The predicate of the general liar’s revenge statement (the “formalism”) is boolean algebra. Thus, it would follow, that an alternative to boolean algebra would seriously undermine the paradox. Moreover, the algorithmic quality of the paradox suggests that any substantial departure from boolean algebra would throw the facticity of the paradox into serious doubt. What, then is the status of boolean algebra?
Boolean algebra is a formal symbolic language – a collection of symbols that signify logical formulae. Boolean algebra would be a formal symbolic language signifying traditional propositional logic. It is possible to speak about a single formal symbolic language for traditional propositional logic “the logic”, traditionally interpreted, and the language that represents that logic. Traditional logic concerns itself with true and false, interpreted normally, and equations performed with true and false.
Because of the algorithmic property of the paradox, any departure from the predicate of boolean algebra serves to undermine the entirety of the paradox (obviously the paradox builds on boolean algebra). If boolean algebra were to fail in some respect, then the formula for the formalism wouldn’t be algorithmic. If the formula for the formalism weren’t algorithmic, then the paradox would fail. The paradox is exact – the paradox is only a rules-based execution of the formula for the paradox with no room for interpretation.
Therefore, “paraconsistent logic” would include any logics that depart from traditional symbolic logic in any way. The classic example of a paraconsistent logic would be one that tries to neuter something called “disjunctive syllogism”, a train of reasoning like this: “A or B, not A, therefore B”. The proponent of this paraconsistent logic would say that the idea that a paradox can prove anything relies on this disjunctive syllogism, so undermining disjunctive syllogism would undermine the principle that a paradox can prove anything, called “the principle of explosion”.
Any departure from boolean algebra would seriously undermine the general paradox as presented. Boolean algebra is structured by true and false, various logic gates such as both “and” statements and “or” statements, and an order of operations. A descriptive analogy would be computer programs. Importantly, boolean algebra is a formal symbolic language – a collection of symbols that describe a logic. Not only that, but boolean algebra can be interpreted as an instantiation of “the” traditional formal symbolic language – the linguistic expression of “the” traditional logic.
It is difficult to talk about alternatives to traditional symbolic logic analytically. If traditional symbolic logic belongs to the domain of philosophy, would it perhaps be preferable to examine the question of alternatives to traditional symbolic logic through other lenses such as art? A paraconsistent logic would need to evaluate itself through its own formal symbolic language, so deductive evaluation of a paraconsistent logic would be problematic.
One way to approach the question of traditional logic vs. paraconsistent logics is via a formalization of the proof: posit that the proof is a formalization of “truth”and that a formalization of “truth” is a traditional symbolic logic. If the proof concerns itself with truth, traditional formal symbolic logic would be the structure of that truth.
How might “truth” be interepreted in divergent ways? Perhaps truth could be an aesthetic phenomenon. Maybe truth would be a social phenomenon. Or truth could be a physical artefact, and it would be impossible to talk about “truth” without discussing particles and fields. Perhaps truth is a question best considered philosophically. Many various potential interpretations of “truth” would probably require a certain level of internal consistency, coherence, or applicability. How would a claim to represent “truth” be evaluated?
On another level, a comparison could be made from the viability of boolean algebra to the operations of disjunctive syllogism. If a diner goes into a restaurant and say they want the soup or the salad, and they don’t want the soup, they would probably not leave a very large tip if the waiter brought them the soup, or a hamburger.
Classical logic is not to be dismissed casually, glossed over, or rhetorically pilloried. If a diner say they want the soup or the salad, and they don’t want the soup,. they probably want the salad. Furthermore, they probably want only the salad, and not a hamburger or an igloo. Alternative solutions to our hypothetical diners dinner order wouldn’t be “the salad”. The other potential items on the list of options our waiter could serve our diner, who quite clearly requested the salad, are probably vague. Perhaps we could venture one step further, and say that when our diner requests simply “the salad”, they want “the salad”, if “the salad” and “the soup or the salad, and not the soup” are read to be equivalent.
If a generalization of “A or B, not A, then B” were to hold, then boolean algebra would hold. The intellectual leap from disjunctive syllogism to boolean algebra is not difficult; a better question would be whether disjunctive syllogism were to hold. The previous hypothetical scenario of a waiter and a diner is an example of disjunctive syllogism. As an example of disjunctive syllogism, what the hypothetical diner wants matters to disjunctive syllogism. What could the patron at the restaurant want other than a salad when that patron says they want a soup or a salad, and that they don’t want the soup?
The direct approach to defending Boolean algebra is the soundest. The theoretical burden would thereby be to show that Boolean algebra were true. First, classical logic makes a lot of sense, prima facie. Second, a reflexive examination of classical logic would imply that Boolean algebra would define itself as true according to itself. Any departure from Boolean algebra would probably cause logic to disprove itself, because it would arrive at a solution other than that proscribed by Boolean algebra. There might be a “silver bullet”, a proof of classical logic, but I haven’t encountered it.
Mathematical proofs assume classical logic as a predicate. Thus, my work meets the same standards as, for example, Gödel’s incompleteness theorem.
May 5, 2025
Updated May 14, 2025 and May 16, 2025