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Notes
See, e.g., Priest (2006), ch. 6.
\(\bot\) can be taken to be governed by the axiom \(\bot\,\vartriangleright\, A, \) or simply replaced with any sentence whatsoever.
Curry (1942).
This is often called Contraction, but I will reserve that name for the structural rule, to be met later.
Meyer et al. (1979).
There is some flexibility about how, exactly, to set up the semantics, and what, exactly, constitutes the logic B. The following is the approach of Routley et al. (1982), chs. 4, 5, except that they allow for a set of base worlds, P, and define x ≤ y as: for some \(w\in P, Rwxy\). This does not affect the set of logical truths. A slightly different approach (the “simplified semantics”) is given in Priest (2008), ch. 10.
In fact, one can establish this using just the sequent rules cited. Details are left as an exercise.
See, e.g., Field (2008).
See the cited reference by Restall.
In standard presentation of linear logic, there are the two notions of conjunction in the language, but only one form of premise combination, the intentional one. This gives rise to certain features, such as the failure of distribution for the extensional connectives. Given the connection between the conjunctions and premise combination, it would seem to me to be much more natural to have both forms of premise combination (and so distribution), in the way that I have set things up.
I omit the details of proof here. They can be extracted from the references cited.
I note that as long as there are no quantifiers in the language, one can interpret the system with the V rules in the one without, simply by translating \(V(\left\langle C\right\rangle \left\langle D\right\rangle \) as \(C\rightarrow D\).
In his (1990), Slaney moots a solution to the sorites paradox. He notes, in effect, that given the premises of a sorites inference, \(\Upsigma=\{A_{0}, A_{1}\rightarrow A_{2},\ldots, A_{n-1}\rightarrow A_{n}\},\) the deductive machinery allows us to establish only that \((\prod\nolimits_{i=n}^{1}(A_{i}\rightarrow A_{i-1})\otimes A_{0})\rightarrow A_{n}\) (associating to the right). Given that we cannot accept A n , we are not entitled to an arbitrary bunch in \({{\mathfrak{B}}(\Upsigma).}\) The premises cannot be taken together, as it were, in the appropriate fashion.
In connection with this, see Bacon (2013).
See Beall et al. (2012) and Priest (201+).
I owe this observation to Stephen Read.
From the present perspective, what happens to the validity curry argument with this notion of validity? The answer is that V1 breaks down, since truth preservation at the base world does not guarantee truth preservation at all worlds. A slightly different version of the argument appeals to the principle \(V3: V(\left\langle C\right\rangle ,\left\langle D\right\rangle )\oplus C\,\vartriangleright\, D. \) Lines 2 and 4 of (**) give us that \(C\,\vartriangleright\, V(\left\langle C\right\rangle ,\left\langle \bot\right\rangle )\oplus\bot, \) and Cut then gives us \(C\,\vartriangleright\,\bot. \) But there is no reason to suppose that V3 holds. What does hold is the same thing with ⊗ instead of \(\oplus; \) and with this principle, the argument will not go through without an illicit contraction. Being valid is a sort of conditional; and conditionals require intentional connection to produce the consequent.
This is very clear in Anderson and Belnap (1970). See pp. 1–29 and (esp.) pp. 473–492.
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Acknowledgments
Thanks go to the participants of the meetings mentioned in the note at the start of this paper for their helpful thoughts and comments, and especially to Greg Restall, Dave Ripley, and Jerry Seligman. A particular thanks goes to Lloyd Humberstone, both for comments at one of the meetings and for substantial written comments. Thanks also go to two anonymous referees for this volume.
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A version of this paper was given at the workshop on Curry’s Paradox at the University of Otago, and to a meeting of the Melbourne Logic group both in August 2012.
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Priest, G. Fusion and Confusion. Topoi 34, 55–61 (2015). https://doi.org/10.1007/s11245-013-9175-x
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DOI: https://doi.org/10.1007/s11245-013-9175-x