Skip to main content
SpringerLink
Log in

Inference and Epistemic Transparency

  • Published:
Topoi Aims and scope Submit manuscript

Abstract

In his paper “Explaining Deductive Inference” Prawitz states what he calls «a fundamental problem of logic and the philosophy of logic»: the problem of explaining «Why do certain inferences have the epistemic power to confer evidence on the conclusion when applied to premisses for which there is evidence already?». In this paper I suggest a way of articulating, and partly modifying, the intuitionistic answer to this problem in such a way as to both answer Prawitz’s problem and satisfy a requirement I argue to be crucial for any epistemic theory of the meaning of the logical constants: the requirement that evidence is epistemically transparent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Prawitz (2015), p. 77.

  2. Prawitz (2015), p. 73.

  3. Prawitz (2015), pp. 77–78. I do not think that Heyting would agree that the legitimacy of introductory inferences is constitutive of the meaning of their conclusions. More on this below.

  4. Prawitz (2015), p. 78.

  5. Prawitz (2015), p. 73. In Prawitz (2011) the problem is introduced and stated in the following terms: «If we think of a proof as a chain of inferences, the crucial problem is thus what it is that gives certain inferences the power to justify their conclusions.» (p. 389).

  6. Prawitz (2015), p. 84.

  7. What I will say here is largely based on Usberti (2015). However, in that paper I represented Prawitz’s ToG as motivated by the desire to grant epistemic transparency to the key notion of the theory of meaning, whereas from Prawitz (2015) it can be surmised that the main motivation was to permit to define proofs as chains of legitimate inferences.

  8. The question of the epistemic transparency of the notion of intuitive evidence for A will be discussed in a moment.

  9. Prawitz (2015), p. 83.

  10. Prawitz (2015), p. 96.

  11. Prawitz (2015), p. 88.

  12. Prawitz (2015), p. 89.

  13. Prawitz (2015), pp. 83–84.

  14. Prawitz does not discuss this very question. He discusses a related one, namely whether possession of an intuitionistic construction of A is epistemically transparent (see below in the text); but intuitionistic constructions are theoretical explicantia of evidence, not intuitive evidence.

  15. Williamson (2000), p. 95.

  16. Firth (1978), p. 218.

  17. Of course there are differences between the case considered by Firth and the ones that are encompassed by ToG (for instance, it is a case of abductive, not deductive, inference; its conclusion is atomic, not logically complex); but they do not seem relevant to the point I am making.

  18. Prawitz (2015), p. 98.

  19. Ibid.

  20. Ibid.

  21. I assume for simplicity that O is defined on grounds for sentences.

  22. Prawitz (2015), p. 89.

  23. Kreisel (1962), p. 202.

  24. Prawitz (2015), p. 87.

  25. e.g. Prawitz (1987): «The usual intuitionistic attempt to explain the logical constants in terms of what counts as proofs of sentences of different logical forms is quite misleading in that respect.» (p. 139).

  26. Usberti (2012), pp. 41–42.

  27. Dummett (1977), p. 6.

  28. Prawitz (2015), p. 88.

  29. Prawitz (2015), pp. 88–89.

  30. There is an important tradition behind this obliteration of the difference between evidence as a mental state and evidence as an object: Curry–Howard isomorphism and the consequent idea that «the question whether a term denotes a ground for an assertion of a sentence A coincides with the question of the type of the term» (Prawitz 2015, p. 89).

  31. This is clearly recognized by Prawitz: «It can be assumed to be part of what it is to make an inference that the agent knows the meanings of the involved sentences» (p. 96).

  32. For the reasons why I consider necessary to define a defeasible notion like justification instead of an indefeasible one like proof, see §5 below. Here I shall make reference to proofs in order to facilitate the comparison with Heyting’s explanation.

  33. Heyting (1955), p. 17.

  34. I have argued for this point in Usberti (2015), pp. 432–433.

  35. Dummett remarks that «we can make no suggestion for what it would be to be given a concept» (Dummett 1973, p. 241). He observes also that «the notion of identifying a concept [⋯] seems quite inappropriate.» (p. 408).

  36. I shall come back to this point below.

  37. Dummett (1977), pp. 391–392.

  38. See Usberti (2012), pp. 41–42.

  39. “Yields” is to be understood as equivalent to “is known to yield”.

  40. The crucial difference between the two notions—and a criterion to distinguish them—is that the former is defined by induction on the number of inferential steps, the latter by induction on the logical complexity of proved sentences.

  41. Here is a typical expression of this idea: «By a proof of A from Γ we may understand either a valid argument for A from Γ or, more abstractly, what such an argument represents; I shall here reserve it for latter use.» (Prawitz 2005, p. 678).

  42. “*” denotes the composition of functions.

  43. Prawitz (2015), p. 66.

  44. Ibid.

  45. There is evidence as well of intuitive conscious phenomena; see Evans (2010).

  46. Of course here “reflective” means no longer conscious, as in Prawitz’ use, but belonging to System 2 as characterized for instance by Evans (2010).

  47. Hadamard (1954), p. 45. Hadamard is quoting in turn Poincaré (1913).

  48. Hadamard (1954), p. 16.

  49. Hadamard (1954), pp. 29–30.

  50. Hadamard (1954), p. 15.

  51. According to Hadamard, «The true process of thought in building up a mathematical argument is certainly [...] to be compared with [⋯] the act of recognizing a person.» (p. 15).

  52. Cp. Hadamard’s remarks on the process of understanding Euclid’s proof (pp. 76–77).

  53. Rizzi (2016), p. 347.

  54. As a consequence, grammaticality judgements of native speakers are an important source of data for the linguist, whereas correctness judgements of subjects about inferences are not a source of data for the logician, although they are for the psychologist.

  55. This corresponds to Hadamard’s fourth stage, ‘precising’/verifying.

  56. Gentzen (1969), pp. 144–149.

  57. In ToG this distinction is blurred, since a proof consists of a chain of conscious inferences.

  58. Within PTS it is a conceptual necessity to define valid and canonical arguments by means of a simultaneous induction; since the definition of canonical argument makes reference to inferential steps, the notion of proof defined in this way is the inferential one.

  59. Prawitz (2015), p. 96.

  60. Actually I hold that it should be extended to possession of grounds for sentences of whatever logical complexity.

  61. Prawitz (2012), p. 893.

  62. Prawitz (2002), pp. 90–91.

  63. Prawitz (2009), p. 186.

  64. Usberti (2015), pp. 420–421.

  65. Cozzo (2015), p. 114.

  66. Prawitz (2015), p. 94.

  67. Prawitz (1980), p. 8.

  68. Usberti (2015), Sect. 3.

  69. Prawitz (2015), p. 89.

  70. According to Prawitz,

    to infer a conclusion A from a set Γ of premisses may be experienced not just as making the assertion A giving a set Γ of premisses as one’s reason but rather as seeing that the proposition asserted by A is true given that the propositions asserted by the premisses of Γ are true. To characterize correct reasoning we may need to give substance to this metaphorical use of seeing. (Prawitz 2012, p. 890).

    My suggestion in the paper quoted in the text was to give a computational substance to the metaphorical use of seeing.

References

  • Cozzo C (2015) Necessity of thought. In: Wansing H (ed) Dag Prawitz on proofs and meaning, Springer, Basel, pp 101–120

  • Dummett M (1973) Frege: philosophy of language. Duckworth, London (1981 2)

    Google Scholar 

  • Dummett M (1977) Elements of intuitionism. Clarendon Press, Oxford (Second Edition 2000)

    Google Scholar 

  • Evans JSt (2010) Thinking twice. Oxford University Press, New York

    Google Scholar 

  • Firth R (1978) Are epistemic concepts reducible to ethical concepts? In: Goldman AI, Kim J (eds) Values and morals. Reidel, Dordrecht, pp 215–229

    Chapter  Google Scholar 

  • Gentzen G (1969) The consistency of elementary number theory. In: Szabo ME (ed) The collected papers of Gerhard Gentzen. North Holland Publishing Company, Amsterdam, pp 132–213

    Google Scholar 

  • Hadamard J (1954) The psychology of invention in the mathematical field, Dover Publications, New York. (Originally published by Princeton University Press, 1945; enlarged edition 1949)

    Google Scholar 

  • Heyting A (1955) Les fondements des mathématiques. Intuitionnisme, Théorie de la démonstration, Gauthier-Villars, Paris. French translation: 1934 (Mathematische Grundlagenforschung, Intuitio-nismus, Beweistheorie). Springer, Berlin

    Google Scholar 

  • Kreisel G (1962) Foundations of intuitionistic logic. In: Nagel E, Suppes P, Tarski A (eds) Logic, methodology and philosophy of science, Stanford University Press, Stanford, pp 198–210

    Google Scholar 

  • Poincaré H (1913) Mathematical creation. In: The foundations of science. The Science Press, New York

    Google Scholar 

  • Prawitz D (1980) Intuitionistic logic: a philosophical challenge. In: von Wright GH (ed) Logic and philosophy. Nijhoff, The Hague, pp 1–10

    Google Scholar 

  • Prawitz D (1987) Dummett on a theory of meaning and its impact on logic. In: Taylor B (ed) Michael Dummett: contributions to philosophy. Nijhoff, The Hague, pp 117–165

    Chapter  Google Scholar 

  • Prawitz D (2002) Problems for a generalization of a verificationist theory of meaning. Topoi 21:87–92

    Article  Google Scholar 

  • Prawitz D (2005) Logical consequence from a constructivist point of view. In: Shapiro S (ed) The Oxford handbook of philosophy of mathematics and logic. Oxford U.P., Oxford, pp 671–695

    Chapter  Google Scholar 

  • Prawitz D (2009) Inference and knowledge. In: Pelis M (ed) The logica yearbook 2008. College Publications, King’s College London, London, pp 175–192

    Google Scholar 

  • Prawitz D (2011) Proofs and perfect syllogisms. In: Cellucci C, Grosholz E, Ippoliti E (eds) Logic and knowledge, Cambridge Scholar Publishing, Newcastle on Tyne, pp 385–402

    Google Scholar 

  • Prawitz D (2012) The epistemic significance of valid inference. Synthese 187(3):887–898

    Article  Google Scholar 

  • Prawitz D (2015) Explaining deductive inference. In: Wansing H (ed) Dag Prawitz on proofs and meaning, Springer, Basel, pp 65–100

    Google Scholar 

  • Rizzi L (2016) Linguistic knowledge and unconscious computations. Riv Int Filos Psicol 7(3):338–349

    Google Scholar 

  • Usberti G (2012) Anti-realist truth and truth-recognition. Topoi 31(1):37–45

    Article  Google Scholar 

  • Usberti G (2014) Gettier problems, C-justifications, and C-truth-grounds. In: Moriconi E, Tesconi L (eds) Second Pisa colloquium in logic, language and epistemology. ETS, Pisa, pp 325–361

    Google Scholar 

  • Usberti G (2015) A notion of C-justification for empirical statements. In: Wansing H (ed) Dag Prawitz on proofs and meaning, Springer, Basel, pp 415–450

    Google Scholar 

  • Williamson T (2000) Knowledge and its limits. Oxford University Press, Oxford

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DISPOC (Dipartimento di Scienze sociali, politiche e cognitive), Università degli Studi di Siena, via Roma, 56, 53100, Siena, Italy

    Gabriele Usberti

Authors
  1. Gabriele Usberti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriele Usberti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Usberti, G. Inference and Epistemic Transparency. Topoi 38, 517–530 (2019). https://doi.org/10.1007/s11245-017-9497-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11245-017-9497-1

Keywords

Search

Navigation