allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘Modern Logic’

Categorical Propositions aren’t Same as Conditional Propositions

Posted by allzermalmer on September 20, 2013

It is sometimes held that categorical propositions are equivalent to conditional proposition. However, at least with propositional logic, this isn’t necessarily true.

Categorical proposition: All X are Y.
Conditional proposition: If X then Y.

In other words, it is sometimes held that, All X are Y if and only if X implies Y.

This can be shown false by a very simple method that Categorical propositions aren’t equivalent to Conditional propositions. All we have to do is replace the variables X or Y with Truth Values, and see what the Truth of the proposition as a whole will come out to.

Suppose that X is True and Y is True. Now we replace those variables with their Truth Values in the Statements.

Categorical Proposition: All True are True= True.
Conditional Proposition: If True then True= True.

Suppose that X is False and Y is False.

Categorical Proposition: All False are False=True.
Conditional Proposition: If False then False=True.

Suppose that X is True and Y is False.

Categorical Proposition: All True are False= False.
Conditional Proposition: If True then False= False.

Here is where the Fallacy comes in of thinking Categorical Propositions are equivalent to Conditional Propositions.Suppose that X is False and Y is True.

Categorical Proposition: All False are True= False.
Conditional Proposition: If False then True=True.

We immediately notice that their truth value’s are not equivalent when each variable has the same truth value. This shows that categorical propositions necessarily say something different from conditional propositions.

The only way that Categorical Propositions will say the same thing as Conditional Propositions is if the Subject of the Categorical Proposition isn’t False & the Predicate isn’t True. In other words, the Subject of the Categorical Proposition must Exist.

All Mermaids are creatures that swim in the Ocean if and only if Mermaids implies creatures swim in the Ocean. Mermaids can’t not exist for this equivalency to hold with the Conditional, while the Conditional doesn’t need that Mermaids exist.

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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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