The claim about the action axiom and UPB being undeniable because of performative contradiction? Refuted.

published Jun 26, 2013

Truth may or may not be universally preferable, independent of whether one affirms the preferability of truth for oneself. Actions may or may not be purpseful, independent of whether one's actions are purposeful. A formal logical study.

Let P(x) be "Truth is preferable for x".

Let us assume two elements a and b in our universe U, where I myself am Mr. a. That is, a ∊ U ∧ b ∊ U.

Now let us express the predicate "Truth is preferable for a" in the following form: P(a).

With this vocabulary, we can say "Truth is preferable for everyone in the Universe" as ∀ x ∊ U P(x). We can also say "Truth is preferable for at least someone in the Universe" as ∃ x ∊ U P(x).

So far, so good. Your claim is that, whenever I am debating whether ∀ x ∊ U P(x) is true, I am affirming P(a) in doing so. I accept that claim P(a).

Finally, we can formally write the "performative contradiction" claim K that "When you imply that truth is preferable, thus you accept the universal preferability of truth" as this: P(a) -> ∀ x ∊ U P(x).

Now, I, Mr. a, propose ∃ q ∊ U ¬P(q). For example, b or some other element in U could very well not comply with the predicate; that is to say, ¬P(b) is true.

Note how I can propose ∃ q ∊ U ¬P(q) (a logical proposition which contradicts ∀ x ∊ U P(x)) without ever contradicting P(a). This is exactly what claim K says can't be done. Yet I'm doing it. Right now. Nanner nanner? :-)

See? Claim K above has certainly been refuted. And now that it's in logical language, itshould be pretty clear that, contrary to K, P(a) most certainly does not imply ∀ x ∊ U P(x). The universality of P is certainly deniable while at the same time affirming P for oneself.

Now, of course, if I wanted to refute ∀ x ∊ U P(x) by empirical counterexample, heh, there remains the "small" problem of proving ∃ q ∊ U ¬P(q) by actually finding b and showing it to you. But that wasn't the point of the exercise -- the point was merely to point out that the "proof" for UPB that supposedly makes it "undeniable", is simply fallacious.

This refutation of the undeniability of "truth is universally preferable" can easily be transferred to refute the undeniability of the action axiom as well. To reuse this refutation against the action axiom, all you have to do is assume P(x) means "Actions performed by are purposeful" rather than "Truth is preferable for x".