The notion of problem, intuitionism and partiality

Pavel Materna



Problems are defined as abstract procedures. An explication of procedures as used in Transparent Intensional Logic (TIL) and called constructions is presented and the subclass of constructions called concepts is defined. Concepts as closed constructions modulo α- and η-conversion can be associated with meaningful expressions of a natural or professional language in harmony with Church’s conception. Thus every meaningful expression expresses a concept. Since every problem can be unambiguously determined by a concept we can state that every problem is a concept and every concept can be viewed as a problem.Kolmogorov’s idea of a connection between problems and Heyting’s calculus is examined and the non-classical features of the latter are shown to be compatible with realistic logic using partial functions.


abstract procedures; constructions; effective procedures; concepts; partiality

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[Bealer 1982] Bealer, G., Quality and Concept, Clarendon Press: Oxford.

[Bolzano 1837] Bolzano, B., Wissenschaftslehre, Sulzbach.

[Chierchia, G. 1989] Chierchia, G., ‘Structured meanings, thematic roles and control”, in: G. Chierchia, B.H. Partee, R. Turner (eds.), Properties, Types and Meaning, Vol. II., Semantic Issues, Kluwer Academic Publishers, pp. 131–166.

[Church 1956] Church, A., Introduction to Mathematical Logic, Princeton.

[Cresswell 1975] Cresswell, M.J., “Hyperintensional logic”, Studia Logica 34, 25–38.

[Cresswell 1985] Cresswell, M.J., Structured Meanings, MIT Press: Cambridge, Mass.

[Fletcher 1998] Fletcher, P., Truth, Proof and Infinity, Kluwer Kluwer Academic Publishers: Dordrecht, Boston, London.

[Jespersen 2003] Jespersen, B., “Why the tuple theory of structured propositions isn’t a theory of structured propositions”, Philosophia 31, 171–183.

[Kirkham 1997] Kirkham, R.L., Theories of Truth, MIT Press: Cambridge, Mass. [Kleene 1952] Kleene, S.C., Introduction to Metamathematics, D. van Nostrand Company, Inc.: New York, Toronto.

[Kolmogorov 1932] Kolmogorov, A., “Zur Deuutung der intuitionistischen Logik”, Mathematische Zeitschrift 35, 58–65.

[Lewis 1972] Lewis, D., “General semantics”, in: D. Davidson and G. Harman (eds.), Semantics of Natural Language, Reidel: Dordrecht, pp. 169–218.

[Materna 1998] Materna, P., Concepts and Objects, Acta Philosophica Fennica, Vol. 63, Helsinki.

[Materna 2004] Materna, P., Conceptual Systems, Loogos Verlag: Berlin.

[Materna 2007] Materna, P., “Once more on analytic vs. synthetic”, Logic and Logical Philosophy 16, 3–43.

[Montague 1974] Montague, R., Formal Philosophy: Selected Papers of R. Montague, R. Thomason (ed.), Yale University: New Haven, London.

[Moschovakis 1994] Moschovakis, Y.N., “Sense and denotation as algorithm and value”, in: J. Väänänen and J. Oikkonen (eds.), Lecture Notes in Logic, #2, Springer, pp. 210–249.

[Sundholm 2000] Sundholm, G., “Virtues and vices of interpreted classical formalisms: Some impertinent questions for Pavel Materna...”, in: T. Childers and J. Palomäki (eds.), Between Words and Worlds. A Festschrift for Pavel Materna, Filosofia, The Institute of Philosophy of the Academy of Science of the Czech Republic, Prague, pp. 3–12.

[Tichý 1968] Tichý, P., “Sense and procedure”. English in Tichý 2004, pp. 78–92.

[Tichý 1969] Tichý, P., “Intension in terms of Turing machines”, Studia Logica 26, 7–25. In Tichý 2004, pp. 94–109.

[Tichý 1978] Tichý, P., “Questions, answers, and logic”, American Philosophical Quarterly 15, 275–284. In Tichý 2004, pp. 294–304.

[Tichý 1988] Tichý, P., The Foundations of Frege’s Logic, De Gruyter, Berlin, New York.

[Tichý 1994] Tichý, P., “The analysis of natural language”, in: Tichý 2004, pp. 802–841.

[Tichý 1995] Tichý, P., “Constructions as the subject matter of mathematics”, in: W. Depauli-Schimanovich, E. Köhler, F. Stadler (eds.), The Foundational Debate: Complexity and Constructuivity in Mathematics and Physics, Kluwer: Dordrecht, Boston, London, Vienna, pp. 175–185. In Tichý 2004, 873–885.

[Tichý 2004] Pavel Tichý’s Collected Papers in Logic and Philosophy, V. Svoboda, B. Jespersen and C. Cheyne (eds.), Publisher jointly by: Filosofia, The Institute of Philosophy of the Academy of Science of the Czech Republic, Prague, Czech Republic, University of Otago Press, Dunedin, New Zealand.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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