Abstract
Informational semantics were first developed as an interpretation of the model-theory of substructural (and especially relevant) logics. In this paper we argue that such a semantics is of independent value and that it should be considered as a genuine alternative explication of the notion of logical consequence alongside the traditional model-theoretical and the proof-theoretical accounts. Our starting point is the content-nonexpansion platitude which stipulates that an argument is valid iff the content of the conclusion does not exceed the combined content of the premises. We show that this basic platitude can be used to characterise the extension of classical as well as non-classical consequence relations. The distinctive trait of an informational semantics is that truth-conditions are replaced by information-conditions. The latter leads to an inversion of the usual order of explanation: Considerations about logical discrimination (how finely propositions are individuated) are conceptually prior to considerations about deductive strength. Because this allows us to bypass considerations about truth, an informational semantics provides an attractive and metaphysically unencumbered account of logical consequence, non-classical logics, logical rivalry and pluralism about logical consequence.
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Notes
We use the terms “informational semantics” and “information-conditional” semantics interchangeably. The former is appropriate because of the analogy with informational content, whereas the latter emphasises the contrast with truth-conditional and proof-conditional semantics.
Although this restriction is primarily for reasons of focus, a more principled exclusion of linear logic can be mounted by, for instance, denying that linear logic (without distribution) should be seen as a formal model of logical consequence in natural language. This seems consistent with Girard’s original motivations for developing linear logic; which he didn’t conceive as a rival to classical logic. See Paoli (2007) for a dissenting opinion.
Other approaches could equally well be described as informational, e.g. the informational interpretation of substructural logic defended by Wansing (1993a, b), the informational reading of sequents mentioned in Paoli (2002, 33), and the Boolean informational conception described by John Corcoran (see e.g. Saguillo 2009).
To see why this holds, just note that for consistent situations \(s \sqsubseteq_c t\) reduces to the information-containment relation \(s \sqsubseteq t\) that is familiar from the Kripke-style semantics for intuitionist logic. We do not pursue this in further detail, but will come back to the issue of intuitionist content in relation to proof-theoretical semantics and the inferential account of consequence.
See also MacKay (1969, 56) on the need to distinguish between the measure of a thing and the thing itself. This worry obviously extends to qualitative individuations of informational content.
This means that in some cases, the semantics is justified in terms of a prior inferential practice (Mares 2009a, 348).
For the interpretation of the first-degree fragment of relevant logic, there is no need to be exceedingly precise about the notion of information that is at play. It could equally well be explained in terms of subjective information (the information stored in a database) as in terms of objective information (the information accessible in an environment). Roughly speaking, the former is closely related to the so-called American plan which uses a four-valued semantics for relevant logic (think, for instance, of Belnap’s “How a computer should think” Belnap (1976) reprinted in Anderson et al. (1992, §81)), while the latter is tied to the Routley-Meyer semantics that is distinctive for the Australian plan.
In virtue of \(s \sqsubseteq_c s'\) iff \(c \sqsubseteq_s s',\) it follow that the information in c also doesn’t have to be contained in the information in s′.
Let 1 be propositional constant such that \(s \,\Vdash 1\) iff \(s \in Log\). Note furthermore that in a Routley-Meyer semantics where the clause for negation is given by \(s \,\Vdash \neg A\) iff \(s^{*} \not\,\Vdash A, \) we have that (1) \(s \sqsubseteq s^{*}\) holds iff s is consistent, (2) s = s * holds iff s is maximal, and (3) the validity of double-negation elimination is ensured by s ** = s. To show that all logical situations are maximal, let s be a logical situation. This can be expressed as \(s \,\Vdash 1\). Furthermore, since all logical situations are consistent, we have \(s \sqsubseteq s^{*}\). Thus we may conclude that \(s^*\,\Vdash 1,\) which is just to say that s * is a logical and a fortiori also consistent. Applying the same reasoning to s * as we did for s, we may conclude that \(s^{*} \sqsubseteq s^{**}\). Because s ** is just s, we thus have \(s^{*} \sqsubseteq s\). Since we’ve already established that \(s \sqsubseteq s^{*},\) we conclude that s = s * as required.
The here described collapse of De Morgan and Intuitionist negation into Boolean negation only works on the assumption of distributivity. As described by Dunn (1993), in the absence of distribution we only obtain Ortho-negation.
A common objection could be that while meanings can in general be discriminated more or less finely, their individuation in terms of their logical properties (with logical equivalence as the only identity criterion) is unique. This objection assumes that there is something like a “logical degree of discrimination,” but as soon as we give up truth-conditions there are only logical and extra-logical differences in discrimination; the former due to differences in the logical vocabulary and the latter due to differences in the extra-logical vocabulary.
Remark that since by merely varying the set of designated elements, one can obtain different logics (compare for instance the strong Kleene 3-valued logic with the paraconsistent logic LP), the number of elements and their ordering do not suffice to fix the degree of logical discrimination.
This is compatible with the fact that some theories of truth will turn a non-classical account of consequence into a non-classical account of truth (but see Mares (2008, sect. 10) on how we may add a truth-predicate), and with the fact that semantic paradoxes can still exclude some combinations of logics and formal theories of truth.
Strictly speaking, if we follow the characterisation of Cook (2010), pluralism still implies a form of relativism in the sense that logical consequence is relative to a prior choice of a degree of logical discrimination.
Paoli (2002, 3.3) proposes such a reading, but because he uses the term ‘information’ to refer to data-types, it is only appropriate in the absence of the structural rules of weakening and contraction. A more open-ended reading is required.
As Stalnaker puts it: “The formalism of possible worlds semantics assumes that possible states of the world are disjoint alternatives, and that everything that can be said within a given context can be said by distinguishing between these alternatives. This assumption of internal completeness is required by the explanation of propositional contents as sets of possible states of the world, and this explanation is motivated by our account of the nature of representation: since to represent the world just is to locate it in a space of alternative possibilities, content should be explained in terms of those possibilities.” (Stalnaker 1986, 118).
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Acknowledgments
This paper was first presented at the Conference on the Foundations of Logical Consequence that was held in 2010 at the University of St Andrews. We would like to thank the organisers and audience of this conference as well as two referees of this journal for their encouragement and valuable feedback.
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Allo, P., Mares, E. Informational Semantics as a Third Alternative?. Erkenn 77, 167–185 (2012). https://doi.org/10.1007/s10670-011-9356-1
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DOI: https://doi.org/10.1007/s10670-011-9356-1