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**11 - 15**of**15**### .1 Natural Deduction

"... 108 Equality Symmetrically, we can also replaces of occurrences of t by s. ` s : = t ` [t=x]A : = E 2 ` [s=x]A It might seem that this second rule is redundant, and in some sense it is. In particular, it is a derivable rule of the calculus with only : = E 1 : ` s : = t : = I ` s : = s : ..."

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108 Equality Symmetrically, we can also replaces of occurrences of t by s. ` s : = t ` [t=x]A : = E 2 ` [s=x]A It might seem that this second rule is redundant, and in some sense it is. In particular, it is a derivable rule of the calculus with only : = E 1 : ` s : = t : = I ` s : = s : = E 1 ` t : = s ` [t=x]A : = E 1 ` [s=x]A However, this deduction is not normal (as defined below), and without the second elimination rule the normalization theorem would not hold and cut elimination in the sequent calculus would f

### Functorial Kripke-Beth-Joyal models of the lambda Pi-calculus I: type theory and internal logic

, 2001

"... We give a categorical account of Kripke-Beth-Joyal models of the - calculus. Kripke models. Emphasize semantics of (terms/representatives/realizers for) consequences. 1 Introduction This paper, \Functorial Kripke-Beth-Joyal models of the -calculus I: type theory and internal logic" (henceforth abb ..."

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We give a categorical account of Kripke-Beth-Joyal models of the - calculus. Kripke models. Emphasize semantics of (terms/representatives/realizers for) consequences. 1 Introduction This paper, \Functorial Kripke-Beth-Joyal models of the -calculus I: type theory and internal logic" (henceforth abbreviated to I), is rst of a sequence of three connected works. It is concerned with the basic model theory of the - calculus considered on the one hand as a system of rst-order dependent function types and on the other as presentation of the f8; g-fragment of minimal rstorder predicate logic with proof-objects. From the point of view of type theory, ... MITCHELL/MOGGI From the point of view of logic, the ... At the core of our denition of Kripke(-Beth-Joyal) models of lies our treatment of comprehension, context extension and (rst-order) dependent function spaces. The essential idea is similar that of earlier work [?, ?, ?]; however, our treatment has the following two advant...

### NATURAL DEDUCTION AND TERM ASSIGNMENT FOR CO-HEYTING ALGEBRAS IN POLARIZED BI-INTUITIONISTIC LOGIC.

"... Abstract. We reconsider Rauszer’s bi-intuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of ..."

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Abstract. We reconsider Rauszer’s bi-intuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized bi-intuitionistic logic (PBL) consists of two fragments, positive intuitionistic logic LJ⊃ ∩ and its dual LJ� � , extended with two negations partially internalizing the duality between LJ⊃ ∩ and LJ� �. Modal interpretations and Kripke’s semantics over bimodal preordered frames are considered and a Natural Deduction system PBN is sketched for the whole system. A stricter interpretation of the duality and a simpler natural deduction system is obtained when polarized bi-intuitionistic logic is interpreted over S4 rather than bi-modal S4 (a logic called intuitionistic logic for pragmatics of assertions and conjectures ILPAC). The term assignment for the conjectural fragment LJ� � exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The duality is extended from formulas to proofs and it is shown that every computation in our calculus is isomorphic to a computation in the simply typed λ-calculus. §1. Preface. We present a natural deduction system for propositional polarized bi-intuitionistic logic PBL, (a variant of) intuitionistic logic extended with a connective of subtraction A � B, read as “A but not B”, which is dual to implication. 1 The logic PBL is polarized in the sense that its expressions are regarded as expressing acts of assertion or of conjecture; implications and conjunctions are assertive, subtractions and disjunctions are conjectural. Assertions and conjectures are regarded as dual; moreover there are two negations, transforming assertions into conjectures and viceversa, in some sense internalizing the duality. Our notion of polarity isn’t just a technical device: it is rooted in an analysis of the structure of speech-acts, following the viewpoint of the