### 6. The Liar paradox (redux)

I dealt with the Liar paradox previously, in my Future Logic, but now realize that more needs to be said about it. This paradox is especially difficult to deal with because it resorts to several different discursive ‘tricks’ simultaneously.

### 1.    First approach

The statement “This proposition is false” looks conceivable offhand, until we realize that if we assume it to be true, then we must admit it to be indeed false, while if we assume it to be indeed false, then we must admit it to be true – all of which seems unconscionable. Obviously, there is a contradiction in such discourse, since nothing can be both true and false. But the question is: just what is causing it and how can it be resolved? We are not ‘deducing’ the fact of contradiction from a ‘law of thought’ – we are ‘observing’ the fact through our rational faculty. We cannot, either, ‘deduce’ the resolution of the contradiction from a ‘law of thought’ – we have to analyze the problem at hand very closely and creatively propose a satisfying solution to it, i.e. one which indeed puts our intellectual anxiety to rest. As we shall see, this is by no means a simple and straightforward matter.

The proposition “This proposition is false” is a double paradox, because: if it is true, then it is false; and if it is false, then it is true. Notice the circularity from true to false and from false to true. The implications we draw from the given proposition seem unavoidable at first sight. But we must to begin with wonder how we know these implications (the two if–then statements) to be true. How do we know that “it is true” implies “it is false,” and that “it is false” implies “it is true”? Apparently, we are not ‘deducing’ these implications from some unstated proposition. We are, rather, using ad hoc rational insight of some sort – i.e. in a sense directly ‘perceiving’ (intellectually cognizing) the implications of the given proposition. But such rational insight, though in principle reliable, is clearly inductive, rather than deductive, in epistemological status. That is to say, it is trustworthy until and unless it is found for some reason to be incorrect. This means, there may be one or more errors in our thinking, here; it is not cast in stone. And indeed there must be some error(s), since it has led to double paradox. Therefore, we must look for it.

Perhaps use of the pronoun “it” is a problem, for it is a rather vague term. Let us therefore ask the question: more precisely what does the pronoun “it” refer us to, here?

At first sight, the “it” in “if it is true, then it is false; and if it is false, then it is true” refers to the whole given statement, “This proposition is false.” In that event, we must reword the double paradox as follows: if ‘this proposition is false’ is true, then ‘this proposition is false’ is false; and if ‘this proposition is false’ is false, then ‘this proposition is false’ is true. Here, the subject of the two if–then statements is more clearly marked out as “this proposition is false,” and so remains constant throughout. But this clarification reveals abnormal changes of predicate, from “true” to “false” and from “false” to “true,” which cannot be readily be explained. Normally, we would say: if ‘this proposition is false’ is true, then ‘this proposition is false’ is true; and if ‘this proposition is false’ is false, then ‘this proposition is false’ is false. The reason we here reverse the predicates is that we consider the original proposition, “this proposition is false,” as instructing such reversal.

However, whereas a proposition of the form “‘this proposition is false’ is true” is readily interpretable in the simpler form “this proposition is false,” a proposition of the form “‘this proposition is false’ is false” cannot likewise be simplified. How would we express the double negation involved? As “this proposition is true”? Clearly, the meaning of the latter is not identical to that of the former, since the subject “this proposition” refers to different propositions in each case. So the formulation of the liar paradox in full form, i.e. as “if ‘this proposition is false’ is true, then ‘this proposition is false’ is false; and if ‘this proposition is false’ is false, then ‘this proposition is false’ is true,” does not make possible the reproduction of the initial formula expressed in terms of the pronoun “it.”

### 2.    Second approach

Let us therefore try something else. If the pronoun “it” refers to the term “this proposition”, then the double paradox should be reformulated as follows: if ‘this proposition’ is true, then ‘this proposition’ is false; and if ‘this proposition’ is false, then ‘this proposition’ is true. But doing that, we see that in each of these two if–then statements, though the subject (“this proposition”) remains constant throughout, the predicate (“true” or “false,” as the case may be) is not the same in the consequent as it was in the antecedent. There is no logical explanation for these inversions of the predicate. Normally, the truth of a proposition P does not imply its falsehood or vice versa.

We might be tempted to use the given “This proposition is false” as a premise to justify the inference from the said antecedents to the said consequents. We might try to formulate two apodoses, as follows:

 If this proposition is true, then it is false (hypothesis), and this proposition is false (given); therefore, this proposition is true (putative conclusion).

 If this proposition is false, then it is true (hypothesis), and this proposition is false (given); therefore, this proposition is true (putative conclusion).

Obviously, in the first case we have invalid inference, in that we try to deny the antecedent to deny the consequent, or to affirm the consequent to affirm the antecedent. In the second case, the putative conclusion does follow from the premises; but we can still wonder where the major premise (the hypothetical proposition) came from, so we are none the wiser. So, this approach too is useless – i.e. it proves nothing.

Alternatively, we might try formulating the following two syllogisms:

 This proposition is false (given), and this proposition is true (supposition); therefore, this proposition is false (putative conclusion).

 This proposition is false (given). and this proposition is false (supposition); therefore, this proposition is true (putative conclusion).

Clearly, these arguments are not quite syllogistic in form; but they can be reworded a bit to produce syllogisms. The first two premises would then yield the conclusion “there is a proposition that is true and false” (3/RRI), which is self-contradictory (whence, one of the premises must be false); the second two premises, however, being one and the same proposition, would yield no syllogistic conclusion other than “there is a proposition that is false and false” (3/RRI), which is self-evident (and trivial). But these are not the conclusions we seek, which must concern “this proposition” and not merely “some proposition.”

A better approach is to look upon the latter two arguments as follows. In the first case, the premises “this proposition is false” (given) and “this proposition is true” (supposition) seem to together imply “this proposition is both true and false;” and the latter paradoxical conclusion in turn indeed suggests that “this proposition is false,” since contradiction is impossible. And in the second case, the premises “this proposition is false” (given) and “this proposition is false” (supposition) agree with each other that “this proposition is false,” and so this is their logical conclusion. Since both arguments conclude with “this proposition is false,” the latter must be the overall conclusion.

However, the latter result is not as conclusive as it seems, because upon closer scrutiny it is obvious that “this proposition is false” and “this proposition is true” do not refer to the same subject, since the predicate changes. The first “this proposition” refers to the proposition “this proposition is false” and the second “this proposition” refers to the proposition “this proposition is true.” So, these two propositions in fact have different subjects as well as different predicates (viz. false and true, respectively). The subjects superficially look the same, because they are verbally expressed in identical words; but their underlying intent is not the same, since they refer to significantly different propositions (propositions with manifestly different, indeed contradictory, predicates). This means that when the predicate changes, the subject effectively changes too. When the predicate is “true,” the subject means one thing; and when the predicate is “false,” the subject means something else. Although the words “this proposition” are constant, their underlying intent varies. That is to say, the term “this proposition” does not have a uniform meaning throughout, and therefore cannot be used as a basis for the inferences above proposed.

### 3.    Third approach

Let us now try another angle. If we examine our initial reasoning in terms of the pronoun “it” more carefully, we can see what is really happening in it. Given that ‘this proposition is false’ is true, we can more briefly say: ‘this proposition is false.’ Also, given ‘this proposition is false’ is false, we can by negation educe that ‘this proposition is not false’ is true, which means that ‘this proposition is true’ is true, or more briefly put: ‘this proposition is true’. In this way, we seem to argue, regarding the subject “this proposition is false,” from ‘it is true’ to ‘it is false’, and from ‘it is false’ to ‘it is true’. But in fact the use of the pronoun “it” or the term “this proposition” as abbreviated subject is a sleight of hand, for the underlying subject changes in the course of the second transition (that ending in “this proposition is true”). When abbreviation is used throughout, we seem to be talking about one and the same proposition throughout as being both true and false. But seeing that this is based on hidden equivocation, the paradoxes disappear.

It is interesting to note that when the reasoning is viewed more explicitly like that, the proposition “this proposition is true” also becomes paradoxical! We can argue: if ‘this proposition is true’ is true, then obviously ‘this proposition is true’. And: if ‘this proposition is true’ is false, then its contradictory ‘this proposition is not true’ must be true, which means that ‘this proposition is false’ is true, i.e. more succinctly: ‘this proposition is false’. Here, superficially, there seems to be no paradox, because we seem to argue, regarding the subject “this proposition is true,” from ‘it is true’ to ‘it is true’, and from ‘it is false’ to ‘it is false’. But if we look at the final conclusion, viz. “this proposition is false,” we see that it corresponds to the liar paradox! And here again, the explanation of the double paradox is that the apparent subject “it” or “this proposition” changes significance in the course of drawing the implications.

Notice that, in both these lines of reasoning, the first leg is ordinary self-implication, mere tautology, while the second leg is the operative self-contradiction, the paradox. If the given proposition (whether “this proposition is false” or “this proposition is true”) is true, we merely repeat the proposition as is (without need to add the predication “is true”). But if the given proposition is false, we cannot drop the additional predication (i.e. “is false”) without changing the original proposition. Thus, we could say that the two propositions, “this proposition is false” or “this proposition is true,” present no problem when taken as true; and it is only when they are hypothetically taken as false that the problem is created. So we could say that the way out of the liar paradox (and its positive analogue) is simply to accept the two claims as true, and not imagine them to be false!

We could furthermore, if we really want to, argue that “this proposition is false” and “this proposition is true” differ in that the former explicitly appears to put itself in doubt whereas the latter does not do so. On this basis, we could immediately reject the former and somewhat accept the latter, even while admitting that the latter is equally devoid of any useful information. That is to say, since the former appears ‘more paradoxical’ than the latter, the latter is to be preferred in extremis. But this, note well, ignores the equally insurmountable difficulties in it. It is better to resolutely reject both forms as vicious constructs.

### 4.    Fourth approach

To grasp the illusoriness of the liar paradox, it is important to realize that the two forms, “this proposition is false” and “this proposition is true,” are not each other’s contradictory; and that, in fact, neither of them has a contradictory! This is a logical anomaly, a fatal flaw in the discourse of the liar paradox; for in principle, every well-formed and meaningful proposition is logically required to have a contradictory. If a propositional form lacks a contradictory form, it cannot be judged true or false, for such judgment depends on there being a choice. We do not even have to limit our propositions to the predicates “true” or “false” – any predicate X and its negation not-X would display the same property given the same said subject. That is, “this proposition is X” and “this proposition is not-X” are not each other’s contradictory, and are therefore both equally deprived of contradictory.

We could, of course, remark that “this proposition is X” can be denied by “that proposition (i.e. the preceding one) is not X,” or even introduce a symbol for the original proposition in the new proposition. In such case, although the subjects would be verbally different, their intents would surely be the same. But the form “that proposition is not X” is more akin to the form “‘this proposition is X’ is not X,” in which the whole original proposition is given the role of subject and its predicate is given the role of predicate. However, though these two forms are somewhat equivalent in meaning to each other and to the original proposition, their logical behavior patterns are not identical with that of the original proposition, as we have already seen. The fact remains that “this proposition is not X” is not the contradictory of “this proposition is X.”

Clearly, any proposition involving the special subject “this proposition” exhibits a very unusual property, and may be dismissed on that basis alone. The reason why such a proposition lacks a contradictory is that its subject refers to the proposition it happens to be in, and that proposition is evidently different when the predicate in it is the term “false” and when it is the term “true” (or more generally, any pair of predicates ‘X’ and ‘not-X’). When the predicate changes, so does the subject; so the subject cannot be pinned-down, it is variable, it is not constant as it should be. The term “this proposition” has a different reference in each case, which depends on the predicate; consequently, each subject can only be associated with one predicate and never with the other (i.e. its negation).

From this we see that when at the beginning we thought, looking upon the statement “This proposition is false,” that if we take it at its word, then it is must be regarded as false, and so we have to prefer to it “This proposition is not false,” i.e. “This proposition is true,” and so forth, we did not realize that we were in fact, due to the ambiguity inherent in the term “This proposition” or “it,” changing its meaning at every turn. This change of meaning passes by unnoticed, because the term used is by its very nature not fixed. The pronouns “this” and “it” can be applied to anything and its opposite without such change of meaning being verbally signaled in them. They are not permanently attached to any object, but are merely contextual designations. In the technical terminology of linguistics, they are characterized as ‘deictic’ or ‘indexical’.

Thus, it appears that the liar paradox arises, however we understand its terms, as a result of some sort of equivocation in the subject. Although we seem superficially to refer to one and the same subject in the antecedent and consequent of our if–then reasoning, there is in fact a covert change of meaning which once we become aware of it belies the initial appearance of contradiction. The suggested impossible implications are thus put in doubt, made incredible. The contradictions apparently produced are thus defused or dissolved, by virtue of our inability to make them stick.

### 5.    Fifth approach

Another, and complementary, way to deal with the liar paradox is to point out the logical difficulty of self-reference. This is a tack many logicians have adopted, including me in my first foray into this topic in Future Logic. The argument proposed here is that the term “this proposition” refers to an object (viz. “This proposition is false” or “This proposition is true”) which includes the term itself. A finger cannot point at itself, and “this” is the conceptual equivalent of a finger. Effectively, the expression “this” has no content when it is directed at itself or at a sentence including it. It is empty, without substance. It is as if nothing is said when we indulge in such self-reference.

Thus, “This proposition is X” (where X stands for false, or true, or indeed anything) is in fact meaningless; and a meaningless sentence cannot be true or false. Such a sentence can reasonably be described as neither true nor false, without breach of the law of the excluded middle, because neither of these logical evaluations is applicable to meaningless sentences. “This proposition is false” looks meaningful because its four constituents (i.e. “this,” “proposition,” “is” and “false”) are separately normally meaningful. But in this particular combination, where one of the elements (viz. “this”) does not refer to anything already existent, the sentence is found to be meaningless.

The apparent contradictions that self-reference produces help us to realize its meaninglessness. And it is through the intellectual realization of the meaninglessness of self-reference that we explain away and annul the apparent contradictions. On this basis, we can say that even though the sentence “This proposition is true” does not at first sight give rise to any paradox (as people think: “if it is true, it is true; and if it is false, it is false”), nevertheless, since it involves self-reference as much as “This proposition is false,” it is equally meaningless and cannot be characterized as true or false. In fact, as I have shown above, “This proposition is true” does also give rise to double paradox.

Someone might object: What about the propositions: “this statement is self-referential” and “this statement is not self-referential”? Surely, we can say that these are meaningful and that the former is true while the latter is false! The retort to that objection is that the two propositions “this proposition refers to itself” and “this proposition does not refer to itself” are not mutual contradictories, because (just like in the liar paradox) their subjects differ radically, each referring to the proposition it is in and not to the other. Thus, while the positive version may seem more self-consistent than the negative one, and therefore to be preferred in extremis, they are in fact both fundamentally flawed, because (just like in the liar paradox) neither of them has a contradictory, and without the logical possibility of negating a discourse it is impossible to judge whether it is right or wrong.

### 6.    Sixth approach

Not long after the preceding reflections, I happened to come across another interesting example of paradoxical self-reference, namely “Disobey me!” This involves the ‘double bind’ – if I obey it, I disobey it and if I disobey it, I obey it. To resolve this paradox, we need to first put the statement in more precise form, say: “you must disobey this command!” We can then disentangle the knot by realizing that the order being given has outwardly imperative form but inwardly lacks content. It does not define a specific, concrete action that is to be done or not-done. If we wished to obey it, or to disobey it, we would not know just what we are supposed to do or not-do! It is therefore an order that can neither be obeyed nor be disobeyed. Ruminating on this case led me to what I now believe is the trump card, which convincingly finalizes the resolution of the liar paradox, even as the preceding reflections all continue to be relevant.

It occurred to me then that this is precisely the problem with the liar paradox. It says “this proposition is false” – but it does not tell us anything about the world that can be judged as true or false. A ‘proposition’ is a statement that makes some claim about the world. If the statement makes no such claim, if it ‘proposes’ nothing, it cannot be logically assessed as true or false. If it refers to nothing – whether physical, mental or spiritual, perceptual, intuitive or conceptual – it has no meaning. A meaningless statement does not qualify as a ‘proposition’. The attributes of ‘true’ or ‘false’ are not ordinary predicates, like ‘white’ or ‘black’, which can be attached to any subject and then judged to be truly or falsely attached. The attributes of ‘true’ or ‘false’ require a precise claim to be made before they can at all be used.

The truth of this explication can be seen with reference to the ‘propositional forms’ used in logic theory. Take, for example, “All X are Y.” Such a propositional form cannot be judged true or false because it manifestly has no content. Only when such an abstraction is given some specific content, such as “All men are mortal,” can we begin to ask whether it is true or false. A propositional form is too vague to count as a proposition. It does not tell us anything about the world, other than implying that there are (or even just that there may be) concrete propositions which have this form. Just as we cannot disobey or even obey an imperative without content, so we cannot judge a purely formal expression true or false.

The same applies to the liar paradox: like a formal proposition, it has no concrete content, and therefore cannot be judged true or false. The liar paradox has no content partly due to its having a self-referential subject (“this proposition”). But the truth is, even if its subject was not self-referential, it would still have insufficient content. This is so, because its predicate “false” (and likewise its opposite, “true”) is not an ordinary predicate; it is more like a formal predicate. It can only be used if another, more concrete predicate has already been proposed for the subject at hand. For example, “this proposition is interesting” could be judged true or false (if it was not self-referential) because it already has a predicate (viz. “interesting”). Thus, the problem with the liar paradox is not only the self-reference it involves but also its lack of a predicate more concrete than the logical predicate “false” (or “true”).

All this illustrates how the ‘laws of thought’ are not axioms in the sense of top premises in the knowledge enterprise from which we mechanically derive other premises. Rather the expression ‘laws of thought’ refers to recurring insights which provide us with some intellectual guidance but cannot by themselves determine the outcome. The individual in pursuit of knowledge, and in particular the logician, is driven by the obviousness or by the absurdity of a situation to look for creative solutions to problems. He or she must still think of possible solutions and test them.

From A Fortiori Logic, appendix 7.4.

              Some logicians have tried to deal with the liar paradox by denying that true and false are contradictory terms, i.e. that not-true = false and not-false = true. Such a claim is utter nonsense; the attempt to shunt aside the laws of non-contradiction of the excluded middle so as to resolve a paradox is self-contradiction in action.

              That ‘this proposition is true’ is implicitly (if only potentially) as paradoxical as ‘this proposition is false’ is, so far as I know, a new discovery. Note well how both paradoxes occur through quite ordinary eductions: viz. if ‘P is Q’ is affirmed, then P is Q; and if ‘P is Q’ is denied, then ‘P is not Q’ is affirmed, then P is not Q (where P stands for ‘this proposition’, and Q for ‘false’ or ‘true’ as the case may be).

              Another objection (which was actually put to me by a reader) would be propositions like “this statement has five words” and “this statement has six words” – even though they contain the demonstrative “this,” the former looks true and the latter false! Here, we might in reply point out that though the propositions “this statement has five words” and “this statement does not have five words,” seem to mean opposite things, they cannot be contradictories, since both appear true. Also compare: “this statement has five words” and “this statement does have five words” – the former is true while the latter is false, though both mean essentially the same. Clearly, the behavior of these propositions is far from normal, due to their unusual dependence on the wording used in them. On one level, we get the message of the proposition and count the number of words in it, and then check whether this number corresponds to the given number: if yes, the proposition is judged ‘true’, and if no, it is judged ‘false’. But at the same time, we have to be keep track of the changing reference of the demonstrative “this,” which complicates matters as already explained, and additionally in this particular context we must beware of the impact of wording. The Kneales give “What I am now saying is a sentence in English” as an example of “harmless self-reference” (p. 228).

              I found this example in Robert Maggiori’s La philosophie au jour le jour (Paris: Flammarion, 1994); the author does not say whether it is his own invention or someone else’s (p. 438).