Abstract
Here is one argument against realism. (1) Realists are committed to the classical rules for negation. But (2) legitimate rules of inference must conserve evidence. And (3) the classical rules for negation do not conserve evidence. So (4) realism is wrong. Most realists reject 2. But it has recently been argued that if we allow denied sentences as premisses and conclusions in inferences we will be able to reject 3. And this new argument against 3 generates a new response to the anti-realist argument: keep 1 and 2, avoiding 4 by rejecting 3. My aim in this paper is to see how much work in the fight against anti-realism this new response can really do. I argue that there is a powerful objection to the response: 2 is in tension with the claim that denied sentences can be premisses and conclusions in inferences. But I show that, even given this objection, the new response has an important role to play.
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Notes
Compare Rumfitt (2000, p. 797).
I follow other participants in this debate in characterising the question about the legitimacy of classical logic as a question about the legitimacy of a classical natural deduction system. See for example Dummett (1991, ch. 11); Rumfitt (2000). There is a debate about whether CNI is better formalised as the stipulation that if both B and ⌜~B⌝ can be derived from the assumption that A, together with X, we may discharge the assumption and move to ⌜~A⌝ from X. The alternative formulation has the advantage of not using ‘⊥’ (which is itself going to require rules for use). It has the disadvantage of using ‘~’ in the specification of the conditions under which ‘~’ may be introduced. It is not possible to enter into the debate about rival formulations of CNI here. For a brief remark on the connection between this debate and the Negation Argument see note 5.
For representative discussion of the ‘inference preserves evidence’ model in Dummett see (1978, especially 307–308, 312–313; 1991, pp. 217-220). There is a hard question about whether the inference-preserves-evidence framework can be sustained in a version which does not require an appeal to the notion of ‘direct’ or ‘canonical’ verification. But I cannot discuss this question here.
Dummett (1991, p. 291). Compare Campbell (2002, pp. 196–197), Rumfitt (2000, pp. 790–791). Any combination of rules for ‘~’ which uses some form of reductio introduction rule, and validates proof by reductio (proof of A from the reduction to absurdity of ⌜~A⌝) will entail that use of ‘~’ does not conserve evidence, and so be subject to a variant of the Negation Argument. The standard intuitionist introduction rule for ‘~’ is CNI. The intuitionist elimination rule is ‘From A and ⌜~A⌝ infer ⊥’. This rule, unlike CNE, is in harmony with CNI. So if CNI is assumed, the Negation Argument is also an argument for preferring intuitionist logic to classical logic.
See, for example Dummett (1991, p. 326; 1993, p. 56). Dummett writes that he considered retrenching from this position on the ground that there may be a distinction between ‘deep’ and ‘shallow’ rejections of bivalence, where only deep rejections are anti-realist, but ended up retaining the Bivalence Characterisation on the ground that a step away from bivalence for a class of statements is a step away from the strongest possible realist view for statements in the class, and, therefore, a form of anti-realism. See (1991, ch. 15), and compare (1993, pp. 265, 467–468). Note that for the case of vague statements Dummett proposes an amendment. See, for example, (1993, p. 468): ‘A realist must…hold that, for every vague statement, there is a range of statements giving more precise information of which a determinate one is true and the rest false’, so that failures in bivalence for vague statements are treated as arising from lack of precision in the description of a determinate reality.
For Dummett stating and defending this principle see, for example, (1991, ch. 4, especially 88–92 and 103–105).
Note that just upholding classical logic is not enough to manifest commitment to bivalence. For example, consider supervaluationism about vague predicates (the view that a sentence containing a vague predicate is (absolutely) true iff true under every precisification and (absolutely) false iff false under every precisification). A supervaluationist can uphold all classical theorems (because a classical theorem will come out as true under every precisification) while denying bivalence (because many sentences containing vague predicates will come out true under some precisifications but false under others). Deployment of classical laws does manifest commitment to the claim that every sentence is either true or false if the notion of absolute truth commutes with both negation and disjunction (so that ⌜It is not true that A⌝ is equivalent to ⌜It is true that not A⌝, and ⌜It is true that (A or B)⌝ is equivalent to ⌜It is true that A or it is true that B⌝). Compare Dummett (1991, pp. 74–75). These conditions combine with ⌜A v ~ A⌝ to entail that there are only two truth values, and ⌜~A⌝ has one iff A has the other, which is abbreviated in the text to ‘⌜~A⌝ is taken to be true iff A is false’. So Dummett’s claim is that reasoning according to CNI and CNE manifests commitment to bivalence iff embedded in this wider inferential environment.
The suggestion that basic logical beliefs are ‘default reasonable’ is a view of this kind. See Boghossian (2000, pp. 238–240) for a summary.
See for example Campbell (2002, pp. 101–104, 198; 2007, p. 12). Rumfitt (2000, pp. 785–787) argues against this kind of position on the ground that a justification for a rule of inference must be concerned not only with the combinations of premisses and conclusion that it treats as legitimate but with the path from premisses to conclusion that it lays down.
The following line of thought forms part of McDowell’s attack on the Bivalence Characterisation in (1998)—see especially Sect. 6 of that paper. Other philosophers have given other arguments against the Bivalence Characterisation. For a summary of such arguments see Hale (1997, pp. 274, 287–288). For extended discussion of the relations between realism, classical logic, and the Principle of Bivalence see Wright (1993, pp. 433–457). See note 6 for references to some places where Dummett defends the Bivalence Characterisation from these attacks.
Compare Horwich (1990, p. 8): ‘What I am claiming on behalf of the minimalist conception of truth is not that it, by itself, will engender realism or anti-realism; but rather that it will make it easier for us to see that the central aspects of the realism debate have nothing to do with truth’. Note that deflationists will also reject the inference-preserves-evidence framework that generates the Negation Argument. See Field (1994, Sect. 3) for deflationist derivations of the introduction and elimination rules for classical logical constants.
For example, Dorothy Edgington argues that our grasp of realist truth conditions is manifested by our deployment of conditionals with untestable antecedents and testable consequents. See Edgington (1985, pp. 44–46).
See Smiley (1996, p. 1) and Rumfitt (2000, pp. 178–179). What follows gives only a basic account of one motivation for bilateralism. There are many others. For example, Smiley (1996) and Rumfitt (1997) explore the move to bilateralism as a way to secure the result that the meanings of the logical constants used in a calculus are given by the introduction and elimination rules they have in a sound and complete deduction system for the calculus. Restall (2005) presents a range of motivations for bilateralism. Price (1983, p. 169) gives another motivation.
It is important to distinguish this claim about the independence of denial from assertion from the claim that denial is ‘independent from’ assertion in that A’s denial is not equivalent to the assertion of its negation. For this alternative independence claim see Parsons (1984, p. 139), Tappenden (1999, Sect. 1) and Restall (2005, p. 2).
This example originates with Frege’s discussion of ‘the Frege Point’—the claim that denial must be collapsed into the assertion of negated sentences in order to account for the validity of the inference from the denial of ‘p’ and the assertion of ‘If not p then not q’ to the assertion of ‘Not q’. See (1984, pp. 384–385). The bilateralist response to this point is that the validity of inferences involving denials is to be accounted for by (bilateralist) rules governing how assertions and denials interact, and that Frege’s claim that it is more ‘economical’ to deal in assertion alone is misguided. See Smiley (1996, pp. 2–4).
This response to the Negation Argument is modelled on elements of Rumfitt (2000, Sects. 5–7). Price and Restall offer alternative bilateralist defences of classical negation. I focus on Rumfitt’s defence because I take it to be dialectically the strongest against the Negation Argument as I have stated it. It is not possible to argue for this comparative point here.
Rumfitt demonstrates that the bilateral rules also satisfy a requirement for rules to be in good order which is specific to bilateral reasoning systems. This is the demand for coordination, where (i) assertion and denial are ‘coordinated’ for a sentence iff assertion and denial of the sentence are never correct simultaneously; and (ii) the rules for a logical operator are ‘coordinated’ iff they ensure that assertion and denial are coordinated for a complex sentence formed using the operator if they are coordinated for the simpler sentences that the complex sentence contains. Gibbard (2002, pp. 299–300) objects that there are coordinated bilateral sets of rules for ‘~’ other than Rumfitt’s rules. I take Rumfitt to be right in replying (2002, p. 309) that even if this were the case it would not count against his claim that if bilateralism is assumed a proof-theoretic justification for the classical ‘~’ rules is available: showing that a set of rules meets the requirements for good order does not require showing that these are the only rules which meet the requirements. Rumfitt also argues that Gibbard’s proposed alternative rules could not be taken to fix the sense of ‘~’ as used in L because they do not fix conditions for denying a negated sentence (2002, pp. 310–311).
Rumfitt (2000, p. 803).
I should stress that I am not suggesting that any of the proponents of bilateralism I have referred to in this paper actually does extend the response to the Negation Argument into a response to the Anti-Realist Argument in this way.
Compare Kripke’s discussion of ‘unicorn’ at (1980, pp. 23–24).
Tappenden uses this kind of case to argue that natural language negation is wide scope (‘external’) negation at (1990, Sect. 4).
Compare Dummett on how non-classical operators may be ‘superimposed’ on classical ones at (1991, pp. 332–337).
For an example of a ‘strong denial’ bilateralist system see Rumfitt (1997).
In Rumfitt’s strong denial system +RAA is replaced by a weakened reductio rule, whose ‘+’ form is ‘If X, +A ├ +B and X, +A ├ −B, and [X] determines [A], then X├ −A’, where [X] ‘determines’ [A] iff every valuation that assigns a truth value to every member of X also assigns a truth value to A. See (1997, p. 230).
See Section 2.2 above, and compare Rumfitt (1997, p. 234).
See Section 1.3 above.
Compare Rumfitt (2000, p. 811).
See note 19.
It turned out that there is no Superweak Force.
‘Strangeness’ was initially introduced as a term for whatever property of kaons leads them to display a characteristic pattern of decay. The definition was later revised to recognise pions as containing both ‘strange’ and ‘anti-strange’ components (which cancel each other out, explaining why pions do not display the same pattern).
The model might distinguish plain denial from strong denial, and distinguish two kinds of negation (‘external’ and ‘internal’ negation) to coordinate with plain and strong denial. See Rumfitt (1997, pp. 232–234) for an account of what a system which incorporates this distinction might look like. The proposal involves introducing an ‘It is true that’ operator, so that ‘−’ is still a sign for strong denial, and ‘A’s plain denial is expressed by ⌜− It is true that A⌝. But the problem I am raising arises for any bilateralist proposal which allows messy plain denials to be premisses and conclusions in inferences.
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Acknowledgements
Thanks to Diana Raffman and to audiences at the University of Western Ontario and Carleton University for discussion of earlier versions. Thanks to Ranpal Dosanjh for a physicist’s perspective on the material in Sect. 3.3. Many thanks to Philip Kremer for extensive comments on an earlier draft and multiple helpful discussions of this subject matter. Many thanks also to an anonymous referee for Philosophical Studies for extremely useful comments.
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Dickie, I. Negation, anti-realism, and the denial defence. Philos Stud 150, 161–185 (2010). https://doi.org/10.1007/s11098-009-9364-z
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DOI: https://doi.org/10.1007/s11098-009-9364-z