The liar paradox

Consider our friend Pinocchio who says, "I am lying". Is his statement a lie? Or does he speak truly? It seems that if he is telling the truth, then he must be lying, just as he says. In which case he is not telling the truth. But if he is not telling the truth, then he is indeed lying. Just as he truly says!

This is known as the liar paradox. It seems that there is no satisfactory answer for whether Pinocchio is lying or speaking the truth. There are many variations of this paradox. For example, "Epimenides was a Cretan who made one immortal statement: 'All Cretans are liars.'". Another is the sentence, "This sentence is false."

In this post I will offer an explanation for the why the paradox occurs that builds on Gilbert Ryle's analysis [1].

The first thing that is usually noticed is that the statement, "This sentence is false." refers to itself. So perhaps self-reference is the problem. But we use self-reference all the time. For example, "I have a blue shirt on" is a perfectly valid statement. As is, "This sentence has five words".

So perhaps the problem is the way that the statement refers to itself. To evaluate whether the sentence is false first requires evaluating the truth value of the referring expression, namely 'This sentence'. That referring expression is just the original sentence. So that sentence, in turn, first requires evaluating the truth value of its referring expression, namely 'This sentence'. We can see that this is a recursive expression that never terminates in an evaluable statement. So it is not truth-apt [2].

But this raise the question as to why the statement, "This sentence has five words" doesn't have the same unending recursion. This statement expands as, "'This sentence has five words' has five words". We naturally stop at this point and just count the words in the inner sentence. But what is the rule for why we should stop? Shouldn't it expand indefinitely as well? One answer is that this sentence is grounded [3]. That is, the recursive expansion terminates in a sentence that doesn't mention truth.

However one can then go on to create strengthened liars such as, "This sentence is false or not truth-apt". If it is true, then it is false or not truth-apt, so it is not true. If it is false, then it is true, so it is not false. If it is neither true nor false then it is not truth-apt. Which is what the sentence says, so it is true! So it seems there is no satisfactory answer to the strengthened liar. However one approach here is to say that since the sentence is demonstrably not truth-apt, then it should not be subsequently read as making a truth-apt claim. It is a kind of illusion - it appears to be truth-apt but it is not.

A shortcoming of the above approach is that that there doesn't seem to be a principled rule to apply to determine whether a liar-style sentence is truth-apt. One just plugs in values and tries to find a custom solution. If a solution is found, then a new strengthened liar is constructed to foil it (these are also aptly called revenge liars!) And we don't seem closer to understanding what it is that makes liar-style sentences defective.

The approach I take instead involves distinguishing between use and mention. My claim (on ordinary use grounds) is that only the use of statements can be truth-apt whereas mentions of statements are not truth-apt (and treating them as if they were is a category mistake). When we mean to mention a statement rather than use it, we enclose the statement in quotation-marks. Consider the sentence, "This sentence has five words". This expands as "'This sentence has five words' has five words". The inner sentence is mentioned and thus (per my claim) is not truth-apt. This isn't an issue here since a truth predicate is not being used and the sentence is readily evaluable with the predicate it does have.

But what about a statement like "'Snow is white' is true"? Since that statement does evaluate as true, it seems that a mention of a statement can be true after all. My claim is that this is not the case. Instead, in ordinary use, a truth-predicate has a disquotation function [4]. That is, it removes the quotation marks and uses the statement. Thus, the statement ""Snow is white' is true" is transformed to "Snow is white" which (when used) is a truth-apt statement. More generally, "'p is q' is true" becomes "p is q" and "'p is q' is false" becomes "Not (p is q)". I will term this the disquotation rule.

Now consider the liar sentence, "This sentence is false". This expands as, "'This sentence is false' is false". Note that the inner sentence is mentioned, not used, so it is not truth-apt. However, since a truth predicate is specified, the disquotation rule can be applied. So applying this rule to the liar sentence, we get:

1. This sentence is false
2. "This sentence is false" is false [mention]
3. Not ("This sentence is false" is true)
4. Not (This sentence is false) [disquotation]
5. Not ("This sentence is false" is false) [mention]
6. "This sentence is false" is true
7. This sentence is false [disquotation]

What emerges is that the liar sentence is cyclic (line 7 is identical to line 1). We can also apply this rule to the truth-teller sentence:

1. This sentence is true
2. "This sentence is true" is true
3. This sentence is true

Thus the truth-teller is also cyclic. We can also test this approach with a conventional self-referring sentence:

1. This sentence has five words
2. "This sentence has five words" has five words
3. True

So the sentence is indeed truth-apt. The nice thing about applying the disquotation rule is that there is no need to plug in values and test each one. We can just follow the logic of the statement and note whether it terminates in a truth-apt value or, else, cycles.

Now consider the strengthened liar:

Let L be the sentence, "This sentence is false or not truth-apt".

1. L
2. "L" is false or not truth-apt
3. "L" is false or "L" is not truth-apt
4. Not ("L" is true) or ("L" is not true and "L" is not false)
5. Not (L) or (not "L" is true and "L" is true)
6. Not L or (not L and L)
7. Not L or false
8. Not L
9. Not (not L) [substituting 8 into 1]
10. L

Thus the strengthened liar is cyclic and so is not truth-apt. But isn't this just what the strengthened liar says? Since it has been demonstrated that it is cyclic, its apparent truth-aptness is an illusion. Just as a straight stick can appear bent when partially submerged in water, so to a cyclic statement can appear truth-apt when it is actually not.

In conclusion, the reason why the liar sentence (and any strengthened liar) is not truth-apt is because the self-reference is cyclic. Conventional (non-cycling) self-reference is fine. [5]

For more paradoxical fun, see Part 2.

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[1] From "Heterologicality", Gilbert Ryle, 1951:

"The same inattention to grammar is the source of such paradoxes as 'the Liar ', 'the Class of Classes ...' and 'Impredicability'. When we ordinarily say 'That statement is false ', what we say promises a namely-rider, e.g. '... namely that to-day is Tuesday'. When we say 'The current statement is false' we are pretending either that no namely-rider is to be asked for or that the namely-rider is '... namely that the present statement is false'. If no namely-rider is to be asked for, then 'The current statement' does not refer to any statement. It is like saying 'He is asthmatic' while disallowing the question 'Who?' If, alternatively, it is pretended that there is indeed the namely-rider, '... namely that the current statement is false', the promise is met by an echo of that promise. If unpacked, our pretended assertion would run 'The current statement {namely, that the current statement [namely that the current statement (namely that the current statement ...'. The brackets are never closed; no verb is ever reached; no statement of which we can even ask whether it is true or false is ever adduced.

Many of the Paradoxes have to do with such things as statements about statements and epithets of epithets. So quotation-marks have to be employed. But the mishandling which generates the apparent antinomies consists not in mishandling quotation-marks but in treating referring expressions as fillings of their own namely-riders."

[2] For a sentence to be truth-apt means that it is either true or false. Sentences like "Hello!" or "Twas brillig, and the slithy toves" are not truth-apt. A cyclic sentence like the liar sentence is also not truth-apt. (Note: an alternative view, called dialetheism, regards the liar sentence as both true and false.)

[3] Saul Kripke, 1975, “Outline of a theory of truth”.

[4] Disquotation is the reversal of the process of quotation. It transforms a quoted statement into an actual statement. That is, it uses the mentioned statement. For example "'Snow is white' is true" just means "Snow is white".

[5] Two more interesting sentences are worked out below:

Let L be the sentence, "This sentence is truth-apt".

1. L
2. "L" is truth-apt
3. "L" is true or "L" is false
4. "L" is true or not "L" is true
5. L or not L
6. True [Law of excluded middle]

Let L be the sentence, "This sentence is not truth-apt".

1. L
2. "L" is not truth-apt
3. "L" is not true and "L" is not false
4. Not "L" is true and "L" is true
5. Not L and L
6. False [Law of non-contradiction]

Hylomorphism

Aristotle's hylomorphism

In a previous post I discussed the problem of universals. Briefly, Plato held that universals (like redness, roundness and numbers) exist in a higher realm of the Forms.

On the other hand his pupil, Aristotle, held that universals are immanent in particular things in the everyday, natural world. In fact, he considered every physical thing (termed a particular or a substance) to be an inseparable compound of matter and form. This view is called hylomorphism, where the Greek word hule means matter and morphe means form.

Consider a coffee cup. It has a round rim. For Aristotle, the roundness of the rim (a universal) is part of the form of the cup. So too are the cup's other identifiable characteristics such as its color and weight. But the cup also has a material aspect - it's made of ceramic. Without that material there would be no cup. Similarly, without the form, there would also be no cup. Thus the cup necessarily consists of both matter and form. It's not possible to extract the roundness of the rim or the weight of the cup - the form is inseparable from the cup.

Nonetheless we can think about the roundness of the rim or the weight of the cup. We can even think about a circle as a geometric shape separate from any particular thing at all, consider its geometric properties, and transform it to, say, an oval. This is a process of abstraction. The form is in the cup, but we can think about it in a purely formal manner separate from the cup. That separation is epistemic, not ontological - there are no formal circles floating around the universe apart from the particular cups, checker pieces and rings that exhibit that form.

For Aristotle, hylomorphism is applicable to all things, from atoms to plants, animals and humans. In the case of human beings, Aristotle is the particular (or primary substance), his body is the matter and his soul is the form. The species and genera are secondary substances. So human being and animal are secondary substances of Aristotle.

It's interesting to compare Aristotle's view to the modern view where the hard sciences are considered to be concerned primarily with matter, hence the prominence of materialism. Aristotle's broader naturalism, on the other hand, fully includes notions of purpose, meaning, morality, intentionality and everyday experience. For Aristotle, all things are a holistic integration of form and matter. And it is in our experience of things that we discover their true form (nature).