Abstract
It has been claimed that contracting connectives are conditionals. Our modest aim here is to show that the conditional-like features of a contracting connective depend on the defining features of the conditional in a particular logic, yes, but they also depend on the underlying notion of logical consequence and the structure of the collection of truth values. More concretely, we will show that under P-consequence and suitable satisfiability conditions for the conditional, conjunctions are contracting connectives for some logics without thereby being conditional-ish.
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Notes
- 1.
We do not intend to endorse (Detachment) as a good definition, much less the right one, of a detachable connective. If a close connection between conditionals and detachable connectives was intended, the definition is too broad, as conjunction satisfies (Detachment) in most logics. The problem is not about specific connectives, though, but rather that as it stands, the definition does not distinguish between ‘self-detachable’ connectives, like conjunction, for which the other premise plays no role, and other connectives for which the additional premise is essential. Proof-theoretically, it might be demanded that both A and \(A\vartriangleright B\) are effectively used to prove (in L) B or, model-theoretically, that \(A\vartriangleright B\) alone has none of its proper sub-formulas as logical consequences (in L). These changes would affect some details of the discussion below, for sure—for instance, whether conditionals are necessarily detachable, and if not, whether what is needed in the definition of a contracting connective is a detachable connective or a conditional suffices—, but we stick to the definition in the literature and leave the discussion of a better definition for another occasion.
- 2.
Although very widespread, this satisfiability condition for conjunction does not encompass all conjunctions. For example, Bochvar’s (internal) conjunction does not fall into its scope.
- 3.
This notion can also be made compatible with logics that take more than one designated value. However, this requires explanations that are unnecessary and even distracting for our purposes. The only point we want to get across is that (T-consequence) is a very natural generalization of our usual definition of validity.
- 4.
A quick note on notation: so far, we have been using \(\Vdash \) as a sort of generic turnstile, meant to be read from left to right, as usual. We will be using \(A \dashv \vdash B\) as shorthand for ‘\(A \Vdash B\) and \(B \Vdash A\)’.
- 5.
And this is a recurrent lesson in many semantic projects. For one of its more recent appearances in proof-theoretic semantics, see Dicher and Paoli (2021).
- 6.
A connective k is contraposable (in L) if and only if, \(A \, k \, B\dashv _{L}\vdash \sim B \, k \, \sim \! A\), where \(\sim \) is a generic negation. In order to evaluate contraposition, we make use of the usual generalized satisfiability condition for negation: \(\sigma (\sim A) = 1\) if and only if \(\sigma (A) = 0\), \(\sigma (\sim A) = 0\) if and only if \(\sigma (A) = 1\), and \(\sigma (\sim A) = *\) otherwise. Our choice reflects nothing beyond the decision to stick with basic, not too deviant, many-valued logical vocabulary. The inclusion of other negation connectives would certainly make for interesting discussion; however, we are also aware that matters might already be complicated enough as they stand. Consequently, we think that the introduction of other negation connectives deserves its own treatment elsewhere.
- 7.
A connective c is non-symmetric (in L) if and only if, either \(A \, c \, B\nVdash _{L} B \, c \, A\) or \(B \, c \, A\nVdash _{L} A \, c \, B\).
- 8.
- 9.
For more on this discussion, see Wansing and Shramko (2008).
- 10.
For an even more elaborate defense of the logicality of P-logical consequence and other non-Tarskian notions of logical consequence, see Estrada-González (2015).
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Acknowledgements
The present work was funded by UNAMs PAPIIT project IN403719 “Intensionalidad hasta el final: un nuevo plan para la relevancia la”.
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Estrada-González, L., Ramírez-Cámara, E. (2021). Non-conditional Contracting Connectives. In: Mojtahedi, M., Rahman, S., Zarepour, M.S. (eds) Mathematics, Logic, and their Philosophies. Logic, Epistemology, and the Unity of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-53654-1_12
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