On the Problem of Deduction and Induction
I agree with C&C that deduction is not problematic, but I disagree with C&C that deduction is just as problematic as induction. In order to argue for this position, I will first explain Hume’s problem of induction. Then I will present a similarly framed problem for deduction. I will argue that neither induction nor deduction can be justified in a non-circular way. But I will also argue that there is a difference between the two forms of reasoning that makes the justification of one problematic, but not the other: namely, that it is impossible to think of counterexamples to the inference rules of deduction, whereas it is possible to do so for induction. The upshot I will draw from this is that some things can be rationally self-evident—that is to say, justified without further justification.
How can inferences be justified? According to Hume, “All the objects of human reason fall naturally into two kinds, namely, relations of ideas and matters of fact ” (p. 11) . Relations of ideas are demonstratively certain, which seems to mean that it is deductive—the conclusion is logically entailed and the premises guarantee its truth. That is to say, it is impossible for the conclusion to be false if the premises are true. Matters of fact, on the other hand, which Hume defines as: “propositions about what exists and what is the case” (p.12) are inductive—the premises in an inductive reasoning only provide a degree of support (or likelihood) to the truth of its conclusion and it is nonetheless possible for the conclusion to be false even though all the premises were true.
Inductive inferences rely on past observations, as evidence that what has happened in the past will similarly happen in the future. Generally, an inductive inference goes along the following lines: “we have observed properties A and B, and some past observations confirms that particular thing A had property B. Many further observations also confirm this. Therefore, all A’s are B (or at least it is likely that the next A will be B).” Hume’s problem is that: if all A’s observed so far are B, what justifies us in inferring that unobserved A’s are B? Observational data does not entail inductive inferences without circularity. Inductive inferences rely on past observations, as evidence that what has happened in the past will similarly happen in the future is presupposing the validity of the inference. Hence, it is circular. Furthermore, a counterexample to inductive inference is the possibility that the course of nature may change, and so past observations will prove to be no longer, if at all, reliable. The problem here is that there appears to be an intermediate proposition that cannot be justified rationally. To put it in context, consider the following:
P1: I have tasted countless lemons in the past, and I have observed that all of them were sour.
C: All lemons are sour. (Implies: The next time I see a lemon I know it will likely be sour.)
The inference claims that based on previous experience and observation, we can foresee that the future observation will be similar to the past. However, our past experience is not quite a reliable support. The intermediate proposition that cannot be justified rationally is that past observations of lemons linked to future observations of lemons are unwarranted. I can only justifiably assert that all the lemons I have tasted in the past were sour, but I cannot justifiably extend such generalization to the future. We are thus not justifiably entitled to rule out that there will be no lemon that is not sour, or that all A’s will be B’s, since its contradiction, that there is some A’s that will not be B, is possible. Consider an attempt to justify the conclusion:
C: All lemons are sour. (Implies: The next time I see a lemon I know it will likely be sour.)
P1: We know C is true because our observation in the past confirms it so.
Hume’s contention is that, if there is a justification for induction, it will be appealing to past confirmation, and in that case it is circular. Accordingly, Hume claims that induction lacks any kind of epistemic justification; in particular, there is no non-circular justification for induction.
Having described Hume’s problem of induction, I will proceed to explain the view that the commentators held as a response to the problem. In the Philosophy of Science: The Central Issues, commentators M. Curd and J.A. Covers (C&C) claim that the problem raised against induction is also a problem that is likewise parallel to deduction, and yet failing to justify deduction seems not at all problematic. Since deduction and induction appear to be in the same boat in terms of lacking non-circular justification, and deduction does not seem problematic, then induction should, in the same way, not be problematic. That deduction cannot be justified in a non-circular way can be shown by this example. Consider the valid deductive inference:
(1) P ∧ Q is true if and only if P is true and Q is true.
(2) Therefore, P ∧ Q entails P.
As said before, the premises logically entail the conclusion of a deductive inference. But it seems that this is the case only because the terms are circular. The premise “P and Q” is true appeals to each term: the truth of P, as well as the truth of Q. It is parallel to the inductive justification that, all A’s are B because past A’s have been B. Plainly, the parallel is: a proponent of induction thinks that induction has a property such that if all premises are true, then the conclusion is likely to be true. But when an inductive skeptic demands for a non-circular justification, we find that the proponent of induction fails to give a non-circular justification. Similarly, a proponent of deduction thinks that deduction has a property such that if the premises are all true, then the conclusion is necessarily true. Likewise, when a deductive skeptic demands a non-circular justification for deduction, the proponent of deduction also fails to give a non-circular justification. Therefore, answering the skeptic in the inductive case is just as problematic as answering the skeptic in the deductive case.
Whether the commentators are right that justifying deduction is not problematic, I believe there are reasons why they are correct. Deductively valid arguments are ones that hold by necessity. The only way in which one could reject deductively valid arguments is when one does not understand the semantics, or meaning of the terms. Therefore, deductive inferences are naturally circular, however this is not to say that such property is a weakness of deduction. Instead, such property can be considered as consistency that supports correctness. So, a priori—without doing any empirical justification—we can know for certain that it is impossible for the conclusion to be false when all the conclusions are true. It is hard to rationally think that deductively sound statements are not infallible. By this I mean, it is logically necessary that a=a, or that proposition that a square is a square is true. Thus, it is arguably the case that deduction is on a firm foundation and is correspondingly not problematic.
Secondly, I don’t fully agree with the commentators that justifying induction is no more problematic than justifying deduction. Unlike deduction, which is self-evident and axiomatic, induction is more problematic since it relies on the basis of likelihood and not of certainty. I think there are reasons why induction and deduction “is on the same boat” for the reason that both of them cannot be justified in a noncircular way. But, there are reasons why induction requires more justification, contrary to the claims of the commentators. For one, induction takes risks by going beyond what we have confirmed. Some of the risks taken by induction are justified, but some may just seem groundless. Perhaps there is no need to further justify deduction simply because of the fact that it is self-justifying. The conclusion of a deductive inference only discovers facts that are already implicitly contained in its premises. But the reason why induction should not be placed on the same pedestal as deduction is because induction is only reliable contingent on the strength of the evidence that confirms it. Induction involves placing confidence on future prediction or anticipation, and in turn considering this to be in some way reliable. Hume raised a very strong point against induction: we cannot have strong grounds for thinking that our inductive inferences are true (p.11). Thus, induction is not established in the same way as the demonstrative kind that is deduction and so it requires more justification.
I contend, then, that some things can be justified even if in not in a non-circular way. (In other words, I hold that some things can be self-evident.) The problem with induction, which is not a problem for deduction, has, then, not to do with this circular justification simpliciter, but rather with its lack of any rational justification. In conclusion, deduction and induction are in the same boat in some aspects but not in others. They are the same in that they are both necessary features of human reasoning and it is equally difficult to show a non-circular justification for either of them. Having a justification for both kinds of inference is important in contributing to the standards of our justified belief. However, as I have argued, deduction is in a firmer foundation and is not itself problematic, but induction is—which is why it requires further justification.