Extensionality and the Fregean Axiom

Mieczysław Omyła

The Extensionality Principle states that coreferential expressions can be mutually interchangeable in all sentential contexts without changing their logical value, symbolically:

(φ = ϕ) ⇒ (α ⇔ α[φ/ϕ])

In predicate logic the Extensionality Principle is embodied in the following logical scheme a = b ⇒ (α[x/a] ⇔ α[x/b])

In classical sentential calculus this principle translates into the following extensionality rules: (α ⇔ β) ⊢ γ[p/α] ⇔ γ[p/β], α ⇔ β, γ[p/α] ⊢ γ[p/β]

These rules result from the fact that Fregean Axiom holds for classical logic. The Axiom states that the semantic correlates of sentences are their logical values. Suszko in [1], [2] rejected Fregean Axiom and following Wittgenstein assumed that the situation presented by a sentence is its semantic correlate.

To formally express the difference between the logical value of a sentence and the situation presented by it, Suszko enriched the language of classical logic by the ”≡” non-truth-functional connective, which defined by means of the following logical schemes:

α_{1} ≡ α_{2}, (where α_{1}, α_{2} differ at most in bound variables)

(α ≡ β) ⇒ (α ⇔ β),

(α ≡ β) ⇒ (¬α ≡ ¬β),

(α ≡ β ∧ γ ≡ δ) ⇒ (α§γ ≡ β§δ), (where § : ∧, ∨, ⇒, ⇔, ≡)

∀v(α ≡ β) ⇒ (Qvα ≡ Qvβ), (where Q = ∀, ∃, and v is any variable sentential or nominal).

As a result Suszko obtained logical calculus which he called Non-Fregean Logic. This logic is a generalization of classical logic, for if we assume that ”≡” = ”⇔”, then we obtain classical logic. Non-Fregean logic is both extensional and logically bivalent, but is not ontologically bivalent.

Non-Fregean Logic is extensional for it includes the following schemes logical theorems: (φ ≡ ϕ) ⇒ (γ[v/φ] ⇔ γ[v/ϕ])

The calculus is logically bivalent, as it includes following logical theorems: p ∨ ¬p, ¬(p ∧ ¬p). But not ontologically bivalent as it is, an important role in it is played, unlike in classical predicate calculus, by sentential variables which run over the universe of situations.

[1] Suszko Roman, Ontology in the Tractatus of L. Wittgenstein, Notre Dame Journal of Formal Logic, 9, (1968), p.7-33.

[2] Suszko Roman, Abolition of the Fregean Axiom, Lecture Notes in Mathematics (453) (1975), p.169-239.

Address: Warsaw University, Institute of Philosophy, Division of Logic, Poland

E-mail: omyla@plearn.edu.pl