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## Proof-Theoretic SemanticsSo far, we gave some kind of answer to what we called Frege's questions stressing the epistemic character of a possibly incomplete set of proven equalities of an agent, in contrast to identity in a model. We will now turn to the idea of proof-theoretic semantics. According to ([8], p. 503), [p]roof-theoretic semantics [assigns] proofs or deductions an autonomous semantic role from the very onset, rather than explaining this role in terms of truth transmission. In prooftheoretic semantics, proofs are not merely treated as syntactic objects […], but as entities in terms of which meaning and logical consequence can be explained. This approach is already quite successfully pursued for the usual logical operations (see [7, 12] and this volume). It is our aim to extend it to some further concepts, like equalities here or necessity in [6]. In the case of From a proof-theoretic perspective, “morning star” and “evening star” should, of course, be different. Their Whatever the concrete axioms might be, they should state that the “morning star” is visible (on some days) in the morning, and the “evening star” in the evening, respectively. As discussed above, the equality between them needs a proof. For the As related approaches we would like to mention here Tichý's TIL does not dispense with possible worlds, but assigns them a secondary rôle in the analysis of senses. These are introduced as abstract procedures, called Such a relation would be given, at least partly, by the Curry–Howard correspondence for Moschovakis's approach. He introduces senses as algorithms which compute (denotational) values. Based on the well-known correspondence of algorithms and proofs, we could adapt Moschovakis's slogan by describing our (broader) approach to intensionality as ## References1. Duží, M., Jespersen, B., Materna, P.: Procedural Semantics for Hyperintensional Logic. Logic, Epistemology, and the Unity of Science, vol. 17. Springer, Berlin (2010) 2. Frege, G.: Begriffsschrift, Nebert, Halle a.d.s. (English translation [4]) 3. Frege, G.: Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik, NF 100 pp. 25-50 (1892) (English translation [5]) 4. Frege, G.: Concept script. In: van Heijenoort, J. (ed.) From Frege to Gödel, Harvard University Press, Cambridge (1967) (English translation of [2]) 5. Frege, G.: On sense and meaning. In: McGuiness, B. (ed.) Collected Papers on Mathematics, Logic, and Philosophy, p. 157-177. Basil Blackwell (1974) (English translation of [3] by M. Black) 6. Kahle, R.: A proof-theoretic view of necessity. Synthese 7. Kahle, R., Schroeder-Heister, P. (eds.) Proof-Theoretic Semantics. Springer (2006). Special issue of Synthese 148(3), 503-743 8. Kahle, R., Schroeder-Heister, P.: Proof-theoretic semantics-introduction. Synthese 148(3), 503506 (2006) 9. Kripke, S.: Naming and Necessity. Harvard University Press, Cambridge (1980) 10. LaPorte, J.: Rigid Designators. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Summer 2011 edn.) (2011), plato.stanford.edu/archives/sum2011/entries/rigid- designators/ 11. Moschovakis, Y.: Sense and denotation as algorithm and value. In: Väänänen, J., Oikkonen, J. (eds.) Logic Colloquium '90, pp. 210-249. Lecture Notes in Logic, vol. 2, Association for Symbolic Logic (1994) 12. Schroeder-Heister, P.: Proof-theoretic semantics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2012 edn.) (2012), plato.standard.edu/archives/win2012/ enties/proof-theoretic-semantic/ 13. Tichý, P.: The Foundations of Frege's Logic. De Gruyter, Berlin (1988) 14. Wehmeier, K.F.: How to live without identity–and why. Australas. J. Philos. |

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