Abstract
In this chapter, I shed light on the issues raised in the above chapter by providing an overview of the concepts of proof and assertion, both of which are central to inferentialist approaches to semantics. In so doing, I shall argue that, in contrast to the prescriptive and monological approach to logic that underlies the problems discussed in the previous chapter, we should think of both proof and assertion in a much more liberal way. In brief, the suggestion is that, to take seriously the dictum that “meaning is determined by use” requires us to provide an account of the processes of our proving activities, which take place in the context of assertion games.
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Notes
- 1.
The concept of “inference tickets” is central to the discussion of tonk [1], though it possibly originates in the work of Gilbert Ryle [2]: ‘Knowing if p, then q is, then, rather like being in possession of a railway ticket. It is having a license or warrant to make a journey from London to Oxford’ (p. 308).
- 2.
In other rejectivist approaches, for example that of Rumfitt [8], there is an emphasis on the distinction between force and content. There, for any content, it is the case that it can be marked with the force of assertion, or the force of denial. This is formalised with “ signed” sentences \(+ \alpha \) and \(- \alpha \), signifying the force-markers on sentential metavariables. This is taken to formally capture the English language form of a question-forming sentence \(\alpha \)?, with the answer “ yes” or “no”.
- 3.
Restall uses the standard turnstile, but I use
to bring the duality to the surface. - 4.
So, importantly, the sequent calculus is to be understood not as a metacalculus for single-conclusion sequents, but as directly marshalling the relation between formulas of a language.
- 5.
- 6.
This is following [15], and these are “quasi”-partitions since we allow that either of \(\Gamma ^{\prime }\), \(\Delta ^{\prime }\) may be empty.
- 7.
The other obvious issue with the symmetric classical solution, for the inferentialist, is that whilst relatively maximal theories (and quasi-partitions) can stand in as syntactic counterparts to truth and falsity, this is only possible by idealization over the entire language.
- 8.
If we try to read the turnstile in terms of consequence, then we quickly run into familiar issues facing multiple conclusion consequence.
- 9.
This is Steinberger’s [16] example.
- 10.
That is, unless structural rules are altered as in (e.g. [14]).
- 11.
- 12.
By way of response to this issue, Lafont suggests that we might restrict sequents to asymmetric form, as in intuitionistic logic, in which case (\(Weak_R\)) will not hold. Lafont also considers linear logic, but I will not pursue that here, returning to this issue in Chap. 5.
- 13.
- 14.
Dummett, in [20, p. 254], calls the fact that every closed derivation in an intuitionistic entailment structure can be reduced to a canonical derivation, the “fundamental assumption”.
- 15.
This account also relies upon the subset of atomic formulas being taken to be “valid” in the sense that evidence is available that they hold. Then, a set of logical rules extends atomic validity to an inference structure as defined in the previous chapter.
- 16.
I have already mentioned the relationship between this and issues of compositionality in the previous chapter. As I said there, I think that the problems facing the classical inferentialist are not assuaged by shifting to intuitionistic versions.
- 17.
As [22] puts it, according to Prawitz, ‘[a]n argument is valid if either it reduces to a non-logical justification of an atomic sentence, or it reduces to an argument whose last inference is an introduction inference and whose immediate subarguments are valid’, (p. 7).
- 18.
We should note that this emphasis is clearly linked to the formalisation of proofs in a natural deduction framework, which, as Schroeder-Heister [23] has explained at length, allow open assumptions only to be placeholders for closed derivations. This is not the case in sequent-calculi, which is one reasons why later the proof-theoretic framework will be developed primarily in that framework.
- 19.
I will not enter into a historical exegesis regarding the two positions (on this, the references above are more than adequate).
- 20.
As such, this approach is usually thought to cohere with the standard Brouwer-Heyting-Kolmogorov (BHK) interpretation of logical connectives as providing intuitive constraints on the complex composition of proofs involving logical constants. But, this is clearer in Kolmogorov’s [28] interpretation of a constructive approach to logic than even Brouwer’s, which argues that constructive logic has to do ‘not with theoretical propositions but, on the contrary, problems’:
In addition to theoretical logic, which systematizes a proof schemata for theoretical truths, one can systematize a proof schemata for solutions to problems [...] In the second section, assuming the basic intuitionistic principles, intuitionistic logic is subjected to a critical study; it is thus shown that it must be replaced by the calculus of problems, since its objects are in reality problems, rather than theoretical propositions [28, p. 58] (Translated in [29]).
- 21.
Kolmogorov’s [28] analysis of the conditional also sheds light on this understanding of the role that conditional plays. There, he argues that a solution to the problem posed by a conditional \(\alpha \rightarrow \beta \) must ‘carry the solution of \(\beta \) back to the solution of \(\alpha \)’ (p. 59). In other words, a solution to the conditional is only partially solved by providing a function, \(f_H\) mapping a hypothetical \(\alpha \) into \(\beta \), which becomes a full solution (\(f_A\)) when it is shown that this carries back to a solution for \(\alpha \). On this construal, a solution of \(\alpha \rightarrow \beta \) is a solution of \(\beta \) with premise \(\alpha \), which implies a tacit appeal to a given solution of \(\alpha \), to which a proof of \(\beta \) would “carry back to”. See [34] for discussion.
- 22.
The quotation has been altered slightly to reflect the fact that I am interested in conditional rather than knowability, but the point is theirs.
- 23.
The notion of a hypothetical proof bears obvious similarity to Girard’s [35] notion of paraproof.
- 24.
The analogy Sundholm makes is that written proofs are like annotations for a game of chess, as opposed to proof-acts, which are like the game itself.
- 25.
See [38] for a similar formulation and extended discussion of deduction theorem.
- 26.
Though, unlike the discussion outlined in the introduction, this distinction should not be taken as requiring the reduction of the former to the latter.
- 27.
There is an excellent discussion of these issues in [41].
- 28.
Edgington [42] goes further still: ‘to assert a conditional is to assert that it is true on condition that it has a truth value. To believe a conditional is to believe that it is true on the supposition that it has a truth value. It has a truth value iff its antecedent is true’.
- 29.
This follows Edgington’s [47] argument.
- 30.
This is discussed below, and in [51].
- 31.
Additionally, it may well be the case that these views can be made compatible with the account proposed below in some way.
- 32.
This also follows Wittgenstein’s suggestion that making an assertion is to make a move in a game [57, Sect. 22].
- 33.
An excellent discussion of these issues can be found in [60].
- 34.
See also the excellent discussion in [61].
- 35.
Pagin, in several places [53, 54, 62, e.g.], makes an argument to the effect that the social account of assertions does not, by itself, provide sufficient conditions on the nature of assertions, whilst he accepts that it may be the case that they provide necessary conditions. The discussion in [52] provides a useful rejoinder, though, in any case I do not think that this is an issue for the view espoused here. For example, the kinds of problems usually thought to face commitment approaches involve examples where assertions are made without explicitly making statements, through nonlinguistic signs, for example. I don’t think that these are problematic for the account given here, since, it seems perfectly acceptable that one might ask for reasons for such signs, thereby clarifying them, in the same way as linguistic statements. A slightly different example given by Nunberg [63], and discussed in [52], is a waitress who states that “The ham sandwich left without paying”. The waitress has made an assertion, though it does not seem correct to say that she has asserted that the ham sandwich left without paying. But, whilst this may seem prima facie problematic for a commitment view, I agree with Macfarlane [52], that, to the contrary, this view fares very well in this respect:
[...] if we wanted to settle, for example, whether Nunberg’s waitress had asserted that a sandwich had left, or that a person who ordered a sandwich had left, we might ask with (if either) of these propositions she meant to commit herself to.
- 36.
- 37.
See also [66] for a similar approach to the relationship between assertion and proving.
- 38.
In Brandom [59] this is related to Socratic method, which, he argues, is:
[...] a way of bringing our practices under rational control by expressing them explicitly in a form in which they can be confronted with objections and alternatives, a form in which they can be exhibited as the conclusions of inferences seeking to justify them on the basis of premises advanced as reasons, and as premises in further inferences exploring the consequences of accepting them (p. 56).
I shall develop the connection between this approach to logic and the dialogical tradition in the following chapter.
- 39.
This will be developed in the context of formalism in Chap. 4.
- 40.
Furthermore, this objectively true notion of the proof of a statement is equivalent with its truth, according to Dummett [67].
- 41.
So, whilst Restall’s work highlights one way in which norms governing the coherence of speech acts can underpin a theory of meaning, devoid of social context, it is difficult to make any sense of how such assertions and denials are supposed to interact with others.
- 42.
There are certain constraints on the structure of these interactions that we shall discuss in the following chapter, and it is these (I shall argue there and in Chap. 4) that provide a justification for logical rules.
- 43.
- 44.
For example, Wittgenstein’s argument in [57] against the force/content distinction is much stronger than the view here.
- 45.
See also [71].
- 46.
If we take Dummett and Tennant’s description of the reasons why harmony is important, then we might say that cut-elimination would therefore ensure that the kind of balance assertions and consequences is maintained across the course of the proof. Of course, it is also the case that cut-elimination allows us to show that a proof-system is consistent, and that it has the sub-formula property - that each formula in the end sequent of a cut-free proof is a sub-formula of one of the premises.
- 47.
I have briefly suggested the shape that these constraints take, in terms of the relationship between assertions and testing, giving and asking for reasons, but I shall put some meat on these bones in the following chapter.
- 48.
Though, as mentioned in the introduction (and to be discussed further in the following chapter), it is plausible to think that even individual reasoning involves an internalised dialogue.
- 49.
This context will be discussed in further detail in the following section, but I note in passing that this would at least require that Restall’s cut rule, which requires coherence to hold unrestrictedly must be jettisoned so that an assertion and a test may be simultaneously present. This alone is enough to think that grounding the account in the notion of incoherence and coherence will not work when the speech-acts are understood within a social context, rather than in terms of an individual.
- 50.
In fact, Dummett [72] suggests that something like this may be required for non-mathematical statements:
[A] proof of the negation of any arbitrary statement then consists of an effective method for transforming any proof of that statement into a proof of some false numerical equation. Such an explanation relies on the underlying presumption that, given a proof of a false numerical equation, we can construct a proof of any statement whatsoever. It is not obvious that, when we extend these conceptions to empirical statements, there exists any class of decidable atomic statements for which a similar presumption holds good; and it is therefore not obvious that we have, for the general case, any similar uniform way of explaining negation for arbitrary statements. It would therefore remain well within the spirit of a theory of meaning of this type that we should regard the meaning of each statement as being given by the simultaneous provision of a means for recognizing a verification of it and a means for recognizing a falsification of it, where the only general requirement is that these should be specified in such a way as to make it impossible for any statement to be both verified and falsified (p. 71).
If we translate the latter stipulation into a BHK-style clause for negation, so that the negation of \(\alpha \) is verified whenever \(\alpha \) is falsified (and vice-versa), then this is analogous to Nelson’s strong negation [73], and discussed in a similar context in [74].
- 51.
Whilst such weakly-negated formulas would obviously fail standard monotonicity conditions (i.e. that once \(\alpha \) is negated, it is always negated), as [75, Sect. 2.4] points out, adding such an operator to standard intuitionistic logic would be conservative since its existence has no impact on the usual interpretation of all other connectives.
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Trafford, J. (2017). Proof and Assertion. 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_3
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