Abstract
Some recent work on logical pluralism has suggested that the view might be in danger of collapsing into logical nihilism, the view on which there are no valid arguments. The goal of this chapter is to argue that the prospects for preventing such a collapse vary with one’s account of logical consequence. Section 1 lays out four central approaches to consequence, beginning with the approaches Etchemendy (On the Concept of Logical Consequence. CSLI: Stanford, 1999) called interpretational and representational, and then adding a Quinean substitutional approach as well as the more recent universalist account given in Williamson (Modal Logic as Metaphysics. Oxford: Oxford University Press, 2013; Semantic Paradoxes and Abductive Methodology. In Reflections on the Liar, ed. B. Armour-Garb, 325–346. Oxford: Oxford University Press, 2017). Section 2 recounts how the threat of logical nihilism arises in the debate over logical pluralism. Section 3 then looks at the ways the rival accounts of logical consequence are better or worse placed to resist the threat.
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Notes
- 1.
- 2.
A note on terminology: in this chapter I will follow the common practice of treating the expressions ‘(logical) consequence,’ ‘entails,’ and ‘valid’ as intertranslatable—an argument is valid just in case its conclusion is a (logical) consequence of its premises, and this is so just in case the premises together entail the argument’s conclusion. Hence the different views of logical consequence that I will be looking at are just as much different views of validity and different views of entailment. There are two other words that it might be tempting to use here: implication (perhaps being Quineanishly careful to use this for entailment rather than for a conditional) and inference. But following Harman (1986) I take inference to be a different topic altogether, and I’ll be cautious about using ‘implication’ because I think Quine’s suggested regimentation (i.e. restricting its use to talk of the entailment relation) is less entrenched than would be ideal for successful communication (Quine 1966, pp. 165–166; Quine 1981, §5).
- 3.
For example, in the title of Etchemendy (1999). I recognise that Etchemendy’s word ‘concept’ is fighting talk in some philosophical circles. I mean to talk—as Etchemendy did—about different views of what features an argument has to have to be valid and have no special commitment to using the word ‘concept’ or to any particular construal of that phrase.
- 4.
Sometimes people respond to this problem by distinguishing different kinds of necessity. 2 + 2 = 4 and Hesperus is Phosphorus, they might claim, are metaphysically necessary, but not logically necessary, and it is the logical modality in terms of which logical consequence is defined. There are two problems with this response. The first is that it replies on a controversial claim about the kind of necessity possessed by the propositions expressed by these sentences. In, for example, Kripke’s modal argument of the necessity of Hesperus is Phosphorus, the ‘□’ that that sentence inherits is the very same one applied to Hesperus is Hesperus (Kripke 1980). The second problem is that the account threatens to be circular. What is logically necessary if it is not the things which hold in virtue of logic alone—the logical truths. But this assumes that we already have an account of logical consequence.
- 5.
Kaplan (1989). Here A is the actuality operator, N the now operator, and α a singular term, so that informal instances of these sentences might be Actually it is snowing if, and only if, it is snowing., if it is snowing now, then it is snowing, I am here now and dthat[the shortest spy]=the shortest spy.
- 6.
An additional problem for the modal slogan is that: necessity and logical truth come apart in the model theory for modal logics. □p may be true at some points in a model and false at others. This would make no sense if logical truth were necessity—necessity might be world relative but logical truth is not.
- 7.
The terminology here is from Etchemendy (1999). I find this terminology somewhat difficult to remember, but one useful mnemonic is to note that the one view is interpretational because it speaks of different interpretations of the linguistic items, while the other is representational because it talks about the things that get represented.
- 8.
Here is one: (i) if Π is an n-place predicate and t1,…,tn are n terms, the Πt1,…,tn is an atomic wff; (ii) if t1 and t2 are terms, then t1 = t1 is an atomic wff; (iii) if A is a wff, then ¬A is a wff; (iv) if A and B are wffs, then A∧B is a wff; (v) if A is a wff and x is a first-order variable, then ∀xA is a wff. If A contains no quantifiers, then all the variables in A are free. If ∀ is immediately followed by a variable ξ, then we say that it is a ξ-binding quantifier. In a wff ∀ξA, the ξ-binding quantifier ∀ is said to bind all free occurrences of ξ in A. A variable which is bound in A is not free in A. Any wff with no free variables is a sentence.
- 9.
That is, nI will be an element of D, where n is a name, PI will be a subset of Dn, where P is an n-place predicate, and fI will be a complete function from Dn to D, where f is an n-place function.
- 10.
See also “Their characteristic [logical truths] is that they not only are true but stay true even when we make substitutions upon their component words and phrases as we please, provided merely that the so-called ‘logical’ words ‘=’, ‘or’, ‘and’, ‘not’, ‘if-then’, ‘everything’, ‘something’, etc., stay undisturbed” (Quine 1950, p. 4).
- 11.
Tarski too considers a variation of the word-test approach:
“If, in the sentences of the class K and in the sentence X, the constants—apart from purely logical constants (like signs being everywhere replaced by like signs), and if we denote the class of sentences thus obtained from K by ‘K′’, and the sentence obtained from X by ‘X′’, then the sentence X′ must be true provided only that all the sentences of class K′ are true.” (Tarski 1936, p. 415)
- 12.
Etchmendy (1999, pp. 68–69; 111–114).
- 13.
It is a bit loose to speak of ‘the’ universal closure of ‘the’ shell, since there will be many trivial variants, for example, ∀Y(Y → Y) and ∀Z(Z → Z) are also universal closures of a shell of P → P. If desired, we might enumerate the variants and count only the first (or the fifth) as the universalisation. Since the issue won’t be of importance here, I won’t mention it again.
- 14.
In his chapter in this volume , Aaron Cotnoir distinguishes two views called ‘logical nihilism’ and focuses on one. The view I have in mind here is the other one.
- 15.
I’ve looked in more depth at one argument for logical nihilism here: (Russell 2017)
- 16.
All of our conceptions of logical consequence agree that this is necessary for logical consequence, though some may require more than this.
- 17.
Admittedly, that argument was intended to be a reductio on someone else’s view. But one person’s reductio ad absurdum is another person’s argumentum ad absurdum.
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- 19.
The problems of higher order vagueness make it clear that this is not sufficient to accommodate and explain vagueness, but there are still many views on which the move is regarded as necessary.
- 20.
And in fact Strong Kleene logic provides counterexamples to all the classical logical truths.
- 21.
We leave to the previous section discussion of what happens if a does not exist.
- 22.
That is, The Liar sentence is true and the Liar sentence is not true or ‘Heterological’ is heterological and ‘heterological’ is not heterological.
- 23.
And one of the reasons things are especially complicated is that the most familiar views can be understood as quantifying over either meanings or bits of the world.
- 24.
It would be especially interesting in future work to examine this pattern against the phenomenon of context-sensitivity. Context-sensitivity is a paradigmatically linguistic, rather than worldly, issue, but there are still views (such as MacFarlane’s non-indexical context-sensitivity) which allow it to go deeper than others. Moreover, context-sensitivity is known to have interesting non-classical effects on logic.
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Russell, G. (2018). Varieties of Logical Consequence by Their Resistance to Logical Nihilism. In: Wyatt, J., Pedersen, N., Kellen, N. (eds) Pluralisms in Truth and Logic. Palgrave Innovations in Philosophy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-98346-2_14
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