Abstract
How do we use language to represent the world? It hardly needs stating that there is an intimate relationship between meaning, logic, and reason. Yet, increasingly, the standard approach to semantics has come under fire in the same moment as the nature of logic itself has been questioned.
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- 1.
For example, for an overview of non-classical logics, see [1].
- 2.
- 3.
In this sense, we may make an obvious allusion to the idea that, as mathematics is the descriptive and explanatory “language” of physical space, so with formal logic and the space of reasons. This would also cohere with Sellars’ distinction between the “manifest” and “scientific” images, which, in brief, have to do with distinct languages of explanation, see (e.g. [19]).
- 4.
A similar argument is due to Wittgenstein [18].
- 5.
As Harman suggests, there are many cases of ordinary reasoning where agents do not make an inference to \(\beta \) from \(\alpha \) because, perhaps, \(\alpha \) is unreasonable given the antecedent context.
- 6.
This distinction between logic (proper) and reasoning is obviously not one that we will accept here, though see, for example, Beall [24] for a position that accepts the distinction with some interesting nuance, in the context of default classicality for paraconsistent logics.
- 7.
The following is folklore, but I particularly follow the account in [26].
- 8.
An additional issue that is worth mentioning here is that any attempt to provide a reductive account of formal rules by means of model-theory brings with it a significant loss of information. A couple of examples suffice to highlight this. Classically, \(\alpha \vee \beta \), and \(\alpha \ddagger \beta \) (not both \(\alpha \) and \(\beta \)), share the same semantic model. But, as Macfarlane [28] points out, they do not express the same kind of information: something will have been lost by their semantic identification. Similarly, a proof of an elementary mathematical conjunction and a proof of a conjunction involving a complex mathematical statement differ in terms of the kind of processes of verification involved, yet they are equivalent from the point of view of their being true or false. We shall also discuss in the following chapter the well-known “categoricity” problem [11, 14, 29–33], which has the result that the standard natural deduction framework for classical logic fails to rule out non-standard semantic models. In particular, that framework is easily shown to be sound and complete with respect to both the classical semantic model, and a model in which every formula is interpreted “true”.
- 9.
I discuss these issues in detail in Chap. 2.
- 10.
In this sense, there is also a clear connection, and also a distinction, with non-monotonic logics as developed to capture defeasible reasoning, artificial intelligence, and logic programming (e.g. [37]). The non-monotonicity of these logics allows for consequence relations that are not required (by monotonicity) to ensure that the addition of premises does not alter the validity of certain inferences, so that it is possible to retract consequences, for example. Whilst the approach to logic by means of interaction and dialogue requires a form of “non-monotonicity”, rather than writing this in to the semantics as is often the case there (with the so-called closed- or open-world assumptions, for example), the approach given here makes aspects of non-monotonicity part-and-parcel with the dynamics of syntax, with the result that semantics that does not require standard completeness results. Resultantly, it is possible to provide a syntactical basis accounting for closed- and open-world assumptions without writing them into the semantics, whilst also dealing with issues arising with default negation, for example (as discussed in (e.g. [38])). I will discuss this in detail in Chap. 5, where I show how the formal framework for interactions given in Chap. 4 can be developed into a framework for interactive proof and refutation search (a similar approach though developing a different set of formal tools is taken in [39]).
- 11.
I marshal more evidence to this effect in Chap. 3.
- 12.
See, for example, the discussion in [41].
- 13.
In the tradition to which Ernest refers, the work of Imre Lakatos (e.g. [43]) is, perhaps, most exemplary, and will be discussed in detail in Chap. 3, where I also discuss how this requires a distinct approach to more standard formal approaches to dialogue and games. For example, Lorenzen-style dialogical logic [44], is effectively a deterministic formula-checking approach that requires each dialogical move subsequent to the first to be determined by a set of rules about which inference rule must be applied. Hintikka’s [45] approach to game-theoretical semantics, on the other hand, has more to do with determining relationships between syntax and objects in semantic models, and, as such, do not really reflect the dialogical approach that we are interested in here. Moreover, just as Lorenzen’s dialogue games are deterministic evaluations of formulae, so too are Hintikka’s winning strategies fixed as soon as the moves are fixed by a set of game rules (see e.g. [46] for an excellent overview of the two approaches).
- 14.
I also take the overarching picture of reasoning presented by Laden [47] to be allied to the view discussed here, particularly since that work “[...] describes reasoning as the responsive engagement with others as we attune ourselves to one another and the world around us” (p. 8).
- 15.
So, rather than derive the denial of \(\alpha \) from the assertion of \(\lnot \alpha \) as Frege has it, rather \(\lnot \alpha \) is explained in terms of the fundamental role of denial.
- 16.
I will discuss this further in Chap. 2.
- 17.
I discuss this further in Chap. 2.
- 18.
This shift in perspective will obviously also bring with it a change in the way in which we think about soundness and completeness proofs, since semantic models can no longer play the role of keeping check on where our reasoning goes astray. As such, we shall discuss internal constraints on reasoning that give rise to an account of validity and local completeness results in Chaps. 4 and 6.
- 19.
This is not dissimilar to the view presented in Lakatos [43] on the nature of mathematical inquiry, but also the view of Per Martin-Löf [54] that a ‘proof is, not an object, but an act [...], and the act is primarily the act as it is being performed, only secondarily, and irrevocably, does it become the act that has been performed. (p. 231)’ There are also connections with the project initiated by Jean-Yves Girard (e.g. [55]), especially the relationship between proof-search and termination. I will situate the project outlined here in relation to all of these in Chaps. 2 and 4.
- 20.
- 21.
See [56] for an overview of the field.
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Trafford, J. (2017). Introduction: Reasoning in Time and Space. In: Meaning in Dialogue. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-47205-8_1
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