Dummett's LC Research Articles

Urquhart's C with Intuitionistic Negation: Dummett's LC without the Contraction Axiom

This paper offers a particular intuitionistic negation completion of Urquhart's system C resulting in a super-intuitionistic contractionless propositional logic equivalent to Dummett's LC without contraction.

Completeness theorem for dummett's LC quantified and some of its extensions

Dummett's logic LC quantified, Q-LC, is shown to be characterized by the extended frame 〈Q+, ≤,D〉, where Q+ is the set of non-negative rational numbers, ≤is the numerical relation “less or equal then” and D is the domain function such that for all v, w ∈ Q+, Dv ≠ φ and if v ≤ w, then D v . D v \( \subseteq \)D w . Moreover, simple completeness proofs of extensions of Q-LC are given.

Hypersequents, logical consequence and intermediate logics for concurrency

The existence of simple semantics and appropriate cut-free Gentzen-type formulations are fundamental intrinsic criteria for the usefulness of logics. In this paper we show that by using hypersequents (which are multisets of ordinary sequents) we can provide such Gentzen-type systems to many logics. In particular, by using a hypersequential generalization of intuitionistic sequents we can construct cut-free systems for some intermediate logics (including Dummett's LC) which have simple algebraic semantics that suffice, e.g., for decidability. We discuss the possible interpretations of these logics in terms of parallel computation and the role that the usual connectives play in them (which is sometimes different than in the sequential case).

A Cut‐Free Calculus For Dummett's LC Quantified

A Cut‐Free Calculus For Dummett's LC Quantified

Quasilinear Posets and some Subsystems Of Dummett's LC

Quasilinear Posets and some Subsystems Of Dummett's LC

On Some Subsystems of Dummett's LC

On Some Subsystems of Dummett's LC

Algebraic Completeness Results for Dummett's LC and Its Extensions

Algebraic Completeness Results for Dummett's LC and Its Extensions

Free L-algebras

Dummett's LC [1] is a system which characterizes all formulas of the propositional calculus which are valid in every chain (for definitions and notation see the first section of [2]). An L-algebra is a Heyting algebra in which (x → y) + (y → x) = 1 for all x, y. L-algebras bear the same relation to LC as Boolean algebras to the classical propositional calculus and Heyting algebras to the intuitionist propositional calculus.

Tsutomu Hosoi. Algebraic proof of the separation theorem on classical propositional calculus. Proceedings of the Japan Academy, vol. 42 (1966), pp. 67–69. - Tsutomu Hosoi. Algebraic proof of the separation theorem on Dummett's LC. Proceedings of the Japan Academy, vol. 42 (1966), pp. 693–695.

Tsutomu Hosoi. Algebraic proof of the separation theorem on classical propositional calculus. Proceedings of the Japan Academy, vol. 42 (1966), pp. 67–69. - Tsutomu Hosoi. Algebraic proof of the separation theorem on Dummett's LC. Proceedings of the Japan Academy, vol. 42 (1966), pp. 693–695. - Volume 33 Issue 1

Two additions to positive implication

Let LIC be the implicational fragment of Dummett's LC, axiomatised by Bull in [1] by adding the axiom (L) CCCpqrCCCqprr to positive implication. The system which Bull in [2] calls OIC, i.e. positive implication plus (0) CCCCpqqrCCCpqrr, is contained in LIC (it is thesis 34 in [1], p. 193); but it does not contain it, as the following matrix (due to Bull) verifies OIC but not (L) (CCC342CCC4322 = 2). Nor is (L) deducible in OIC supplemented by intuitionist axioms for negation, since the latter are all verified by the above matrix with Cp5 for Np.

The implicational fragment of Dummett's LC

It has been shown in Dummett [1] that the system LC given by Heyting's Calculus plus the additional axiom ACpqCqp is complete with respect to the model having as truth values the integers 0, 1,2, … and ω, 0 being the designated value, with In this system CCCpqrCCCqprr is equivalent to ACpqCqp, and the purpose of this paper is to show that the implicational fragment of Heyting's Calculus plus the additional axiom CCCpqrCCCqprr yields the implicational fragment of LC.