This Proposition Is Not True

Very many philosophers believe that (i) every proposition is either true or not true and (ii) no proposition is both true and not true.  However, this raises a problem.  Consider this proposition:

(a) This proposition is not true.

Is (a) true or not true?  If (a) is true, then it must be not true, for that is what it says.  Yet no proposition is both true and not true, per (ii).  If (a) is not true, then it must be true, for (a) says that it is not true.  Yet, once again, no proposition is both true and not true, per (ii).  However, per (i), (a) must be either true or not true.  However, we have just shown that it cannot be either one without violating (ii)!  Consequently, we must reject either (i) or (ii).

The above line of reasoning is known as the Strengthened Liar Paradox (the regular old Liar Paradox uses the values ‘true’ and ‘false’ rather than ‘true’ and ‘not true’).  Peirce offered two different solutions to the Liar Paradox (in 1865 and 1869); however, he rejected them both (in 1869 and 1903, respectively).

This paper develops a conditional solution to the Strengthened Liar Paradox.  I argue that if we accept Peirce’s theories of truth, assertion, and assent (that’s the condition, for I do not defend Peirce’s theories), then we can avail ourselves of a novel solution to the paradox.  The solution works as follows: The premises of any argument are assertions.  However, for a proposition to be asserted, it must be assertible.  That is to say, we must able to take responsibility for the truth of the proposition and what it entails.  “This proposition is not true” is not assertible, as the paradox shows.  Therefore, it cannot be the premise of an argument.  However, the line of reasoning above requires us to treat “This proposition is not true” as the premise of an argument, and that is the source of the problem.

Unfortunately, a variety of assertions made in the paper entail the conclusion “This proposition is not true.”  This seems to be a problem, for if the conclusions of arguments are assertions then “This proposition is not true” must be assertible.  However, consistent with Peirce’s view, I argue that the conclusions of arguments are not assertions but propositions to which we give our assent.  I identify two other cases where the conclusions of arguments are not assertions: cases of doxastic incontinence (when we believe in the teeth of evidence) and abductive reasonings.

Also included in this paper are: (a) a detailed account of Peirce’s first two solutions and why he rejects them and (b) a discussion of the distinctions among mere saying, assertion, belief, and assent.

This essay won the 2010 Charles S. Peirce Society Essay Contest.