"For every liar Sentence, there exists some person for whom the sentence is either self-referential or (purely) negative."
I am a behavior analyst with an undergraduate degree in philosophy and some training in first-order logic. I am interested in learning more about logic both (a) to apply it to behavior analysis in order to further advance the field of behavior analysis and (b) to apply behavior analysis to logic, namely, in explaining the behavior of logicians. One way I hope to do this is by attempting a behavior-analytic approach to the Liar Sentence/Paradox, but I want to inform it with the language of formal logic.
I wish to demonstrate that the problem with the liar Sentence/Paradox is not merely one of syntax or semantics, but also a problem of pragmatics, e.g., the behavior of logicians in their choosing to stipulate that the Liar Sentence be both self-referential and negative thus leading to the Liar Paradox, but which seems like an arbitrary stipulation. It seems that as long as a person does not make that stipulation, no paradox exists. Just because that stipulation can be made, and the paradox thus be constructed, does not mean it should be.
Informally, in English, the Liar Sentence is "This sentence is false," from which the Liar Paradox is generated. In Tarski's approach to the Liar Sentence/Paradox, he took the route of arguing that the concept of truth is ill-defined, specifically ill-defined within the object language in which the Liar Sentence is constructed, and that one must venture outside the object language into a meta-language in order to evaluate the truth of the Liar Sentence. Others have pursued similar routes of approaching the Liar Sentence/Paradox by questioning the nature of truth, such as arguing that the truth-value of the sentence is indeterminate. But, of course, there are problems with both of these lines of reasoning. To my understanding, both lines of reasoning rely on a truth-gap.
I want to take the opposite route of arguing that it is not the concept of truth that is necessarily ill-defined or the truth-value of the Liar sentence that is necessarily indeterminate, but rather the concepts of "This" and “negation.” If the word "This," which traditionally is stipulated as referring to the Liar Sentence itself, is ambiguous, that is, could refer to the Liar Sentence itself or to some other sentence, then it's debatable whether the Liar Sentence is self-referential. And if the concept of negation is impossible without the possibility of assertion, then it’s debatable whether the liar sentence is even negative.
Similar sentences to the Liar sentence can be constructed without invoking the concept of truth. For example:
This sentence does not mean what you think it means.
This sentence does not refer to itself.
This sentence does not refer.
These sentences demonstrate that it is language itself—and not the concept of truth—that is indeterminate and that language only becomes determinate in the act of reading and I want to argue that this is true no less for formal languages than for informal languages. What I want to argue is that the bearers of truth are not the usual candidates of sentences and propositions but rather pragmatics (behaviors) and people.
Thus, whichever side a person comes down on--whether the Liar Sentence is both self-referential and purely negative or either self-referential or purely negative--is a matter of choice, i.e., behavior. And thus, the Liar Sentence/Paradox is arguably a behavioral problem. And I believe the above statement is key in being able to demonstrate this. For people who insist the liar sentence be both self-referential and purely negative, the paradox persists. For those who don’t, there is no paradox.
The above, of course, is all informal. Out of respect for the literature on the Liar Paradox, I want to turn the above into a formal argument, starting with translating the above statement into formal logic.
My thinking is influenced by C. S. Peirce. While Peirce does not make the above argument (at least, not that I know of), he does insist on the importance of pragmatics and the necessity of people ("interpretants") in any theory of language and truth. Tarski's T Convention of truth (e.g., "Snow is white" is true iff snow is white) seems devoid of the pragmatic element. By including the perspectives of different people/interpretants/readers in my statement above, I am attempting to develop an argument that the Liar sentence is read qua a Liar sentence to some people and not qua a Liar sentence to other people, depending on whether one insists on the stipulation that the Liar sentence be both self-referential and purely negative.
In other words, if the Liar sentence is taken to be either syntactically correct or semantically correct, then it is pragmatically correct, e.g., useful. But if the Liar sentence is taken to be both syntactically correct and semantically correct, then it is pragmatically incorrect, e.g., useless, and as such, in behavior-analytic terms, it should be neither reinforced (affirmed) nor punished (denied) but extinguished (ignored without further argument, mentioned but not used).
In arguing this, I am essentially seeking to challenge Tarski’s T Convention. What I want to argue is that the issues with the Liar sentence / paradox are resolved if we modify his T Convention to something like: “Snow is white” is true iff, for some members of the verbal community, “Snow is white” is taken to mean snow is white and at the same time snow is white. For example:
The Liar sentence as purely negative but not self-referential:
“This sentence is false” is true iff, for some members of the verbal community, “This sentence is false” is taken to mean “This sentence, ‘Snow is black,’ is false,” and at the same time this sentence, “Snow is black,” is false.
The Liar sentence as self-referential but not purely negative:
“This sentence is false” is true iff, for some members of the verbal community, “This sentence is false” is taken to mean “This sentence is false or this sentence is true” and at the same time this sentence is false or this sentence is true.