A Simpler Demonstration About the Nonexistant Liar Paradox

I remain convinced that syllogistic logic is immune to the liar. I don’t know why its immune to the liar, but there is simply no way to form a valid syllogism in which the liar makes an appearance. Here is where the confusion arises in my view.

In symbolic logic based on the works of Frege, a letter represents either judgable or unjudgable content. For example, the noun “house” is not judgable content. “The house exists” is judgable content. If A is judgable content then,

⊢A

is valid. In traditional form, it would simply be “A is true.” However, if A is unjudgable content, then

⊢A

is invalid. This would amount to “house is true,” which is nonsense.

Frege’s goal was to put logic into terms where it could more easily be analyzed. Modern systems are based on his. In Russell’s Prinipia Mathematica, he writes the law of excluded middle as

⊢. p V ¬p.

In traditional logic, Aristotle said, “Of two contradictory judgments, the one must be true and the other false.” (Metaphysics, Book III, chapter 8) In symbolic logic, this would amount to

⊢A⊕⊢¬A

which is different than Russell’s formulation. In the truth table

A ¬A
T T
T F
F T
F F

the first and last rows are excluded. In Russels formulation, only the last row is excluded. However, since the law of contradiction applies, the first row is also excluded in his system, so its no problem for Russell’s system. The point is, all this his only makes sense if A is judgable content. If it were not, then the truth table could not be applied to A.

For some reason, the liar paradox is often given in regular language. We can try to construct the paradox by saying

All said by the liar is false.

A is said by the liar.

A is false.

 

Through an incorrect version of the law of excluded middle we have

If A is false, A is true.

A is false.

A is true.

 

Now we have a contradiction. However, this is not the correct law of excluded middle. Applying the correct law of excluded middle we still would still have

All said by the liar is false.

A is said by the liar.

A is false.

 

With the correct law of excluded middle we have

If A is false, ¬A is true.

A is false.

¬A is true.

 

Since ¬A was never asserted by the liar, this is where the train ends. There is no paradox.

 

Mathematical symbols are also used in instances explaining the liar paradox. We may construct

A is false.

A = “A is false”.

Through substitution we have

“A is false” is false.

A is true.

Hence, a contradiction. However, in traditional logic, there is no principle allowing the transition from the second to third step. In traditional logic, “is equal to “A is false”.” is predicated of A. This is not the same as a mathematical “=” sign. The copula “is” has a much different signification in traditional logic. Its meaning is entirely different. In traditional logical form, it would be expressed as a classic figure 3 syllogism.

A is false.

A is equal to “A is false”.

Some (that which is) false is equal to “A is false”.

or, if the premisses are switched around,

Some (that which is) equal to “A is false” is false.

There is no paradox here. The reason that the subject is not plainly “false” or plainly “equal to “A is false”” is because in traditional logic, subjects are always substances, and they are never accidents. This distinction was invented by Aristotle, for more information go here: http://plato.stanford.edu/entries/aristotle/#Hyl You could have also added parenthesis in the premises if desired. A is (that which is) false. They have identical meanings. Its needed in the conclusions because, if omitted, the sentence sounds awkward, but it would still be technically correct, grammatically, if the parenthesis was omitted.

Substitution is valid in Fregian function and argument type constructions. e.g. f(x) and f(y). However, the function and argument never strictly replaced the subject and predicate. For all intents and purposes for analysis, it did replace the subject and predicate form, but the subject and predicate did not cease to exist. Instead, Frege ingeniously subsumed them within his system for the sake of brevity. For easier and clear analysis, he lumped all content into a subject. If the content was judgable, then he merely asserted it, or not. As explained in Begriffsschrift, “Such a language would have only a single predicate for all judgments, namely, ‘is a fact’.”

 

Gupta and Belnap in The Revisionist Theory of Truth, express the liar paradox on page 5 as follows

“(The Liar) The Liar is not true.
Now it is easily seen that the T-biconditionals for English imply a
contradiction. For we have
(2) The Liar = ‘The Liar is not true’,
and the T-biconditional for The Liar is
(3) The Liar is not true’ is true if and only if The Liar is not true.
By substitutivity of identicals we deduce from (2) and (3) that
(4) The Liar is true if and only if The Liar is not true,
which immediately yields a contradiction.”

However, the principle labeled “substitutivity of identicals,” is never explained. Obviously, the error here is that this principle is invalid or that this principle is imaginary.

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