Intermediate logic

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In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent intermediate logic, whence consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic).

Specifically, a superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties:

  1. all axioms of intuitionistic logic belong to L;
  2. if F and G are formulas such that F and F → G both belong to L, then G also belongs to L (closure under modus ponens);
  3. if F(p1, p2, ..., pn) is a formula of L, and G1, G2, ..., Gn are any formulas, then F(G1, G2, ..., Gn) belongs to L (closure under substitution).

Such a logic is intermediate if furthermore

  1. L is not the set of all formulas.

There exists a continuum of different intermediate logics. Specific intermediate logics are often constructed by adding one or more axioms to intuitionistic logic, or by a semantical description. Examples of intermediate logics include:

  • intuitionistic logic (IPC, Int, IL, H)
  • classical logic (CPC, Cl, CL): IPC + p ∨ ¬p
  • the logic of the weak excluded middle (KC, Jankov's logic, De Morgan logic): IPC + ¬¬p ∨ ¬p
  • Gödel–Dummett logic (LC, G): IPC + (p → q) ∨ (q → p)
  • Kreisel–Putnam logic (KP): IPC + (¬p → (q ∨ r)) → ((¬p → q) ∨ (¬p → r))
  • Medvedev's logic of finite problems (LM or ML): defined semantically as the logic of all frames of the form \langle\mathcal P(X)\setminus\{X\},\subseteq\rangle for finite sets X ("Boolean hypercubes without top"), as of 2008 not known to be recursively axiomatizable
  • realizability logics
  • Scott's logic: IPC + ((¬¬p → p) → (p ∨ ¬p)) → (¬¬p ∨ ¬p)
  • Smetanich's logic: IPC + (¬q → p) → (((p → q) → p) → p)

The tools for studying intermediate logics are similar to those used for intuitionistic logic, such as Kripke semantics. For example, Gödel–Dummett logic has a simple semantic characterization in terms of total orders.

[edit] Semantics

Given a Heyting algebra γ, the set of propositional formulas that are valid on γ is an intermediate logic. Conversely, given an intermediate logic it is possible to construct its Lindenbaum algebra which is a Heyting algebra.

An intuitionistic Kripke frame F is a partially ordered set, and a Kripke model M is a Kripke frame with valuation such that \{x\mid M,x\Vdash p\} is an upper subset of F. The set of propositional formulas that are valid in F is an intermediate logic. Given an intermediate logic Σ it is possible to construct a Kripke model M such that the logic of M is Σ (this construction is called canonical model). A Kripke frame with this property may not exist, but a general frame always does.

[edit] Relation to modal logics

Main article: Modal companion

Let A be a propositional formula. The Gödel–Tarski translation of A is defined recursively as follows:

  • T(p_n) = \Box p_n
  • T(\neg A) = \Box \neg T(A)
  • T(A \and B) = T(A) \and T(B)
  • T(A \vee B) = T(A) \vee T(B)
  • T(A \to B) = \Box (T(A) \to T(B))

If Λ is a modal logic extending S4 then ρΛ = {A | T(A) ∈ Λ} is a superintuitionistic logic, and Λ is called a modal companion of ρΛ. In particular:

  • IPC = ρS4
  • KC = ρS4.2
  • LC = ρS4.3
  • CPC = ρS5

For every intermediate logic Σ there are many modal logics Λ such that Σ = ρΛ.

[edit] References

  • Toshio Umezawa. On logics intermediate between intuitionistic and classical predicate logic. Journal of Symbolic Logic, 24(2):141–153, June 1959.
  • Alexander Chagrov, Michael Zakharyaschev. Modal Logic. Oxford University Press, 1997.
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