Abstract
This paper concerns the connection between speech act theory, especially the theory of assertion, and deduction, especially Natural Deduction.
From a very abstract point of view, an assertion of a content p can be described as the ascription of the property of being p to the actual index, or point of evaluation. This is the abstract characterization of assertoric force. Let’s assume that the actual index is a possible world, namely the actual world. Thus, the conclusion of a closed argument, as an act, is an assertion, and thereby characterized as the ascription of the conclusion content as a property to the actual world.
The question that will concern us in this talk is how this idea extends to the status of other acts in the practice of Natural Deduction. In these terms, what is the force of an inference that depends on one or more open assumptions? What is the force of the assumption itself? What is the force of an assertion that depends on an entire derivation? Do we need to ascribe a force to the derivation as a whole? Is there a coherent complete theory of act forces of Natural Deduction along these lines? What is the relation of such a theory to the theory of validity of an argument?
This paper has been presented at History, Epistemology and Logic of Justification Practices Summer School, University of Tübingen, Friedrich von Weizsäcker Zentrum, in August 2022, at a CLLAM seminar at the Department of Philosophy, Stockholm University, and at the Philosophy Seminar at KTH, The Royal Institute of Technology, Stockholm, both in September 2022. I am grateful to many participants at those seminars, in particular Cesare Cozzo, Ansten Klev, Tor Sandqvist, Dag Prawitz, Per Martin-Löf, Valentin Goranko, and John Cantwell. I have also benefitted from additional comments from Dag Prawitz, Dag Westerståhl, Ansten Klev, and Kathrin Glüer. The paper finally improved in important respect thanks to comments from two anonymous referees.
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Notes
- 1.
The same goes for the German original, ‘Anerkennung’. Thanks to Kathrin Glüer here.
- 2.
Cf. Pagin (2001).
- 3.
The general appeal to grounds was part of a project to characterize inference and argument validity more broadly, beyond Natural Deduction and beyond formal systems in general. That project is ongoing, but in the most recent years, it has changed direction. Instead of quantifying over grounds, Prawitz now appeals to more structural principles that relate the concepts of validity of an inference and validity of an argument. See Prawitz (2019) and Prawitz (2023).
- 4.
We should not confuse the force of an assumption with the force of an utterance such as ‘Assume that A’. Such an utterance can be seen as signaling that the speaker is making an assertion, or as proposing that an assumption be made in a conversation, or yet something else. But signaling that one is making an assumption, or proposing to make an assumption, or urging someone to make an assumption, are all different from merely making an assumption.
- 5.
The objection can be pushed further by supposing that A contains demonstrative expressions which get definite values in some particular context of utterance. The result will depend on general views about the nature of demonstratives and contexts. If a context does not contain a fully determined index, then the demonstrated object is available for demonstration at more than one index. If contexts are fully determinate with respect to all index elements, then the interpretation of A fixes the index of the context. But that still allows that A is evaluated for truth value at another index, for instance at another possible world.
In David Kaplan’s theory of demonstratives (Kaplan 1989, 552), an assertion of a sentence A is evaluated for truth value at the index \(i_{c}\) of the context of use c of A. This is the actual index from the point of view of the context. But an evaluation of \(\Diamond A\) at \(i_{c}\) leads to an evaluation of A at distinct possible worlds, even if A contains demonstratives. And, of course, in an utterance of \(\Diamond A\), A is not asserted; the assertion of \(\diamondsuit A\) may be true even if A is false at c. All sentences will be evaluated at non-actual indices when embedded under modal operators, or temporal operators, or other index-shifting operator, regardless of containing demonstratives. Hence, any sentence can be so evaluated. There is thus no problem in principle in allowing an assumption of A to be evaluated at other indices than the actual one.
- 6.
The question of the force of assumption is raised also in Sundholm (2006, 627ff.). Sundholm there proposes a separate force for assumptions, and suggests indicating it with a reversed turnstile, ‘\(\dashv \)’ (2006, p. 629). He himself then partially rejects the idea; not for the reason that assumptions lack force, but for the reason that a more comprehensive theory is needed that also provides the force for the conditional assertion made under assumptions.
- 7.
I have suggested this basic idea before, in Pagin (2007). Is conditional force a kind of force? I would say no. It is a force-related status, distinct from categorical force, and distinct also from merely lacking (categorical) force.
- 8.
This idea has been developed by Belnap (1973). I have criticized it in Pagin (2007). In fact, Sundholm (2006, p. 630) proposes exactly this, transposed into the inferential relation: in an argument from the assumption A to the conclusion B, B is asserted on the condition that A is true. He says so also with reference to Belnap, so it is clear that he sees his view close to that of Belnap’s.
- 9.
Is there a stable correlation between on the one hand endorsing the inference and on the other hand the disposition to (being prepared to) assert the conclusion if one (is prepared to) assert the premises? This is a substantive question. There can of course be irrational discrepancies: the speaker may be so wired up as to assert the conclusion of an argument \(\Sigma \) if he asserts the premise even if he does not endorse \(\Sigma \).
There is also the issue of the vacuous conditional: it may be true that I would assert B if I were to assert A, simply because I would never assert A, and this would hold whether or not I endorsed the inference from A to B. This was pressed by Tor Sandqvist. The general connection is of course not vacuous, since it is schematic and it has infinitely many non-vacuous instances. In the particular vacuous instance, the conditional analysis does not give the right result, and something must be said about that. This is not a problem of principle, however.
More interestingly, some inferences seem to be unacceptable even though the disposition is rational. Consider the inference from A to There is a verification of A. This inference is not valid. But when I am prepared to assert A, this is because I have some kind of verification of A, but when I do, I am also prepared assert that there is a verification of A (if you like, change the example to the support of your choice). Hence, there is a discrepancy between the two notions that should not be overlooked. It depends on reflecting exactly what turns an assumption into an assertion. What nevertheless remains is that a rational thinker who asserts all the premises and endorses the argument also asserts the conclusion. The opposite also holds, with special exceptions, noted above.
(This example also has a bearing on the intuitionistic notion of a valid inference cf. an exchange between myself and Prawitz: Pagin 1998, Prawitz 1998, Pagin 2009).
In discussion, Prawitz has also emphasized, as in many of his writings (cf. Prawitz 1977, 1980, 1985), that an argument may fail to be valid because it lacks the epistemic justification, even though the truth of the premises do necessitate the truth of the conclusion. This is an important point, but not directly relevant in the present context, since we are concerned with a thinker’s disposition to assert the conclusion given that he asserts the premises.
We will come back to the idea of inferential force in Sect. 10.5.
- 10.
Sufficient conditions for correctness would be of two kinds. Either there is for every (relevant) object a term that refers to it, such that every such substitution instance would be correct to assert, or else there exists an argument with x free that concludes with Ax and licenses a universal introduction to \(\forall x Ax\). Thanks to an anonymous reviewer for pressing the point.
- 11.
There is the sequent format of ND, and the distinctly different sequent system LK, which allows multiple formulas after the arrow. This distinction is not relevant here, as I am mainly concerned with the issue of using sequents instead of formulas as the unit in derivations.
- 12.
Sundholm (2006, 631ff) comes to the opposite conclusion. Sundholm holds that the conditional form B is true (A is true), i.e. that B is true on condition that A is true, captures the force both of assumptions and conditional assertions, and that therefore “Gentzen’s standard Natural Deduction turns out to be nothing but the Sequent Calculus version in disguise: as a matter of semantical fact, at each node the assumptions are carried along.” (p. 631). For reasons already given, I reject this view.
- 13.
Schröder-Heister (2003) notes that standard Natural Deduction is heavily assertion oriented and gives assumptions a subordinate role. Schröder-Heister undertakes the project of making their roles more equal, and uses the SC format for the task. The first basic idea is that assertions are governed by right-hand rules of inference, modifying the succedent, while assumptions are governed by left-hand rules, modifying the antecedent. The second main idea is that this requires a change of the specificity of assumptions. In standard ND, there are no restriction at all on what formula can be used as an assumption, while in the new system, there are restrictions in particular contexts of derivation on what assumptions can be made there, requiring more specific assumptions. The cost of the change is that the derivability relation no longer is transitive by default. Rather, transitivity needs to be guaranteed by certain more general restrictions on derivations. Schröder-Heister discusses several alternatives.
By contrast, here I have accepted the assertion oriented nature of deduction.
- 14.
- 15.
This use of ‘premise’ as covering entire derivations goes beyond orthodoxy. Prawitz (1965, p. 23) employs scare quotes when extending the use that way.
- 16.
Easily shown by induction.
- 17.
For a concise introduction, see chapters 1 and 2 of Troelstra and Schwichtenberg (2000).
- 18.
Prawitz (2012, 191–194) uses a parallel notation for grounds of assumptions, grounds for assertions under assumptions (unsaturated grounds), and grounds for categorical assertions (closed grounds).
- 19.
By contrast, in Martin-Löf’s Intuitionistic Type Theory (ITT), the format \(\ulcorner t:A \urcorner \) is used for expressing judgments. ITT employs the propositions-as-types interpretation: propositions are the types of their proofs. A judgment \(t:A\) says that t is of type A and again that t is a proof of A. The locus classicus of ITT is Martin-Löf (1984). For a recent introduction, see Dybjer and Palmgren (2020).
Sundholm (2019) uses the free variable notation for propositional assumptions, which he also characterizes as an alethic or ontological assumption, and writes it “\(x: \mathrm {Proof}(A)\)” (2019, 557). He contrasts this with epistemic assumptions, which are assumptions that a proposition has been proved. This is written “\(a: \mathrm {Proof}(A)\)” with a closed proof term a. Here it is the judgment that is assumed rather than the proposition. What it amounts to that a judgment is assumed is in the paper taken as informally understood.
It may be noted that Sundholm characterizes ND derivations (the Gentzen 1932 format) under assumptions as dependent proof-objects \(\Pi \) of the form \(\Pi : C(x_{1}:A_{1}, \ldots , x_{k}:A_{k})\), saying that \(\Pi \) is a proof of C under the assumptions that \(x_{1}, \ldots , x_{k}\) are proofs of \(A_{1}, \ldots , A_{k}\), respectively. This has a clear affinity with the characterization of hypothetical assertion in terms of proof terms given here.
- 20.
To make the picture complete, we also have to consider empirical—typically inductive—inferences from non-empty sets of premises, exemplified by “The streets are wet. Hence, it has been raining.” In these cases the inference step adds uncertainty even in a single premise inference. The main point of the present section does not require us to take this into account.
- 21.
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Pagin, P. (2024). Assertion, Assumption, and Deduction. In: Piccolomini d'Aragona, A. (eds) Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Synthese Library, vol 481. Springer, Cham. https://doi.org/10.1007/978-3-031-51406-7_10
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