Abstract
My aim here is a modest one: to note another example in which the theory of validity and the theory of ‘inference’ naturally come apart. The setting is multiple-conclusion logic. At least on one philosophy of multiple-conclusion logic, there are very clear examples of where logic qua validity and logic qua normative guide to inference are essentially different things. On the given conception, logic tells us only what follows from what, what our ‘choices’ are given a set of premises; it is simply silent on which, of the given ‘choices’, we select from the (conclusion) set of options.
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Notes
I note that while the setup here closely follows the setup in the cited paper (Beall 2011), not only the aim but also terminology—particularly ‘strict-choice validity’—is different here, and the terminology much-improved (I think) from the previous paper. I also note that, after Beall (2011) was published, Arnon Avrum pointed me to some of his pioneering work on sequent systems for \(\hbox{LP}^+\) and neighboring systems (Avron 1991, 2003), which predates my work on \(\hbox{LP}^+\).
LP was first advanced, under a different label, by Asenjo (1966) as a philosophical logic for paradoxes. Later, the logic was independently discovered and widely applied by Priest (1979, 2006), who gave it the title ‘LP’ for ‘logic of paradox’. For basic discussion of the familiar family of non-classical logics in which LP sits, see Beall (2010), Beall and Fraassen (2003), Priest (2008), and Restall (2005).
In this case, we sometimes say that A is ‘true on/according to v’ or the like.
In what follows, I often use common abbreviations: the comma abbreviates union; ‘X, A’, for example, abbreviates ‘X ∪ {A}’; and ‘\(X\vdash B\)’ abbreviates ‘\(X\vdash\{B\}\)’. Importantly, ‘\(X, A_1, \ldots, A_n\nvdash B_1, \ldots, B_n, Y\)’ should be understood in the usual negated-universal way: there’s some X, Y, A i , B i such that the given \(X, A_1, \ldots, A_n\) fail to imply the given B i , Y (according to the given consequence relation—logic—expressed via the turnstile).
Given the monotonicity of \(\hbox{LP}^{+}\) (or, in proof-theoretic terms, ‘weakening on both sides’), it is convenient to refine things even a bit more: we might focus only on those strict-choice validities \(\langle X,Y\rangle\) such that \(\sigma(Y)\subseteq\sigma(X),\) where σ(Z) contains all subsentences of Z. But this refinement is unnecessary for current purposes.
I am grateful to a referee for prompting this remark.
References
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Acknowledgments
This paper is related to my talk at the very enjoyable and productive conference on logical pluralism in Tartu, Estonia. I am particularly grateful to Daniel Cohnitz, Peter Pagin, and Marcus Rossberg for making the conference so productive and enjoyable. I am grateful to two referees for prompting some useful clarifications on various points.
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Beall, J. Strict-Choice Validities: A Note on a Familiar Pluralism. Erkenn 79 (Suppl 2), 301–307 (2014). https://doi.org/10.1007/s10670-013-9479-7
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DOI: https://doi.org/10.1007/s10670-013-9479-7