Abstract
Contemporary philosophers generally conceive of consequence as necessary truth-preservation. They generally construe this necessity as logical, and operationalize it in substitutional, formal or model-theoretic terms as the absence of a counterexample. A minority tradition allows for grounding truth-preservation also on non-logical necessities, especially on the semantics of extra-logical constants. The present chapter reviews and updates the author’s previous proposals to modify the received conception of consequence so as to require truth-preservation to be non-trivial (i.e. not a mere consequence of a necessarily true implicatum or a necessarily untrue implicans) and to allow variants of the substitutional, formal and model-theoretic realizations of the received conception where the condition underwriting truth-preservation is not purely formal. Indeed, the condition may be contingent rather than necessary. Allowing contingent non-trivial truth-preservation as a consequence relation fits our inferential practices, but turns out to be subject to counterexamples. We are left with an unhappy choice between an overly strict requirement that non-trivial truth-preservation be underwritten by a necessary truth and an overly loose recognition of non-trivial truth-preservation wherever some truth underwrites it. We need to look for a principled intermediate position between these alternatives.
Bibliographical note: This chapter was originally published under the same title in Studies in Logic, Grammar and Rhetoric 16 (2009), 137–158.
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Notes
- 1.
I write ‘untru e’ rather than ‘false’, in order to leave open the possibility of a conclusion’s being neither true nor false.
- 2.
For simplicity, I am taking sentences to be the relata of the consequence relation. Nothing in this article should depend on this decision. The same points about consequence could be made if one takes entities other than sentences to be the primary truth-bearers—e.g. utterances, statements or propositions.
- 3.
Here and elsewhere, I use the exact translation into English by Magda Stroińska and myself of the Polish version of Tarski ’s paper (Tarski 1936a), which I argued in (Tarski 2002) is more authoritative than the German version, also written by Tarski ( 1936b), which was used as the basis of the previous rather inexact translation of the paper into English (Tarski 1956 and 1983, pp. 409–423).
- 4.
At least, so we suppose. Research on aging may lead to techniques of preventing human aging, in which case human immortality would become physiologically possible. But the “may” here is epistemic. At the moment, as far as we know, it is physiologically inevitable that every human dies.
- 5.
- 6.
Correction in the present republication: The original article had a definition of the soundness of a formal system rather than of its completeness.
- 7.
The preceding paragraph summarizes and adapts (Hitchcock 1998, pp. 24–25).
- 8.
These revised conceptions adapt the conceptions found in Hitchcock (1998, p. 26), with the additional constraint that the set of content expressions is exhaustive.
- 9.
For a proof with respect to Bolzano ’s substitutional conception, applied to the language of classical propositional logic, see George (1983). The qualification that the implied sentence contains an extra-logical constant is needed to accommodate cases where the consequence relation obtains but the implied sentence contains no extra-logical constants. For example, the sentence ‘there is at least one object’ follows from the sentence ‘there are at least two objects’ on any of the three revised conceptions, even though it contains no extra-logical constants.
- 10.
Added in the present republication: Consider the following arguments: ‘Grass is green, so snow is white.’; ‘Grass is white, so snow is black; ‘Grass is black, so snow is white.’ None of these arguments has a true premiss and a false conclusion. Thus their associated material conditional is true. So, with respect to a set of content expressions that does not include any content expression in these arguments, such as the set consisting of the name ‘Jupiter’, the first condition is met, on any of the revised conceptions of consequence. However, one or other of the two remaining conditions is not met. The second and third arguments have no parallel in which the premiss is true; the first and third arguments have no parallel in which the conclusion is untrue.
- 11.
Correction in the present republication: The original article had ‘true’ and ‘untrue’ reversed.
- 12.
In testing the applicability of my conception of good inference to actual arguments that scholars advance, I used an even more expanded conception that allowed for probabilistic and presumptive inferences, underwritten respectively by for-the-most-part and ceteris paribus covering generalizations. In the present chapter, I do not discuss this further expansion of the concept of consequence.
- 13.
The reflections in the preceding paragraph were stimulated by an article by Pinto (2006) and by subsequent correspondence with Pinto and James B. Freeman .
- 14.
In the preceding sentence, I use the word ‘consequence’ in an inferential rather than a causal sense. To be a consequence of something in the inferential sense is to be legitimately inferable from it.
- 15.
Added in the present chapter: Examples of each type: every conditional with a true antecedent has a true consequent; every bachelor is unmarried; every physical object attracts every other physical object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them; the square of the sum of any two numbers is the sum of the square of the first number and twice the product of the two numbers and the square of the second number; every bird is a theropod dinosaur; every president of the United States is a natural-born citizen.
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Hitchcock, D. (2017). Non-logical Consequence. In: On Reasoning and Argument. Argumentation Library, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-53562-3_7
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