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]

A meeting of minds

Having tackled "pain" and "belief" it's time to move on to the big fish, the mind itself. As with the concepts of pain and belief, the concept of mind is a publicly communicable abstraction. It encompasses all the human capabilities that we observe in ourselves and others such as thinking, feeling, perceiving and desiring. Ludwig Wittgenstein captures the concept aptly when he says, "The human body is the best picture of the human soul". [1]

But, you may ask, how about when we introspect? Aren't we also privately observing our pains and beliefs and therefore our own minds? In a word, no. We feel pain and we have beliefs. As such, we can observe them in our everyday behavior. Once we observe and identify our pains and beliefs, we can subsequently reflect on them and examine them. It is from these kinds of everyday observations that we know that we, and others, have minds.

(A possible objection: What if we hide our pains or beliefs? Even if we do, they are still manifested in behavior although of a different kind (e.g., stoic, reserved or deceptive behavior). While hiding your headache pain or political beliefs makes them more difficult to detect, they are still observable in principle. And, importantly, you can easily observe your own behavior here, since it takes conscious effort to conceal what you think and feel.)

So the important thing to note is that introspection is not a kind of observation. Instead, introspection is an examination of one's thoughts or feelings that are observed in one's own behavior. Unfortunately, the belief that introspection is a kind of observation leads to the idea that our mind is a private theater [2] where we observe on the stage the characters of our inner life, such as pains and beliefs, desires and purposes. On this unfounded view, our mind is a ghost in a machine. [3]

It can be a difficult ghost to exorcise. In my next post, I'll look at a scientific experiment that highlights some pertinent issues around consciousness and the mind.

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[1] Philosophical Investigations, Part II, iv, Ludwig Wittgenstein.
[2] "The Concept of Mind", p222, Gilbert Ryle. Similarly, Daniel Dennett calls it the "Cartesian Theater" in "Consciousness Explained", p17.
[3] Gilbert Ryle calls this view the "dogma of the Ghost in the Machine" in "The Concept of Mind", p15.

Behavioral problems

The philosopher Gilbert Ryle is probably best known for his book The Concept of Mind which argues against Cartesian Dualism. This includes the proposition that much of what we consider private mental life is inextricably linked to perfectly ordinary and publicly observable behavior. For this reason he is usually considered to be a Behaviorist. However a more careful reading of his work shows that he rejects both the Dualist and Behaviorist extremes and instead adopts a common-sense position that rejects the errors of both sides.

To illustrate this, in Thinking and Saying, Ryle gives the example of two boys - one writing a letter and the other boy mimicking him. Both the Behaviorist and the Dualist agree that the boys actions would differ only if the mimic performs additional actions that his victim does not perform. Since no extra actions were witnessed, says the Behaviorist, their actions were therefore identical. Since no extra actions were witnessed, says the Dualist, the mimic was performing extra internal, unobservable actions.

Ryle's view, in contrast, is not that there were unobservable actions or that their actions were identical, but instead that the seemingly identical actions that we did observe were of different sorts. The broader context shows that their intentions and skills were different and so therefore their actions were different. This conclusion avoids both the "Nothing But ..." of Behavioral Reductionism and the "Something Else As Well ..." of Cartesian Dualism.

 

Thinking about thinking

What is Rodin's The Thinker doing?

Ponder it for a while. Perhaps in trying to figure it out, you are doing what The Thinker is doing.

Presumably he is having thoughts. Perhaps talking to himself. Maybe, but is that all that he is doing? A greenback dollar is a piece of paper but also more than that. What exactly is he thinking about? What is he trying to achieve?

Suppose a teacher has set you a math problem. You are trying to figure out the answer - an answer that is already known by the teacher. But is The Thinker a student? If not, then who would know the answer to his problem?

Maybe the problem has no known answer, or is not well-defined or has never been posed before. If so, then The Thinker is more like an explorer. Perhaps he doesn't know what the hoped-for destination looks like. But maybe he's had some experience in this kind of territory in the past. He makes many wrong turns but also sees some promising avenues.

Either way, he has not yet reached his destination. The discovery of any new place first requires a journey.

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A reflection on Gilbert Ryle's The Thinking of Thoughts: What Is 'Le Penseur' Doing? and Thinking and Self-Teaching.