem devoid of a foundation, as if they hang in thin air. Even in the simplest of cases, such as conjunction elimination, we can’t see how entailment works: how does “and” contrive to produce the consequences it does? Not by its form but by its meaning: but how does its meaning license its entailments? Don’t say its meaning is defined by its entailments, because the question is how meaning and entailments are related: how can the meaning of “and” generate the entailments of propositions containing it? What kind of thing could that meaning be? The problem concerns the metaphysics of meaning (propositions) and we lack a conception of meaning that explains how deductive consequence works (just as we lack a conception of causation that explains how causation works). What is meaning such that logical consequences flow from it?
The temptation is to adopt a “constant conjunction” view of entailment: there are no intrinsic necessary connections between propositions (compare the regularity theory of causation) but we find that truth-values are constantly conjoined, and that is all there is to it. Whenever one proposition is true certain others are, but it is not that the former proposition has the power to produce truth in other propositions; there is no such inter-propositional necessitation, merely a general coincidence of truth-value. We mistake this coincidence for an actual connection or linkage or dependence. Propositions don’t entail each other; they simply agree in truth-value in regular ways. It is just that whenever a proposition of the form “p and q” is true p is true and q is true, but the former proposition doesn’t make the other propositions true. What we call entailment is just a word for regular agreement in truth-value (there is no consequence). But this kind of position, though tempting, is massively implausible and only attracts us because we find it hard to ground deduction in anything substantial. As so often in philosophy a common sense platitude is challenged by a type of skepticism that undermines our habitual confidence. Deduction is no more immune from this than anything else.
 I am alluding here to Wittgenstein’s discussion of meaning and rule following in Philosophical Investigations and Kripke’s treatment of it in Wittgenstein on Rules and Private Language. I assume this as background to my remarks here. Doubts about deduction arise naturally from more general doubts about meaning. See also my “Knowledge of Entailment”.