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

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, it*should* 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 *x *are purposeful" rather than "Truth is preferable for *x*".