The Liar's Paradox: Propositions and Contradictions

    The Liar's paradox is a well known conundrum to anyone with a general knowledge of philosophical topics. You are probably already familiar with the paradox and may not even realize it. Take the proposition, "this sentence is a lie." If this proposition is true, then it is true that it is a lie, which makes it false; yet, if this proposition is false, then it is true because it is true that the proposition is false, etc. The truth of this proposition entails its falsity and its falsity entails its truth, meaning that it is both true and false simultaneously. The point of the paradox is to call into question the validity of the Law of Noncontradiction, a classical law of thought which states that a proposition cannot be both true and false in the same sense or at the same time, by providing an exception to it. But is this truly an exception?

     I contend that it is not. There are two responses to the paradox that I am familiar with. One is to say that propositions are not essentially truth-apt, which is a position that I reject out of hand. Propositions are precisely that which serve as the pre-linguistic basis for the shared meaning of token sentences such as "the snow is white" and "der schnee ist weiss" and as such are essentially truth-apt. Another response is to deny that propositions are self-referential, which is the route that I take. What I mean by self-referential is straight forward: that the referent of a proposition is that proposition itself, which is the case with "this sentence is a lie."

    The issue here is when we look at the subject-verb-object structure. Let's provide some example propositions. Proposition P1 states: "The sky is cloudy." Proposition P2 states: "this proposition has five words." There is no issue with P1 being about the sky. In P1, the subject is "sky," the verb is "is," and the object is "cloudy". The verb and object are strictly distinct from the subject, and the subject is complete; it has determinate content in and of itself. Thus, P1 is about something other than itself.

     Look at P2. To take the same structure, the subject would be "this proposition," the verb would be "has," and the object is "(five) words." In this case, the verb and object aren't distinct from the subject, but are included in the subject, and they must be included in the subject in order for the subject to be complete. That's the issue, because connecting a subject with an object through a verb with the typical subject-verb-object structure presupposes that there is some complete subject available to be connected with, a subject that has content in its own right apart from an object and a verb that connects it to that object, but without "has" and "five words," "this proposition" is not a complete subject, and thus, it has indeterminate content.

    And so, the problem of indeterminate content arises because the proposition is self-referential. But a proposition which has indeterminate content cannot be said to be truth-apt, since there has to be some analyzable content being presented by a proposition in order to judge whether that content could be true or false in principle. This entails that a proposition which has indeterminate content is no proposition at all (given that propositions are essentially truth-apt), and thus, there can be no such thing as a self-referential proposition. 

    In conclusion, if there are no self-referential propositions, then what is presented in the Liar's Paradox is not a proposition at all, and therefore, it cannot be used to show an exception to the Law of Noncontradiction.