Three Paradoxes -- One Error

Here are three paradoxes based on a single error.

1. The Liar Paradox (aka Epimenides Paradox): “This sentence is false.” If we assume it's true, then it's false. If we assume it's false then it's true. But, by the Principle of Excluded Middle, it must be true or false.

This has given rise to an entire literature of solutions including discussions by Buridan, Tarsli and Kripke. Many of these focus on the self-referential nature of the paradox.

We can avoid self reference by using a different form:

2. Jourdain’s Paradox: This is a card, on the front of which is printed, “The statement on the other side is true.” On the other side is printed, “The statement on the other side is false.” Again, no matter what truth value we assign to either statement, we can't find a self-consistent set of truth values.

We notice, however, that both paradoxes involve statements about statements. So, perhaps that is the core issue.

The next paradox brings in real world situations, showing that this alone is insufficient:

3. Kripke’s Paradox:

Saul A. Kripke, “Outline of a Theory of Truth,” Journal of Philosophy 72 (19):690-716 (1975):Consider the ordinary statement, made by Jones:
(1) Most (i.e., a majority) of Nixon's assertions about Watergate are false. Clearly, nothing is intrinsically wrong with (I), nor is it ill-formed. Ordinarily the truth value of (1) will be ascertainable through an enumeration of Nixon's Watergate-related assertions, and an assessment of each for truth or falsity. Suppose, however, that Nixon's assertions about Watergate are evenly balanced between the true and the false, except for one problematic case,
(2) Everything Jones says about Watergate is true. Suppose, in addition, that (1) is Jones's sole
assertion about Watergate, or alternatively, that all his Watergate-related assertions except perhaps (1) are true. Then it requires little expertise to show that (1) and (2) are both paradoxical: they are true if and only if they are false.

What do we see in common here? First, despite Kirpke’s best efforts, all three paradoxes involve sentence about sentences. Second, the assumption that we can assign binary truth values to such sentences does not work out. Thus, a possible solution is to discard the Principle of Excluded Middle.

Before discarding Excluded Middle, let’s examine Kirpke’s case more closely. Note that "Everything Jones says about Watergate is true," is not a statement about the reality of Watergate, but one about Jones' statements. Similarly, "Most of Nixon's assertions about Watergate are false," is not a statement about Watergate, but about Nixon's locutions. Thus, it cannot be counted among "Nixon's assertions about Watergate."

To see this more clearly, consider the Watergate event constrained by all of Jones excepted statements, i.e. “all his Watergate-related assertions except” (1). If we change anything about Watergate not covered by Jones' excepted statements, the change will not alter the supposed truth or falsity of either Jones' or Nixon's claims. Thus, they are not “about Watergate” at all.

So, Kripke's paradox is just like Jourdain's. While either statement in it, or in Jourdain's paradox, taken in isolation could be ultimately referential, in the scenario given, they are not.

Why should this matter? Because the Principle of Excluded Middle with respect to statements is not, and cannot be, fundamental. We have seen in our three paradoxes that assuming it to apply to all sentences leads to contradictions. So, its universal applicability is disproven by a reductio ad absurdum.

What is fundamental, is the Principle of Excluded Middle with respect to existence. Either a putative reality is, or it is not. If we want to reason, salve veritate (in a truth preserving way), about reality, our reasoning needs to reflect this principle of being. So also do the sentences expressing our reasoning.

On the other hand, if we aren’t discussing reality (and none of the three paradoxes are) there's no reason to expect the Principle of Excluded Middle to apply to our sentences.

So, the fundamental error here is assuming that truth and falsity can be assigned to statements which can't be cashed out existentially. Truth is the adequacy of our thought to reality and so is intrinsically relational. Thus, not all statements have to be true or false. Some can be non-referential – having no relation to reality, and so neither true not false. Accordingly, it is a category error to assert that non-referential statements are true or false.

“Truth value,” then, is quite different from truth. While is a relational concept abstracted from referential thinking, is a non-relational construct supposed to be a property of propositions taken in isolation. Such constructs can be useful in limited ranges of verified application, but there is no warrant for thinking they are of universal value.