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Notes
I prefer, in other contexts, to talk of assertion and denial, for it seems to me that logical norms apply to acts of assertion and denial whether they correlate with what I accept or reject. However, nothing important will ride on the difference here [7].
This is one of the morals of Lewis Carroll’s tale of Achilles, the Tortoise and modus ponens. The force of modus ponens is not merely to recommend to us a statement \(((p\,{\to}\,q)\,{\land}\,p\;\to\;q)\) for our acceptance, but to render the combination—accept p → q, accept p, reject q—as out of bounds [2].
In the model construction, we add to the domain an open sentence (paired with a tuple of objects for each variable other than x free in ϕ(x)) as the denotation of the property term. The work in the construction is taken up in fixing the interpretation of instantiates in such a way that (NC) is satisfied.
This question is not an artefact of the notation I have introduced. It may be asked without that notation in a more long-winded manner. When do we have \((\exists y)(\exists y')(\forall x)( (x \hbox{ is } y\leftrightarrow \phi(x)) \land (x \hbox{ is } y'\leftrightarrow \psi(x)) \land y=y' )?\) This question gets to the same issue: the identity conditions for properties.
These two properties differ in the model construction Field gives, since \(Fx\land Gx\) and \(Gx\land Fx\) are different sentences.
A case Field discusses (see page 74)—a statement A for which we take there to be no possible evidence for or against—might be thought to be a counterexample to the rule [Cut]. After all, I may be committed to not accept A, and committed not to reject it, since there is no possible evidence for or against it. So, suppose X and Y collect all that I accept and all that I reject, including those commitments upon which my judgement about the evidential status of A is based. Isn’t it the case that I should not accept A and I should not reject A? Aren’t accepting A and rejecting A both ruled out on logical grounds, relative to accepting X and denying Y? Isn’t this a counterexample to [Cut]?
No, they aren’t. There need not be anything incoherent in accepting A (or in rejecting A) while accepting that there is no evidence for or against A. Accepting A is something we may (hypothetically) consider when we try adding A as an hypothesis and see what this addition does to our commitments. If that were inconsistent with what we had granted, this would form the basis of a reductio argument against A. Yes, there is a tension in accepting A while granting that there is no evidence for it. However, it is readily apparent that we may hypothetically consider what would follow if A were actually the case.
Remember, in these cases, we do not merely take accepting A or rejecting B to be unacceptable for going past the evidence. It is ruled out by those other claims we accept and reject.
I am tempted to add →0 where A →0 B is B, but I agree, that is not so much a conditional connective. The connectives → n where n ≥ 1 each act like a conditional in that they support modus ponens, and some form of the deduction theorem if that theorem is satisfied by the ground level conditional →1. If →1 satisfies the inference of weakening (A ⇒ B →1 A) then the connectives → n are progressively weaker (A → n B ⇒ A →n+1 B).
This is a simple example of more general principles concerning identity:
$$ {\frac{X,\phi(a)\Rightarrow Y}{X,\phi(b),a=b\Rightarrow Y}} [{=}L_1] \qquad {\frac{X\Rightarrow \phi(a),Y} {X,a=b\Rightarrow \phi(b),Y}}[{=}L_2] $$we need only the very simple form [\(=\) L] so we do not need to discuss the nature of the contexts ‘ϕ( )’ in which an identity may be validly substituted.
But not very coarse: recall, we do not mean to identify necessarily coextensive properties, merely logically coextensive ones.
References
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Acknowledgements
This research is supported by the Australian Research Council, through grant DP0343388, and Tenebrae’s Mother and Child. Discuss this paper at http://consequently.org/writing/stp/.
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Thanks to the Logic Seminar at the University of Melbourne, an audience at the Fourth World Congress for Paraconsistency, where some of these ideas were discussed. Thanks, too, to Hartry Field for enjoyable discussions on these matters.
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Restall, G. What are we to accept, and what are we to reject, while saving truth from paradox?. Philos Stud 147, 433–443 (2010). https://doi.org/10.1007/s11098-009-9468-5
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DOI: https://doi.org/10.1007/s11098-009-9468-5