Abstract
We naturally take for granted that by performing inferences we can obtain evidence or grounds for assertions that we make. But logic should explain how this comes about. Why do some inferences give us grounds for their conclusions? Not all inferences have that power. My first aim here is to draw attention to this fundamental but quite neglected question. It seems not to be easily answered without reconsidering or reconstructing the main concepts involved, that is, the concepts of ground and inference. Secondly, I suggest such a reconstruction, the main idea of which is that to make an inference is not only to assert a conclusion claiming that it is supported by a number of premisses, but is also to operate on the grounds that one assumes or takes oneself to have for the premisses. An inference is thus individuated not only by its premisses and conclusion but also by a particular operation. A valid inference can then be defined as one where the involved operation results in a ground for the conclusion when applied to grounds for the premisses. It then becomes a conceptual truth that a valid inference does give a ground for the conclusion provided that one has grounds for the premisses.
An early draft of this paper was presented at a seminar arranged by professor Wansing to discuss the contributions to this volume and at another seminar at the Department of Philosophy at Stockholm University. I thank the participants for many valuable comments, many of which have influenced the final version of this paper; special thanks to Per Martin Löf for remarks about references and to Cesare Cozzo for carefully commenting a late version of the paper.
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- 1.
- 2.
Chrysippus (c. 280–207 BC) considers a story of the following kind: Running along a road following his master, who is ahead out of sight, a dog comes to a fork. The dog sniffs one of the roads. Finding no trace of the master, he suddenly sets off along the other road without sniffing.
- 3.
- 4.
Like Frege, and unlike Martin-Löf (1985), I do not take an expression of the form “it is true that \(\ldots \)”, where the dots stand for a declarative sentence, to be the form of a judgement. To assert a sentence of the form “\(\ldots \) is true”, where the dots stand for the name of a sentence \(A\), is to make a semantic ascent, as I see it, and is thus not the same as to assert \(A\).
- 5.
One of the lessons of Carroll’s (1895) regress is that we never get to a wanted conclusion if we see an inference as the assertion of an implication (see further fn 12 and 13).
- 6.
It is sometimes argued against this view that an inference may result not in the acceptance of the conclusion but in the rejection of a former belief. But such a belief revision is better analysed as consisting of a series of acts, the first of which is an inference in the present sense, resulting in an assertion being made or a belief being formed, which is then found to be in contradiction to another belief already held. Instead of using these two beliefs as premisses in another inference that would result in the categorical assertion of a contradiction, some of the former beliefs are reclassified as assumptions. Under these assumptions, a contradiction is inferred, and the resulting hypothetical assertion is the premiss of a last inference (a reductio ad absurdum) in which the negation of one of the assumptions is inferred.
- 7.
Some writers, e.g. Chateaubriand Filho (1999), point out that a flawless sequence of inferences may fail to carry conviction while a geometrical drawing may convince us completely of the truth of an alleged theorem. As long as this is a psychological question about what subjectively convinces us, I am not concerned with it here. If the drawings are regimented in such a way that it can be claimed that we get conclusive evidence for some logically compound assertions by observing the drawings, it may amount to a theory of what it is to have a ground for an assertion, alternative to the one developed in Sect. 3.5.
- 8.
Pagin (2012) develops a view of what has to be required of a good or valid inference that is different from mine, but he makes essentially the same point that truth is not enough; as he puts it, it has to be arrived at by a reliable method.
- 9.
As Cellucci (2013) points out we are often interested in new proofs of what is already known to be true, which he sees as an argument against the idea that knowledge is the aim of inference. New proofs are interesting when they give new grounds, so this observation reinforces the idea that the primary aim of inference is to acquire grounds or evidence.
- 10.
As argued by Corcoran (1974), this is a central aim of Aristotle’s logical work.
- 11.
Etchemendy (1990) argues in effect that if validity is defined in the way of Bolzano and Tarski, then knowing an inference to be valid is not sufficient for it to be legitimate, while if validity is defined in the traditional way it is sufficient. As will be seen, I do not think that there is such a difference.
- 12.
Carroll’s is thereby a philosopher who has raised the problem why a subject gets conclusive evidence for the conclusion (or as he puts, gets forced to accept the conclusion) of an obviously valid inference whose premisses she accepts; his point being that it does not help that she accepts the validity of the inference. See also fn 5.
- 13.
The dialectical situation that we are in, after having conceded that the subject \(S\) does not get evidence for the conclusion \(\mathcal {A}\) merely because of the inference \(I\) being valid, is quite similar to the one that Carroll describes, where \(S\) (the Tortoise) asks why she should accept the conclusion \(\mathcal {A}\) of an inference \(I\) whose premisses she accepts. The advocate of the relevance of validity (Achilles) asks \(S\) to accept that the sentence occurring in \(\mathcal {A}\) must be true if the sentences occurring in the premisses are, in other words, that \(I\) is valid, as defined traditionally. \(S\) agrees to this, and let us say that she now knows that \(I\) is valid. She has thus an extra premiss, which coincides with our clause (1\(^\prime \)), from which to infer that she has evidence for \(\mathcal {A}\), but she still asks why it follows that she has evidence for \(\mathcal {A}\). Admittedly, there is a valid inference to \(\mathcal {A}\) from the now available premisses, but there was one already from the original premisses, and it has been conceded that the mere validity of an inference is not enough in order to infer that a subject who performs the inference gets evidence for the conclusion. It could now be argued that the new inference is not only valid but is surely known by \(S\) to be valid, and to argue so (which is what Achilles does) is to take a second step in the regress that Carroll describes.
- 14.
From this one may be tempted to draw the conclusion that proof and evidence are equivalent notions, as Martin-Löf (1985) affirms, saying, “proof is the same as evidence”. But against this speaks that evidence is a much more general and basic notion. As already noted, there are kinds of assertions for which evidence is got first of all by other means than proofs.
- 15.
This point has been argued for by Detlefsen (1992). Even if evidence for asserting \( A{ \& }B\) is said to consist of a pair whose elements are evidence for asserting \(A\) and evidence for asserting \(B\), we have still to form this pair to get evidence for asserting the conjunction.
- 16.
Cesare Cozzo (1994) has worked out an inferentialism according to which the meaning of an expression is determined only by those primitive inferences that concern the expression in a genuine (rigorously defined) sense. To its advantage the resulting meaning theory becomes compositional.
- 17.
A notion of validity along these lines was first defined for natural deductions (Prawitz 1971, Appendix A1), and was then generalized to arbitrary arguments (Prawitz 1973); the definition given here is essentially as stated there except for letting the justifications be assignments to occurrences of inference figures instead of inference forms. Other variations occur in the definitions of validity given by Dummett (1991) and Schroeder-Heister (2006). None of these variations has relevance to the main question discussed in this section. I investigate in a forthcoming paper (Prawitz 2014) how these different notions of validity are related to each other and to the notion on intuitionistic proof discussed in the next section. It should be mentioned that previously I have also considered a variant form of validity that followed Tait’s (1967) definition of convertible terms and Martin-Löf’s (1971) definition of computable deduction in taking all normal derivations to be valid, which I used to prove normalizability or strong normalizability. But I now concur with Peter Schroeder-Heister saying that this notion does not explicate the idea that the introduction rules is meaning constitutive and it is better not called validity.
- 18.
“Die Intention geht \(\ldots \) nicht auf einen als unabhängig von uns bestehend gedachten Sachverhalt, sondern auf ein als möglich gedachtes Erlebnis (Heyting 1931, p. 113)”.
- 19.
Heyting stresses this character by contrasting his explanation with classical explanations in terms of transcendental state of affairs. (To assert a proposition is “la constatation d’un fait. En logique classique c’est un fait transcendant; en logique intuitionniste c’est un fait empirique” (Heyting 1930, p. 235)).
- 20.
Kleene’s realizability in terms of recursive functions is the first systematic interpretation of intuitionistic sentences. Kreisel (1959) considered an interpretation that was instead in terms of effective operations of higher types introduced by Gödel in connection with his system T (used for an interpretation deviating form Heyting’s). Kreisel (1962) called his interpretation “general realizability” (later renamed “modified realizability” by Troelstra (1973)). He also suggested another interpretation in terms of an abstract notion of construction Kreisel (1962a), deviating from Heyting’s explanations in a way that is of interest for the discussion here (see fn 22). The well-known acronym BHK is used for two different interpretations, the Brouwer-Heyting-Kreisel-interpretation due to Troelstra (1977), and the Brouwer-Heyting-Kolmogoroff-interpretation due to Troelstra and Dalen (1988). They are quite informally stated, the first one being inspired by Kreisel (1962a).
- 21.
I have here essentially followed an earlier presentation of mine (Prawitz 1970, 1971), where a homomorphic mapping of intuitionistic natural deductions into an extended lambda calculus is defined. By applying ideas of Howard (1980, privately circulated from 1969) one can get an isomorphic mapping by considering a finer type structure.
- 22.
Kreisel (1962a) proposes that a proof of \(A\supset B\) or \(\forall xA(x)\) is a pair whose second member is a proof of the fact that the first member is a construction that satisfies clause 2) or 3) above. Thus, he presupposes that we already know what a proof is; it is thought that the second proof establishes a decidable sentence and that a reduction has therefore taken place. Troelstra (1977) first BHK-interpretation follows Kreisel’s proposal saying that a proof of \(A\supset B\) or \(\forall xA(x)\) consists of a construction as required in clause 2) or 3) together with the insight that the construction has the required property. The second BHK-interpretation by Troelstra and Dalen (1988) drops the additional requirement of insight without any comments.
(Dummett (1977), pp.12–13) maintains that clauses 2) and 3) do not correctly define what an intuitionistic proof is, and says that a proof of e.g. \(A\supset B\) is “a construction of which we can recognize that applied to any proof of \(A\), it yields a proof of \(B\)”; thus, a proof is not a pair that contains a proof, but the recognizability in question is a required property of a proof.
Sundholm (1983), who analyses in detail the difference between Heyting’s and Kreisel’s views of proofs, differentiates between constructions in the sense of objects and in the sense of processes, and suggests that it is not from the construction \(c\) of \(A\) but from the construction of this construction that it is to be seen that \(c\) is a construction of \(A\).
Although Martin-Löf (1998) denies that intuitionistic proofs are proofs in an epistemic sense, knowing them are according to him what entitles us to make assertions (but see also fn 26). This is related to his view of the two kinds of knowledge that occurred in the discussion above. In his view knowledge of a truth is to be analysed in the end as knowledge how; knowledge that “evaporates on the intuitionistic analysis of truth” (Martin-Löf 1985).
- 23.
Whether it is merely a mental state, as Williamson (2000, p. 21) claims knowledge to be, need not be discussed here.
- 24.
This answers questions raised by Cesare Cozzo in this volume and by Pagin (2012) whether one has to decompose the inference schemata Barbara and Modus ponendo tollens, respectively, into a chain of natural deduction inference schemata in order to show that they are legitimate. As will be seen (Sect. 3.6), there is no such need: when the operations Barb and Mtp are assigned to the inference schemata in question, valid forms of inferences arise.
- 25.
As Cesare Cozzo remarks in his contribution to this volume, previous definitions of inference that I have given had the defect that they did not allow one to make this distinction.
- 26.
Martin-Löf (1984) type theory contains rules for how to prove such assertions, or judgements, as they are called there, written \(a\in A\) (or \(a:A\) in later writings). There are also assertions of propositions in the type theory, but they have the form “A is true” and are inferred from judgements of the form \(a\in A\), where \(a\) corresponds to what I am calling a ground. Thus, the assertions in the type theory are, as I see it, on a meta-level as compared to the object level to which the assertions that I am discussing belong. (But compare also fn 22, which has a more general setting than type-theory.)
References
Boghossian, P. (2001). How are objective epistemic reasons possible? Philosophical Studies, 106(1–2), 340–380.
Boghossian, P. (2003). Blind reasoning. Proceedings of the Aristotelian Society, Supplementary, 77(1), 225–248.
Boghossian, P. (2012). What is an inference? Philosophical Studies, 169(1), 1–18.
Carnap, R. (1934). Logische syntax der sprache. Vienna: Springer.
Carroll, L. (1895). What the tortoise said to achilles. Mind, 4(14), 278–280.
Cellucci, C. (2013). Philosophy of mathematics: Making a fresh start. Studies in History and Philosophy of Science, 44(1), 32–42.
Chateaubriand Filho, O. (1999). Proof and logical deduction. In E. H. Hauesler & L. C. Pereira (Eds.), Proofs, types and categories. Rio de Janeiro: Pontifícia Universidade Católica.
Corcoran, J. (1974). Aristotle’s natural deduction system. In J. Corcoran (Ed.), Ancient Logic and its Modern Interpretation (pp. 85–131). Dordrecht: Reidel.
Cozzo, C. (1994). Meaning and argument. A theory of meaning centered on immediate argumental role. Stockholm: Almqvist and Wicksell International.
Detlefsen, M. (1992). Brouwerian intuitionism. In M. Detlefsen (Ed.), Proof and knowledge in mathematics, (pp. 208–250). London: Routledge.
Dummett, M. (1973). The Justification of Deductions. London: British Academy.
Dummett, M. (1977). Elements of intuitionism. Oxford: Clarendon Press.
Dummett, M. (1991). The logical basis of metaphysics. London: Duckworth.
Etchemendy, J. (1990). The concept of logical consequence. Cambridge: Harvard University Press.
Gentzen, G. (1934–1935). Untersuchungen über das logische schließen I. Mathematische zeitschrift, 39(2), 176–210.
Heyting, A. (1930). Sur la logique intuitionniste. Académie Royale de Belgique, Bulletin de la Classe des Sciences, 16, 957–963.
Heyting, A. (1931). Die intuitionistische grundlegung der mathematik. Erkenntnis, 2(1), 106–115.
Heyting, A. (1934). Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie. Berlin: Springer.
Heyting, A. (1956). Intuitionism, An Introduction. Amsterdam: North-Holland Publishing Company.
Howard, W. (1980). The formula-as-types notion of construction. In J. R. Hindley & J. P. Seldin (Ed.), To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (pp. 479–490). London: Academic Press.
Kreisel, G. (1959). Interpretation of analysis by means of constructive functionals of finite types. In A. Heyting (Ed.), Constructivity of Mathematics (pp. 101–128). Amsterdam: North-Holland Publishing Company.
Kreisel, G. (1962a). Foundations of intuitionistic logic. In E. Nagel (Ed.), Logic methodology and philosophy of science (pp. 198–212). Stanford: Stanford University Press.
Kreisel, G. (1962b). On weak completeness of intuitionistic predicate logic. Journal of Symbolic Logic, 27(2), 139–158.
Martin-Löf, P. (1971). Hauptsatz for the intuistionistic theory if iterated inductive definitions. In J. E. Fenstad (Ed.), Proceedings of the Second Scandinavian Logic Symposium (pp. 179–216). Amsterdam: North-Holland Publishing Company.
Martin-Löf, P. (1984). Intuistionistic type theory. Napoli: Bibliopolis.
Martin-Löf, P. (1985). On the meaning of the logical constant and the justification of the logical laws. Atti degli Logica Matematica 2 (pp. 203–281). Siena: Scuola de Specializzazione in Logica Matematica: Dipartimento de Matematica, Università di Siena.
Martin-Löf, P. (1998). Truth and knowability: on the principles C and K of Michael Dummett. In H. G. Dales & G. Oliveri (Eds.), Truth in Mathematics (pp. 105–114). Oxford: Clarendon Press.
Pagin, P. (2012). Assertion, inference, and consequence. Synthese, 187(3), 869–885.
Prawitz, D. (1970) Constructive semantics. In Proceedings of the 1st Scandinavian Logic Symposium Åbo 1968 (pp 96–114). Uppsala: Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet.
Prawitz, D. (1971). Ideas and results in proof theory. In J. E. Fenstad (Ed.), Proceedings of the Second Scandinavian Logic Symposium (pp. 235–307). Amsterdam: North-Holland Publishing Company.
Prawitz, D. (1973). Towards a foundation of general proof theory. In P. Suppes (Ed.), Logic, Methodology and Philosophy of Science IV (pp. 225–250). Amsterdam: North-Holland Publishing Company.
Prawitz, D. (2011). Proofs and perfect syllogisms. In E. R. Grosholz & E. Ippoliti & C. Cellucci (Eds.), Logic and Knowledge (pp. 385–402). Newcastle on Tyne: Cambridge Scholar Publishing.
Prawitz, D. (2014). On relations between Heyting’s and Gentzen’s approaches to meaning. In T. Piecha & P. Schroeder-Heister (Eds.), Advances in Proof-Theoretic Semantics, forthcoming.
Prior, A. (1960). The runabout inference ticket. Analysis, 21(1), 38–39.
Ross, W. D. (1949). Aristotle’s prior and posterior analytics. Oxford: Oxford University Press.
Smith, N. (2009). Frege’s judgement stroke and the conception of logic as the study of inference not consequence. Philosophy Compass, 4(4), 639–665.
Sundholm, G. (1983). Constructions, proofs and the meaning of logical constants. The Journal of Philosophical Logic, 12(2), 151–172.
Sundholm, G. (1994). Existence, proof and truth-making: A perspective on the intuistionistic conception of truth. Topoi, 12(2), 117–126.
Sundholm, G. (1998a). Inference versus consequence. In T. Childers (Ed.), The LOGICA Yearbook 1997 (pp. 26–35). Prague: Filosofia Publishers.
Sundholm, G. (1998b). Proofs as acts and proofs as objects. Theoria, 54(2–3), 187–216.
Sundholm, G. (2012). “Inference versus consequence” revisited: Inference, consequence, conditional, implication. Synthese, 187(3), 943–956.
Tait, W. W. (1967). Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2), 198–212.
Troelstra, A. S. (1973). Mathematical Investigations of Intuistionistic Arithmetic and Analysis. Lecture Notes in Mathematics. Berlin: Springer.
Troelstra, A. S. (1977). Aspects of constructive mathematics. In J. Barwise (Ed.), Handbook of Mathematical Logic (pp. 973–1052). Amsterdam: North-Holland Publishing Company.
Troelstra, A. S. & van Dalen, D. (1988). Constructivism in mathematics. Amsterdam: North-Holland Publishing Company.
Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press.
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Prawitz, D. (2015). Explaining Deductive Inference. In: Wansing, H. (eds) Dag Prawitz on Proofs and Meaning. Outstanding Contributions to Logic, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-11041-7_3
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