allzermalmer

Truth suffers from too much analysis

Logic without Existential Import or Free Logic

Posted by allzermalmer on July 31, 2013

Aristotle’s Logic
[(x)(F(x) → G(x)] → Ε(x)(F(x) & G(x))

Modern Logic
[(x)(F(x) → G(X)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))

Modern Logic can discriminate between inferences whose validity requires an existence assumption or doesn’t require an existence assumption.

Required Existence Assumpition
[(x)(F(x) →G(x)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))

Non-Required Existence Assumption
(x) F(x) → Ε(x) F(X)

When move from Quantification Theory to Identity Theory, Modern Logic’s new formula doesn’t hold with Identity Theory, because there is a counter example.

Assume that [(x)(F(x) → G(X)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))
Assume that F=y
then (x) [x=y → G(x)] → [Ε(x) (x=y & G(x))]

So Modern Logic Quantification Theory view of Existential Import would imply that all statements in Identity Theory of the form (x) [x=y → G(x)] carry Existential Import of Ε(x) (x=y & G(x)).

The source of this error in Modern Logic comes from Particularization.

(1) (x) [x=y → G(x)] → [Ε(x) (x=y & G(x))]
is deducible from valid formula
(2) [(x)(F(x) → G(X)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))
by substituting “=y” for “F” and detaching with
(3) (E(x) (x=y)
(3) is valid in conventional Identity Theory and deducible from Identity Axiom
(4) y=y

Particularization is [Fy → (E(x)) (x=y)]

So Free Logic comes about to get rid of the Existential Importation that both Aristotle’s and Modern Logic allow for. This is a logic Free of existential import or assumption. It is built off of a modification of Quantification Theory by altering some axioms.

(x)(F(x) → F(y)) is an axiom of Quantification Theory which is replaced in Free Logic.

A1) (y)(x) (F(x) → F(y))
A2) [(x)(F(x) → G(x))] → [(x)(F(x) → (x)(G(x)]
A3) x=x
A4) (x=y → (F(x) → F(y))
A5) (E(x)(F(x) → F(x))

From these axioms, (x) F(x) → E(y) F(y) isn’t derivable. This means that existential import is not derivable from the axioms of the Free Logic in Quantifier Theory.

However, what can be derived is [F(x) & (E(x) (x=y)) (x)] → (E(y) (F(y)).

(x) [(F(x) → G(x)) → ((E(x) F(x)) → (E(x)G(x)))] can also be derived from the system.

(X) (x=y) can’t be derived and neither can [(x)(x=y → G(x))] → [(E(x)(x=y & G(x))].

This Free Logic allows the differentiation between singular inference patterns where the existence assumption is relevant or not.

Singular Inference Patterns Existence Assumption:
(F(x) & (E(x) (x=y & G(x))) infer (E(x) F(y))

Singular Inference Patterns Non-Existence Assumption:
x=y infer (F(x) → F(y))

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