Dialetheism and the Graphic Liar

I Introduction

A Liar sentence is a sentence that, paradoxically, we cannot evaluate for truth in accordance with classical logic and semantics without arriving at a contradiction. For example, consider L

If we assume that L is true, then given that what L says is ‘L is false,’ it follows that L is false. On the other hand, if we assume that L is false, then given that what L says is ‘L is false,’ it follows that L is true. Thus, L is an example of a Liar sentence.

Several philosophers have proposed that the Liar paradox, and related paradoxes, can be solved by accepting the contradictions that these paradoxes seem to imply (including Priest 2006, Rescher and Brandom 1980). The theory that there are true contradictions is known as ‘dialetheism’ and we may call this the ‘dialethic solution.’ One standard response to the dialethic solution to the Liar paradox and related paradoxes has been to attempt to develop new ‘revenge’ versions of the paradoxes that are not subject to the dialethic solution (e.g. Parsons 1990, Restall 2007, Shapiro 2007). Recently, attempts have been made to produce such a revenge paradox by phrasing the paradox in terms of semantic value. As has been argued by Everett (1993), Smiley (1993), Bromand (2002) and Littmann and Simmons (2004), such revenge Liars threaten dialetheism with triviality.

Since dialetheists identify sentences as being true, false, or both true and false, dialetheism makes use of the three semantic values (T & ~F), (~T & F), and (T & F). The first value may be called ‘true and not false.’ This is the semantic value that everyone, dialetheists included, would ascribe to the sentence ‘1 + 1 = 2.’ The second value may be called ‘false and not true.’ This is the semantic value that everyone, dialetheists included, would ascribe to the sentence ‘1 + 1 = 3.’ The third semantic value may be called ‘true and false.’ This is the semantic value that the dialetheist would give to a Liar sentence such as ‘This sentence is false.’

Once these three values have been identified, it is possible to construct an extended Liar in terms of them, by having our Liar say of itself not simply that it possesses the value F but that it has the semantic value (~T & F). A representative argument might go as follows. Consider S:

It is easily seen that S is a Liar sentence and cannot be classically evaluated without contradiction. If we label S as false without also labeling it as true, then it follows that S is true, since all that S states is that the semantic value of S equals (~T & F). On the other hand, if we label S as true, then S has the semantic value (~T & F), since that the semantic value of S equals (~T & F) is all that is stated by S. Since S has the semantic value (~T & F), S is ~T. So S is (T & ~T). We arrive at contradiction.

The dialethic response to such paradoxes of self-reference is to embrace the contradiction and allow that the paradoxical sentence is both true and false; that is, that the semantic value of S equals (T & F). However, if the semantic value of S equals (T & F) then S is true (as well as false). Since S is true—and what S states is that the semantic value of S = (~T & F)—the semantic value of S = (~T & F). The road to triviality is clear. If the semantic value of S = (~T & F) and the semantic value of S (T & F), then the semantic value (~T & F) = the semantic value (T & F). Now let’s consider any claim C. C must be either true and not false (T & ~F) or both true and false (T & F) or false and not true (~T and F). If C is true and not false, then C is true. If C is both true and false, then C is true. Let’s assume, then, that C is false and not true; that is, that it has the semantic value (~T & F). Since C has the semantic value (~T...