Cicero's Liar Paradox

As a follow up to my previous post and a quick exercise, I will briefly consider a more classic formulation of the Liar's Paradox. This one come from Cicero's Academica (2.30), though he attributes it to Chrysippus. Cicero (an Academic Sceptic) is responding to Lucullus (follower of Antiochus who practically became a Stoic). Here he casts doubt on the validity of logical inference.

Cicero puts its thus: Si dicis te mentiri verumque dicis, mentiris; dicis autem te mentiri verumque dicis; mentiris igitur. Translation: "If you say that you are lying and are speaking truly, you lie; but you do say that you are lying and are speaking truly; therefore you lie."

The following is valid, though not really a syllogism as such because it doesn't give us new knowledge (since the minor premise is the same as the conclusion): Si mentiris, mentiris; mentiris autem; mentiris igitur. If you are lying, you are lying; you are lying; therefore you are lying. Cicero states this later as if it was equivalent to the paradox stated above. It is not however.

Lets take the first premise of the paradox:  "If you say that you are lying and are speaking truly, you lie." What are you saying that you are lying about? If about another proposition and you were in fact lying, then you would indeed be speaking truly. But of course there is no paradox here. You would not be claiming that you were lying about this proposition. The idea is that the liar is lying about his own statement about his lying. 

But if you say you are lying about the very statement you are making (sc. that you are lying), your statement is actually meaningless. In order to be lying about something, you have to be lying about something. If you say that your are lying, what are you lying about? If the answer is that you are lying, then again it can be asked what that lying is about. If that is about lying and that is about lying and so on, we have an infinite regress. The statement "I am lying" only makes apparent sense. But if it just about itself, it doesn't make sense.

Secondly, since every proposition implies its own truth, if I say "I am lying" then I am affirming the truth of my lying. But at the same time I am affirming the falsity of my lying because I am lying. Therefore, the statement is incoherent and absurd. This doesn't mean that it is both true and false. But rather that it is neither. It is absurd. If you both affirm and deny the same thing, you aren't making sense. Likewise a pseudo-proposition that affirms and denies the same thing doesn't actually make any claim. Therefore, it is not a proposition. A proposition affirms an identity between two beings or denies such an identity. If I try to both of these things, then I end up doing neither. I have cancelled myself out. Therefore, I have made my statement absurd. I have deprived it of intelligible value. 

Self-Evident Truth and the Liar's Paradox

Sophists and sophistry have always been around. It's a hallmark of sophistical fallacies to challenge some obvious truth with a supposedly convincing but fallacious argument. One might think how such truths can be doubted. For example, how can someone doubt the principle of non-contradiction? Isn't it self-evident? Yes, the principle of non-contradiction is self-evident but that doesn't entail that nobody can signify or want to doubt it. The wilful rejection of reality is as old as the Devil's non serviam.

Anyway, what does self-evident mean? It refers to some truth which is known through itself (per se nota). In particular, we are focusing here on that which is per se nota to us (quoad nos), rather than the other kind, per se nota in itself. God's existence is self evident in itself (secundum se) because the predicate in "God exists" is included in what it means to be the subject (in ratione subiecti). But we do not behold the essence of God in this life immediately, so we cannot see immediately that God exists. God must exist by His very nature and, in fact, it is not just that we would know the fact of his being (existentia) if we saw him (like we can know the fact of chair existing if we perceive the chair), but we would see immediately his act of being (esse / actus essendi) when we saw immediately what he is (his essentia) because in Him esse and essentia are identical. But what is self-evident to us is what can be immediately known when the terms are known. If I know what wholes are and I know what parts are, I know immediately the truth of the statement that a whole is greater than its parts. If I know what 2 is and what 1 is, I know that 1 + 1 = 2. 2 literally just is 2 1's. Boethius points out that some things can be self-evident only to the learned. So, for example, if I know what an incorporeal substance is and what space is, I know that an incorporeal substance is not extended in space. 

Now, the most fundamental and basic of self-evident truths is the truth that 'being is.' A rock might not know that. But we rational creatures that have come into contact with being cognitionally know it. It follows from 'being is' that 'being is not non-being.' This is one formulation of the principle of non-contradiction. Indeed, 'follows' is said loosely here because unlike our knowledge that God exists, the PNC does not have to be derived from a series of prior premises. It is a principle in itself. It does follow from 'being is' but it also just follows from itself because 'being is' 1) is present in it and 2) is present in it in such a way that if PNC were not the case, it would not be the case. Now, 1) is actually true of every statement. Every statement affirms that 'being is.' 'Being is' is not just the first premise in every argument but is present in every statement. But wait, what if I say the King of France is bald?! Shock horror. Well yes, that statement too implies 'being is.' If I say that the King of France is bald, then I am affirming the being of the King of France's baldness. And if I affirm a way of being, then I am affirming being. If there is no existing King of France (unless one counts Louis XX), then all I've done is give a false statement. Such things...do in fact happen. I have affirmed a way of being that isn't actually there. No paradox here. 

So if I know what 'being' is and I know what 'non-being' is (the negation of being), then I know that they are opposed. I know that being is not non-being. By consequence I know that I cannot correctly both affirm and deny a particular way of being in the same respect at the same time. I cannot coherently claim both that the King of France is bald and that he is not bald. I would have contradicted myself. Ordinary people and sane philosophers call that an error.

Of course someone can fashion a sentence that denies the Principle of Non-Contradiction but the sentence would be incoherent and absurd. Maybe some people (weird people) might want to deny the Principle and so fashion such a sentence. We don't just have intellects. We have a will and emotions. But, if the intellect grasps what is meant by being and non-being, then the intellect will just have to assent. Likewise, if I grasp what is meant by truth and falsity, my intellect will have the recognise their incompatibility. There is a convertibility between truth and being. Truth is being known through an intellect.

But what if I subtly redefine truth and falsity? What if, inspired by the wanderings of analytic philosophy, I lose track of what truth and falsity really mean. What if I confound grammar and logic, while insisting on a firm divide between logic and metaphysics? What if I forget, deny, or fuzzy truth's identity with being, and think of truth and falsity superficially in terms of mere 'statements'? What if I have a retarded (literally - I'm not just thrusting forward an insult) understanding of language?

What is a statement? Is it a set of words? What is its meaning? If I'm talking about a grammatical statement, then it is a collection of conventional signs representing thoughts. These thoughts refer to and present to us reality ie. being. Grammatical signs can have different meanings and they have the meaning we assign to them. There is a contingent relationship here. But there is not such a contingent relationship between the mind knowing something and the something known by the mind. There is a cognitional identity here.

Let's take the Liars' Paradox. According to supposed philosopher Graham Priest, this paradox shows that statements can be both true and false. If we had a good idea of what truth and falsity mean and what statements are, we can see how this is ridiculous. If a statement is true, the mind is in conformity with reality, with being. I am affirming (mentally, not just grammatically) a way of being that is actually there (or denying one that isn't there). If it is false, my mind is at odds with reality. I am affirming a way of being that isn't actually there (or denying one that is there). Being is being. Being is not non-being. If I am affirming some way of being, that being is either there or not there. It cannot both be and not be. 

One formulation of the Liars's Paradox goes like this: "This statement is false." It is then pointed out that if I say the statement is true, then the statement is false, because the statement literally says that it is false. But if I say that it is false, then that would make the statement true. Or would it?

The truth is that the statement is incoherent. It is like saying "square triangles tend to quizzle quazzle." Arthur Prior got close to the truth when he said that the statement is in fact false. As he pointed out, every statement implies its truth. As I said before, if I say that the King of France is bald, then I am affirming the being of the King of France's baldness. I am affirming that it is true that the King of France is bald. I do not have to preface it with "it is true that...." Given that then, Prior pointed out that saying "this statement is false" is equivalent to saying "this statement is true and false." Prior then concluded that such a statement would be false. Therefore, the statement "this statement is false" is false and there is no contradiction here. I do not make it true if I affirm its falsity because the statement also claims its own truth. To emphasise: every statement of itself claims its own truth. 

But I would say that rather than concluding that the statement is false, we might instead conclude that the statement is incoherent and fundamentally meaningless. If I say that "square triangles tend to quizzle quazzle" I am not really saying anything either true or false because I am not really saying anything at all. The terms don't refer to intelligible content. I can say that "squares aren't triangles" because "square" and "triangle" do have intelligible content. I can even say that "square triangles don't exist" because what I'm doing there is denying the possibility of identity between squares and triangles, which both have intelligible content. But if I say "square triangles surf" where square-triangles is taken as a whole and is put in the position of something intelligible, then I have a meaningless proposition. If I say "this statement is false" then I am simultaneously affirming the truth and falsity of the statement. Hence, the statement is absurd. The lesson to take from this is that I can signify a simultaneous denial of the truth or falsity of a statement. But such a statement can't actually be both true and false. It cannot both correctly affirm or deny the same being or way of being.

Anyway, what being or way of being am I actually supposed to be referring to? Hopefully, you can see where logical puzzles can get ridiculous. This is the result of trying to separate logic too far from reality. Logic deals immediately with beings of reason. True. But beings of reason in turn deal with real being. 

"This statement is false" is meaningless. In fact even "this statement is true" is meaningless by itself. What statement? The one that goes "this statement is true"? So "this statement is true is true." But I still don't understand what "this statement" means because all I've been told is that is true! Telling it twice won't solve anything! When I predicate "is true" of something, I have to predicate it of something that it can actually be meaningfully predicated of. So for example: 'The sun is hot' is true. Or I can say "that statement is true" referring to another statement. In short, I must be predicating being of something. If I'm simply saying words, I'm not accomplishing anything.