Talk to be given at the Workshop on the Philosophy of Non-Classical Logics (UNILOG 2015), and at the Logic Colloquium 2015.
Abstract The problem of accounting for acceptable uses of classically valid but paraconsistently invalid arguments is a recurrent theme in the history of paraconsistent logics. In particular, the invalidity of the disjunctive syllogism (DS) and modus ponens (MP) in, for instance, the logic of paradox LP, has attracted much attention.
In a number of recent publications, Jc Beall has explicitly defended the rejection of these inference-forms, and has suggested that their acceptable uses cannot be warranted on purely logical grounds [1], [2]. Some uses of DS and MP can lead us from truth to falsehood in the presence of contradictions, and are therefore not generally or infallibly applicable [3].
Not much can be objected to this view: if one accepts LP, then MP and DS can only be conditionally reintroduced by either
- opting for Beall’s multiple-conclusion presentation of LP (LP+) which only gives us A, A ⊃ B ⊢LP+ B, A ∧ ¬A and ¬A, A ∨ B ⊢LP+ B, A ∧ ¬A, or
- by treating MP and DS as default rules.
The latter strategy was initiated by inconsistency adaptive logics [4], [5], and implemented for the logic LP under the name Minimally inconsistent LP or MiLP [6].
The gap between these two options is not as wide as it may seem: The restricted versions of MP and DS that are valid in LP are the motor of the default classicality of MiLP. The only difference is that the restricted version only give us logical options (Beall speaks of ’strict choice validities’), whereas default classicality presupposes a preference among these options (unless shown otherwise, we must assume that contradictions are false).
A cursory look at the debate between Beall and Priest [3], [7] may suggest that not much can be added to their disagreement. However, if we focus on the contrast between the mere choices of LP+ and the ordering of these choices in MiLP, we can tap into the formal and conceptual resources of modal epistemic and doxastic logic to provide a deeper analysis [8]. We can thus develop the following analogy:
LP+ is motivated by the view that logical consequence is a strict conditional modality, and is therefore knowledge-like. Using a slightly more general terminology: all logical information is hard information.
MiLP is motivated by the acceptance of forms of logical consequence that are variable conditional modalities, and are therefore belief-like. With the same more general information: some logical information is soft information.
This presentation still gives the upper-hand to Beall’s stance (shouldn’t logical consequences be necessary consequences?), but only barely so. The upshot of this talk is to motivate the views that (i) the soft information that underlies the functioning of MiLP can be seen as a global as well as formal property of a logical space, and is therefore more logical than we may initially expect, and that (ii) adding a preference among logical options can be seen as a legitimate and perhaps even desirable step in a process of logical revision.
References
[1] J. Beall, "Free of Detachment: Logic, Rationality, and Gluts", Noûs, no., p. . n/a, 2013.
[2] J. Beall, "Strict-Choice Validities: A Note on a Familiar Pluralism", Erkenntnis, 79, no. 2, p. . 301-307, 2014.
[3] J. Beall, "Why Priest’s reassurance is not reassuring", Analysis, 72, no. 3, p. . 517-525, 2012.
[4] D. Batens, "Dynamic Dialectical Logics", in Paraconsistent Logic - Essays on the inconsistent, no., G. Priest, R. Routley and J. Norman, Editor. München / Hamden / Wien: Philosophia Verlag, 1989, p. . 187-217.
[5] D. Batens, "Inconsistency-adaptive logics", in Logic at Work - Essays dedicated to the Memory of Helena Rasiowa, no., E. Orlowska, Ed. Heidelberg / New-York: Springer, 1999, p. . 445-472.
[6] G. Priest, "Minimally inconsistent LP", Studia Logica, 50, no. 2, p. . 321, 1991.
[7] G. Priest, "The sun may not, indeed, rise tomorrow: a reply to Beall", Analysis, 72, no. 4, p. . 739-741, 2012.
[8] J. Van Benthem, Logical dynamics of information and interaction, no. Cambridge University Press, 2011.
