This is a paradox that, as I, a logic purist (If such a label exists anywhere) have read, troubles me, for it seems to strike a jab at one of the three most foundational laws of logic, which is that of noncontradiction (A or not A, or formally, A v ~A). If such a principle of logic was not necessarily true, or else, in every instance of a possible world, it does not hold in at least one of them, then utterly absurd things about reality could be true, such as that it is possible that I am writing this post and that I am not writing this post at the same time. Indeed, it would, if the law of noncontradiction is not necessarily true, be possible for me to be both fat and not fat, for the sun to also not be the sun, and for me to know English (Necessary for me to write this very sentence) and to not know English (And so I shouldnāt be able to write this sentence), both simultaneously with one another, not merely as a sequential change of state from, say, Not A into A (For example, for me to go from not knowing English to knowing English). This seems completely wrong, and it should be. But then here comes the liarās paradox, stated thus:
This sentence is false.
Interesting, for if it is the case that it is true that the sentence is false, then it is false despite it being true. And if it is false that the sentence is false, then the sentence must be true, despite it being false, which is an outright logical contradiction, for this means, then, that the sentence is, depending on what truth value you assign to it, both true and false at the exact same time. What should follow from this, then, is that certain propositions, such as āThis sentence is falseā, do not follow the law of noncontradiction, and so cannot simply be either true or not true, and if this is the case that some propositions donāt follow the law of noncontradiction, then it follows that the law of noncontradiction does not hold in every possible world, and if the law of noncontradiction does not hold in every possible world, or else is contingently true, then it is possible, in some world, for me to know English and to also not know English. Yikes.
But something seems wrong with this Paradox, and it can be unearthed if we dissect what āThis sentence is falseā actually means. And we can do it with this simple question: What is āThis sentenceā? Is āthis sentenceā the full proposition as stated, āThis sentence is false,ā which can be expressed as A, or else is it āThis sentenceā by itself, expressed as A, and then the āis falseā expresses a negation of āThis sentenceā (A), which would look like ~A? If it is the former, then what would be true is that if it is true that āThis sentence is falseā, or it is true that A, then all that would happen is that it turns out that A is asserted to be a true proposition, and that the only problem, then, would be to characterize what A actually means, or else it would be up to us to characterize what the contents of A have, or what their predicates are, to determine if the assertion that A is true is either a true assertion or a false one. For example, if I assert that it is true that I dropped my ice cream, then it would be up to others to determine whether or not my assertion of the truth of me dropping ice cream, or else the truth of me declaring A, is a true assertion or not a true assertion, on the basis of the existing evidence in support of the assertion, and this would not be a contradictory, nor an unreasonable, way to characterize āThis sentenceā. Now if the latter possibility is true, which is that āThis sentenceā means the first part of the sentence, āThis sentence,ā is the proposition, or A, and that the last part of the sentence, āis falseā is a negation of the first part of the sentence, or A, then āthis sentenceā can be expressed as Not A, or formally, ~A. So if we defined āthis sentenceā this way, then if we assert that āthis sentence is falseā is true, then we would be saying, instead, that it is true that Not A, or it is true that ~A. Therefore, if defined this way, then this would be analogous to me asserting that it is true that I didnāt drop my ice cream, or it is true that Not A, or ~A, and this would not be a contradictory way to define āthis sentenceā.
What I think follows from this argument, then, is that the liarās paradox, which is āThis sentence is false,ā is not a true instance of a proposition that can violate the law of noncontradiction, but instead is an instance of a proposition that, depending on how specifically it is defined, can lead to instances where the truth value assigned to the specific definition can be preserved or negated. Therefore, since definitions arenāt fixed, but the objects which propositions can be assigned to are fixed, then the paradox seems, from this, to be a larger problem in linguistics than it does for logic itself.
To be frank, writing this post kind of melted my brain, and Iāve not looked into other solutions to the problem very deeply other than the problem itself, so if something seemed to be of question in this post (This is my first post), then say something.