By Radford E.M.

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7. 1, it is understood that that if a formula is deducible from the laws of intuitionistic logic, being derived from its axioms by way of the rule of modus ponens, then it will always have the value 1 in all Heyting algebras under any assignment of values to the formula’s variables. However, one can construct a Heyting 48 A Functorial Model Theory algebra in which the value of Peirce’s law is not always 1. Consider the 3-element algebra {0, ½,1} as given above. If we assign ½ to P and 0 to Q, then the value of Peirce’s law ((P → Q) → P) → P is ½.

Given mappings f : X → Y and g : Y → Z, the composition g ◦ f : X → Z is the mapping x → g(f(x)). A first order vocabulary L consists of a set of finitary relation symbols, function symbols, and constant symbols. We use A, B,... to denote L-structures with universe sets A, B,.... By the cardinality of A we mean the cardinality of its universe set A. , xn) is true in A when each xi is interpreted by the corresponding ai. The notation h : A→B means that h is a homomorphism of A into B, that is, h maps A into B and each atomic formula which is true for a tuple in A is true for the h-image of the tuple in B.

Consider the equivalence relation FG induced by the preorder F≼G. (It is defined by FG if and only if F≼G and G≼F. In fact, ∼ is the relation of intuitionist logical equivalence. Let H0 be the quotient set L/∼. This will be the desired Heyting algebra. We write [F] for the equivalence class of a formula F. Operations →, , and ¬ are defined in an obvious way on L. Verify that given formulas F and G, the equivalence classes [F→G], [FG], [FG] and [¬F] depend only on [F] and [G]. This defines operations →, , and ¬ on the quotient set H0=L/.