Does Deduction really rest on a more secure epistemological footing than Induction?

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1 Does Deduction really rest on a more secure epistemological footing than Induction? We argue that, if deduction is taken to at least include classical logic (CL, henceforth), justifying CL - and thus deduction altogether - by means of showing that it preserves truth faces problems analogous to Hume s Problem of Induction. First, we refute the claim that CL needs no justification, and then show that such justification cannot be satisfactorily provided either deductively or inductively. Inductive attempts fail because they cannot establish that CL is truth-preserving, whereas deductive ones cannot avoid circularity. Ergo, deduction cannot be justified, and is therefore, at best, as epistemically secure as induction. Most of our knowledge of the world is inductive: we observe certain regularities in nature - for instance, all observed F s have been G s - and infer from them general laws - e.g., all F s are G s. Such inferences are to be distinguished from deductive ones, which are standardly regarded as non-ampliative 1 and truth-preserving: valid deductive arguments are deemed to always lead from true premises to true conclusions [7], [8]. Per contra, inductive arguments are defeasible: their conclusion is supported, but not strictly entailed, by the premises, and adding new premises may well invalidate previously drawn conclusions. Accordingly, inductive inferences seem to lack the especially secure epistemological status which deduction purportedly enjoys. We then need independent grounds to determine which, if any, of our inductive inferences are warranted and how. Any attempt at providing such justification, however, seems to fail because of what is known as Hume s Problem of Induction [5], [6], which can be sketched as follows. Any inference is warranted either deductively or inductively. Consider the first option. Inductive arguments are not deductively valid, for, in inductive arguments, the negation of the conclusion is logically consistent with the premises. So, a deductive justification of induction would require additional premises - e.g., If all observed F s have been G s, then all F s are G s, the inclusion of which would be warranted only if we already knew that induction were truth- 1 The conclusion says nothing which was not already in the premises. 1

2 preserving [2], [8]. On the other hand, an inductive justification of induction would run into a vicious circle, for it would employ the very same reasoning at stake to solve the riddle [2], [5], [6]. Hence, no justification of induction seems to be at hand. Some - notably, Popper [7] - have therefore claimed that induction should be abandoned altogether (at least within the context of scientific inquiry), in favour of an exclusively deductive method. But what justifies this faith in deductive reasoning? At a closer look, any attempts at justifying deduction seem to incur into problems analogous to those derived for induction. First, any such justification can be either deductive or inductive. Yet, as we clarify later, a deductive justification would be circular, whereas an inductive one would fail to satisfy the truth-preservation requirement, for one would be at best entitled to claim that deduction usually leads from true premises to true conclusions. After all, if induction is not truthpreserving, an inductive argument cannot conclusively establish that deduction is. This is the Problem of Deduction [2]. Next, we consider, and dismiss, some putative solutions to it. While there is no universal consensus over the precise meaning of deduction, it appears there are at least certain inferences that are considered unambiguously deductively valid 2 (e.g., instantiations of modus ponens [1], [2]). Following [2], we can advance two different notions of deductive validity: (i) An argument A 1,...A n 1 = A n is (i-semantically 3 ) deductively valid IFF it is impossible, when the premises A 1,...A n 1 are true, that the conclusion A n be false; (ii) Let L D be a formal system. Then, an argument A 1,...A n 1 A n is (syntactically) deductively valid in L D IFF the conclusion A n is derivable from the premises A 1,...A n 1, and the axioms of L D, if any, in virtue of the rules of inference of L D. It seems to trivially follow from (i) that deductively valid arguments are truthpreserving. Hence, some might argue that deduction, by definition, needs no justification: truth-preservation supplies the desired warrant [2]. Then, if we upheld (i), there would be no problem of justification. Yet, this solution is epistemically vacuous: it solves the problem of justifying deduction at the price of not knowing which inferences are deductively valid, or if any deductively valid inferences exist at all [2]. This is because the old problem of justifying deduction reappears in a new form: how do we know that any particular unambiguously deductive inference that we deem truth-preserving is indeed deductive in the sense of (i), and thus genuinely truth-preserving? After all, (i) does not provide 2 Consider how often we define inductive arguments by contrasting them with arguments which we already consider deductively valid [10], [8]. 3 The term i-semantic is introduced to avoid confusion with the technical meaning of semantic validity in the model-theoretic sense. The difference between these notions will be discussed later. 2

3 any reasons to think that i-semantically valid arguments include any specific class of arguments we might want to justify (e.g., those sanctioned by a classical consequence relation, or otherwise specified by some formal system L D ). But this seems to be the question that any satisfactory justification of deduction should address [2]. The same issue can be seen more directly if we adopt definition (ii). Here, L D is a formal system codifying those inferences that we call deductive. Then, the problem of the justification of deduction is, again, that of explaining why deductively valid inferences in the syntactic sense of (ii) should be considered truth-preserving. This is the question we shall tackle throughout this paper. It remains, however, to specify which formal inference systems might be substituted for L D. This is not straightforward, given the vagueness of the term deduction. Yet, if anything, CL appears to be a suitable candidate: using CL as an inference method seems to be a clear-cut instance of deductive reasoning in the sense of (ii) - so much seems to be granted, or implicitly assumed, in most of the relevant literature [2], [4], [7], [10]. We then assume that deduction stands for some class of formalised inference systems (L D1, L D2,...) at least including CL. Next, we address the issue of whether CL is indeed truth-preserving - and thus warranted. This is because, if deduction at least includes CL, a justification of deduction should be a justification of CL, too. Note, however, that there is no a priori reason why our arguments could not also hold for suitably formalised inference systems other than CL. Now that we restricted the focus on CL, one might immediately retort that soundness and completeness theorems for CL provide a vindication of the correspondence between i-semantic and syntactic validity [3]: by showing that the definition of syntactic validity coincides with the model-theoretic one 4, one can endow deductive inferences with warrant via their being truth-preserving. Yet, this objection mistakenly presumes the equivalence between (i) and the model-theoretic definition of deductive validity for CL. Such equivalence is, again, just the one we wish to ascertain: i-semantic validity is an informal notion which bears no straightforward resemblance with the technical characterisation of model-theoretic validity. In model theory, the validity of an inference depends on various mathematical properties of interpretation functions and their domains. So, whether model-theoretically valid arguments are truthpreserving crucially depends on whether we can justifiably discriminate between those properties of domains/interpretations which hold and those which do not. 4 A model is a structure M, such that M = d, I, where d is the domain of M, and I is an interpretation function which assigns, to each non-logical terms of L, an element in d. We then define the relation of satisfaction between an interpretation function and the formulae of L: an interpretation I satisfies a formula ϕ (in symbols, I = ϕ) IFF ϕ under interpretation I holds in d. Then, an argument Σ, α in the formal language L is model-theoretically valid IFF there is no interpretation which satisfies each element of Σ and fails to satisfy α. We then write: Σ = α [9]. 3

4 But any mathematical argument as to whether any such properties hold - e.g., whether a given interpreted formula holds in a domain - will itself be formulated within CL. We are then back where we started: all we can assert is that modeltheoretic validity coincides with i-semantic validity IFF arguments sanctioned by CL are truth-preserving. Then, CL does indeed need an independent justification. Since justifying CL amounts to showing it is truth-preserving, most attempts at providing such justification hinge on the idea that we might still somehow avoid circularity when employing CL to justify itself. We will therefore bypass inductive justificatory attempts, for induction simply cannot establish the truth-preservation requirement. Yet, we will briefly return to the possibility of justifying deduction inductively at the end of the paper, when suggesting the questionability of the truth-preservation requirement. First, let us consider the possibility that inference rules might be employed at the meta-level to justify themselves at the object-language level. The inference rule from A, A B infer B (modus ponens) in the object-language, for instance, might be warranted at the meta-level because, given the truth-table for, we know that if A is true and A B is true, then B must be true, too. This latter argument is clearly an instance of modus ponens itself - we could even rewrite it as: suppose C ( A is true and A B is true). If C, then D (if [ A is true and A B is true], then B is true). Therefore, D ( B is true) [2]. Yet, the warrant problem invariably reappears up the ladder of meta-languages, for what entitled us to employ modus ponens at the meta-level in the first place? 5 One might reply that it was modus ponens at the meta-meta-level, but it is now clear that this strategy would lead to a hopeless infinite regress, in which the warrant of each step could be challenged. Moreover, one can show this solution renders inference rules which we do not consider classically valid justified, as well. Consider, for instance, the rule from A B, B infer A (modus morons). We can once more construct an argument based on modus morons in the meta-language to justify the rule at the object-language level: assume B and A B are true. Then D (If A B is true, then B is true). If C, then D (if A is true, then [if A B is true, then B is true]). Therefore, C ( A is true) [2]. The last step of this argument clearly employs modus morons. Hence, one could equally well justify CL-invalid inference rules in the object-language through CL-invalid arguments in the meta-language, thereby invalidating this self-supporting strategy. Perhaps, one might instead construct a justification-chain of meta-arguments, each of which justified by a higher-level one, to warrant CL-inferences formulated in the object-language. Such arguments could be expressed equally well in 5 Given our previous discussion of model theory, one could even ask what justified our definition of the truth-table for in the first place. 4

5 CL or some other suitably formalised systems we consider deductive. Both options, however, would succumb to the same counterarguments: (i) if the number of meta-arguments available were finite, we could not avoid, along the chain, some self-supporting ones, which we have just shown unwarranted; (ii) even if we had an infinite amount of employable meta-arguments, each step in the justification-chain would need to be demonstrably truth-preserving to provide the previous steps with warrant. In other words, showing that an argument is truth-preserving itself requires a truth-preserving meta-argument. In turn, any attempt at justifying these meta-arguments by showing they are truthpreserving could not be inductive. It would necessarily rely on deductive arguments formulated in some formal system L D that we consider deductive (CL included). Then, this same reasoning should be iterated for any subsequent higher-level argument, thus necessarily generating an infinite justification-chain. One might question whether such an infinite justification-chain can be constructed at all. If so, one might well have reasons to be wary of infinite justification-chains. And even endorsing them would leave one in an uncomfortable position: that of not conclusively knowing, at any point, whether one s accepted inferences are indeed truth-preserving. This is because the justificationchain we just discussed has a very particular structure. We do not have an infinite chain of propositions (...a i 1, a i, a), where each a i is a reason to accept a i+1, and hence justifies the acceptance of a ( argument A is truth-preserving ). This may in fact do, by infinitist standards. Our chain, instead, presents a serious difficulty, in that one cannot know whether any step from a i to a i+1 is ever warranted: any such step, in fact, itself requires an infinite justification-chain in which every step..., and so forth. Hence, we cannot determine whether any reason to accept any a i is indeed a reason. Why should anyone accept this as a valid justification? After all, if one endorsed an infinite justification-chain in which no single step is warranted, one could justify any infinite set of purportedly mutually supporting arguments, in particular circular chains (where A justifies B, B justifies A, etc.). Lastly, consider that this justification-chain is itself an argument aimed at showing that classically valid (or some other deductive) inferences are truthpreserving. Thus, even regarding it as providing a valid justification for CL would imply that we already considered it prima facie truth-preserving. And this is clearly circular, given our previous discussion. A non-circular, unproblematic deductive justification of CL and, accordingly, a deductive justification of deduction as a whole - whichever inference systems we may allow in that class beside CL - seem then impossible. We have then established that deduction, assumed to encompass at least CL, does not rest on a more secure epistemological footing than induction. Any attempts at justifying it by appealing to the notion of truth-preservation have been shown unsatisfactory: inductive justifications, by their very nature, are too weak to establish the truth-preservation requirement, while deductive ones cannot do so non-circularly. As a final remark, we notice that much of the 5

6 difficulties in justifying CL - and, perhaps, any method of inference as such - appear to stem precisely from the truth-preservation requirement, which might be too stringent. Moreover, if one were to abandon it, an inductive justification of CL would be at least conceivable. References [1] Carroll, L. (1895/1995). What the Tortoise said to Achilles, Mind, New Series, Vol. 104, No. 416: [2] Haack, S. (1976). The Justification of Deduction, Mind, New Series, Vol. 85, No. 337: [3] Haack, S. (1982). Dummett s Justification of Deduction, Mind, Vol. 95, No. 362: [4] Howson, C. and Urbach, P. (2006). Scientific Reasoning: The Bayesian Approach, Peru, Illinois: Carus Publishing. [5] Hume, D. (1777/1975). Enquiries concerning Human Understanding and concerning the Principles of Morals, reprinted by L. A. Selby Bigge, Oxford: Clarendon Press. [6] Hume, David ( ). Treatise of Human Nature, edited by L. A. Selby Bigge, Oxford: Clarendon Press. [7] Popper, K. R. (1959). The Logic of Scientific Discovery, London: Hutchinson. [8] Salmon, W. C. (1967). The Foundations of Scientific Inference, Pittsburgh: Pittsburgh University Press. [9] Shapiro, S. (2005). Logical Consequence, Proof Theory, and Model Theory, in The Oxford Handbook of Philosophy of Mathematics and Logic, ch. 21, , New York: Oxford University Press, Inc. [10] Strawson, P. F. (1952/2000). The Justification of Induction, reprinted in Probability and Confirmation, , NY: Garland Publishing Inc. 6

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