The liar paradox (Part 2)

In my previous post I discussed the liar paradox and argued that it should not be considered either true or false because it is cyclic. That is, it never ends up successfully asserting anything about the world.

Interestingly, the Stanford Encyclopedia of Philosophy (SEP) entry for the Liar Paradox outlines an argument that the liar sentence implies a contradiction. The argument uses two inference rules, as follows:

• Capture: A implies "A" is true
• Release: "A" is true implies A

Taking these two rules together, the terms A and "A" is true are intersubstitutable [1]. The SEP argument, reproduced in plain English [2], is:

Let L be the sentence, "This sentence is not true".

    1. "L" is true or "L" is not true [Law of Excluded Middle]
    2. Case One:
         a. "L" is true
         b. L [2a: release]
         c. "L" is not true [2b: definition of L]
         d. "L" is true and "L" is not true [2a, 2c: conjunction introduction]
    3. Case Two:
         a. "L" is not true
         b. L [3a: definition of L]
         c. "L" is true [3b: capture]
         d. "L" is true and "L" is not true [3a, 3c: conjunction introduction]
    4. "L" is true and "L" is not true [1-3: disjunction elimination]

Line 1 assumes that the liar sentence conforms to the Law of Excluded Middle. That is, it assumes that the liar sentence is either true or not true and not some other value. Lines 2 and 3 analyze each disjunct as separate cases. In each case a contradiction is reached which is then inferred in 4.

The conclusion that the liar sentence implies a contradiction depends on the first premise being truth-apt. However, as argued previously, the liar sentence is not truth-apt and so therefore the first premise can't be either. Consequently logical inference rules and truth evaluation aren't applicable to it. A contradiction is only reached via a false assumption of truth-aptness.

The rule of thumb would be that a sentence is only truth-apt if it is grounded in a state of the world either directly or else indirectly via other sentences. Note that this condition fails for both the simple liar (where not true means false) and the strengthened liar (where not true means false or not truth-apt).

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[1] This is Alfred Tarski's T-Schema: 'S' is true if and only if S (e.g., 'snow is white' is true if and only if snow is white).

[2] For example, I've replaced the corner symbols that indicate quasi-quotation with ordinary quotes.

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]

Do ducks lay eggs?

Aristotle - De Anima (On the Soul)

Ducks lay eggs. True or false? Most people regard the statement as unquestionably true.

Sentences like this are called generics and the interesting feature of generics is that they tolerate exceptions.[1] For example, male ducks don't lay eggs and neither do female ducks that lack the capability due to defects. Yet most of us accept that the statement conveys real information about ducks as a species, despite such exceptions in individual ducks.

Generic sentences are problematic on a nominalist view since universal terms (such as "duck") are considered to be nothing more than a collective name for individuals.[2] So the sentence "ducks lay eggs" would seem to imply a hidden "all" quantifier and thus really means "all ducks lay eggs". So on a nominalist reading "ducks lay eggs" is considered either strictly false or perhaps not a legitimate knowledge claim at all. Whereas realists about universals hold that generic statements differ in meaning from statements that include quantifiers. That is, generic statements refer to universal categories or kinds, not to the individual members of those categories or kinds.

Greg N. Carlson explains the key feature of generics as follows:

Finally, to re-emphasize a point made by Goodman (1955) and more recently by Dahl (1975) (among a host of others), the truth of generics depends on a notion of non-accidental generalization for their truth. The world contains in its extension all manner of possible patterns and convergences, many of which we judge to be purely accidental, but others of which we take to be principled. Only the principled patterns are taken to support true generics.

Truth-Conditions of Generic Sentences: Two Contrasting Views

In Aristotelian terms, this generic meaning is captured by the distinction between essence and accident. One familiar example is the Aristotelian definition of humans as the rational animal. This definition successfully distinguishes between humans as a species and other animal species that lack that rational capability.[3] This is so even though individual exceptions exist. For example, a two-week old embryo, a comatose or sleeping human, or a human acting irrationally.

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[1] For a brief overview, see Generic Sentences and Predication.

[2] See my previous post on the problem of universals.

[3] For an interesting analysis of this definition, see Essentially Rational Animals.

Is color real? (Part 2)

Are tiles A and B the same color?

In my previous post I raised the question of whether color is a property of things in the world (such as paint, strawberries and fire-engines) or whether it is a property of the way that humans see things.

In everyday usage, we generally associate color with things in the world. We suppose that the fire-engine is red not merely when we look at it but also when we are not.

That has raised some scientific and philosophical issues. One issue is that sometimes people name colors differently because of vague boundaries. That seems to be the case with my disagreement with Jason. The color of the stool was on the boundary of red and orange. Since we learn about objects and their colors by looking at them and categorizing them, it is possible that people categorize colors slightly differently.

Another issue is that physical differences in our eyes can cause perceptual differences. This is the case with color blind people who can't differentiate between red and green.

A further issue is that background lighting influences the color we perceive. This is primarily the case with "the dress" phenomenon where some people report the dress as being blue-and-black and others that it is white-and-gold. It turns out that people see the actual dress as blue-and-black so the phenomenon only occurs when viewing an image of the dress. The colors people see generally depends on how much background is shown in the image and also the external lighting when viewing the image on a computer screen.

Then we get the issues of light reflection. If a paint can is closed (or it is open at night with the lights off), then paint would not appear red since color perception requires light to reflect off the surface of the paint onto our eyes. And, of course, a blind person never sees the paint as red.

Finally, there are the wavelengths of light. Visible light covers a small range of wavelengths in the electromagnetic spectrum and constitutes most of the light that the sun radiates. So color can be associated directly with the wavelength as well.

So there are a lot of things to consider. From a pragmatic point-of-view, we don't want to have to deal with all these issues (and others we haven't thought of) every time we refer to color in our everyday lives. We just want to say that the fire-engine is red regardless of the background lighting or the perceptual capabilities of different people.

The way we have achieved this with our everyday concept of color is by abstracting away all the complicating details. We simply associate color with the object [1]. If there are differences in perception due to lighting or a person's perceptual capabilities, then we use the word "appears" to make that distinction. Thus, the fire engine is red, but it doesn't appear red at night or to a blind person. The paint in the closed can is also red even though there is no light reflecting off the paint inside the can.

The benefit to abstracting in this way is that not only is the concept simple to learn and intuitive to use, it is also independent of any particular physical explanation for why color appears as it does. If future scientific discoveries are made that replace our best current theories of light, the reflective properties of surfaces, our eyes, or how our brain implements perception, then our everyday language around color would not need to change. We would just know more about color than we did before. [2]

So this is why color is a property of things and not a property of how we perceive things.

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[1] We learn concepts ostensively by paradigmatic examples. In this case, we learn colors by pointing at objects like fire-engines under normal lighting and comparing and contrasting them with other colored objects. Color is real (it is about objects in the world, not our perception of objects) and abstract (it need not carve nature at its joints).

[2] A further consideration is that other creatures perceive colors differently to humans as a consequence of their different brain and eye structures. So the scope and context of color concepts needs to be recognized.

Yes, they are the same color - they just appear different

Is color real?

I had a metaphysical discussion with my two-year old son this morning. I put a stool back in the room where it belonged (the chair kind of stool, not the other kind).

My son says, "Want orange stool."

So I retrieve the stool and say, "Jason, it's not an orange stool, it's a red stool."

He says, "Orange."

I go and get an obviously red drum stick and an obviously orange toy building. Showing him the drum stick, I say, "Is this red?"

"Yes."

Pointing to the toy building, I say, "Is this orange?".

"Yes."

Pointing to the stool, I say, "So Jason, see how the color of the stool is like the color of the red drum stick? So the stool is red."

"No, orange."

So my son and I are clearly having a serious metaphysical disagreement. Surely the stool has to be one color or the other. Can the stool be orange for him merely because he thinks it is orange?

Before you take the objective or subjective side of the dispute, here is a further puzzle about color. When we buy a can of paint, it has the name of a color on the label, such as "Red". However what is the color of the paint inside the can when the lid is on? (Note: there is no light inside the closed can). Also, when the can is opened under normal lighting, what is the color of the paint for a blind person?

For my solution, see Part 2.

Free to choose

(This post follows on from Beyond Belief.)

In our everyday lives we are familiar with making choices. We choose what to wear and what to have for breakfast. We are also aware of the constraints on our choices. You can't choose to wear a red shirt if you don't have one.

All familiar and obvious stuff. Until the philosophers weigh in...

Suppose, they say, that all our actions can be fully described in cause-and-effect terms by the laws of physics. Wouldn't that mean that the outcome of our choices is inevitable? And therefore that the freedom of our choices is illusory?

The answer is that, no, in a physically determined world, our choices would be neither inevitable nor illusory.

The hidden assumption in the free-will dilemma is that the determination of an outcome on one mode of description (physics) excludes a determination of that outcome on another mode of description (the intentional, which is expressed in our everyday language of purpose and choice). So for the Determinists free will is impossible. For the Free-Willers physical determinism is impossible. However, as I will demonstrate below, the above assumption is false: the physical description does not compete with the intentional description but instead is logically complementary to it. [1]

As an example of complementary modes of description, consider a computer that calculates the sum of two numbers. At one level of description, the answer that the computer gives is fully determined by the rules of arithmetic. At another level of description, the answer is fully determined by physical cause and effect in the computer. Not surprisingly, since the computer has been programmed to follow the rules of arithmetic, the physical and arithmetic descriptions are complementary, not mutually exclusive.

In a similar vein, our choices are free on the intentional mode of description because there cannot exist, even in principle, a physical description of the future state of our brain that we would be logically compelled to accept as correct. While we would be correct to accept a physical description that includes the change in brain state entailed by our acceptance of it, we would also be correct to reject it. Why? Because our rejection of it would result in a physical brain state that differed from the physical description we were originally considering. Therefore we would also be correct in our rejection of it. So both possibilities are always logically open to us. It's not merely that we wouldn't know whether the physical description is correct or not (an epistemological issue). It's that no physical description of our brain state can possibly exist that would be inevitable for us to choose (an ontological issue). So the shared hidden assumption of the Determinists and Free-Willers is false as a matter of logic. [2]

When we employ our everyday language concepts of purpose and choice, we are operating at a different level of abstraction to that of scientists when they describe the physical laws of the universe. The free-will dilemma is dissolved when we realize that it is our familiar everyday contexts that define our choices and constraints, not the mathematical laws of physics (where intentional concepts are inapplicable). The dilemma only arises when the terms that apply to one mode of description are taken out of their context and applied to a different mode of description.

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[1] This view is known as Compatibilism. According to a 2009 survey, the ratio of opinions among philosophers is: compatibilism 59.1%; libertarian free will 13.7%; no free will 12.2%; other 14.9%.
[2] This insight is originally from physicist Donald M MacKay ("What Determines My Choice" in his book "The Open Mind and other essays").

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.

Beyond belief

(A companion post to a thorny issue.)

What is a belief? Per the dictionary, it is the acceptance that something exists or is true. So, for example, you might believe that it is raining. One reason that beliefs are important is because they guide our everyday actions. So if you want to go outside, you might first grab an umbrella or raincoat. Or you might decide to stay inside until the rain stopped. Everyone from their early years, by observing and learning to identify that pattern of behavior in others and in themselves, becomes competent at using the word "belief" in their everyday communication.

Beyond this everyday usage, there are many philosophical, scientific and folk theories about what a belief (and the mind more generally) consists of, ranging from an immaterial substance through to identification with physical brain states. However, as with the concept of pain discussed previously, the concept of belief is a publicly communicable abstraction. Like Wittgenstein's beetle, whatever it is that "belief" ultimately refers to drops out of consideration as irrelevant in normal conversation.

Suppose it were discovered that our beliefs were the result of a tiny gnome in our heads that records what we see and reminds us about it when it matters. Such a discovery, while surprising (unless you read Terry Pratchett), wouldn't change the everyday meaning and usage of the word "belief" at all. We would still want to grab our umbrella when it's raining.

So what does this mean for theories of the mind? For one thing, it means that theorizers have almost limitless scope for creative exploration while still being able to retain compatibility with our everyday language concepts. For another, it means that theories of mind (in principle at least) will always be publicly communicable since they will expand on rather than replace our existing conceptual framework.

One notable example of this in philosophy is with the perennial question of free will. In ordinary usage, the concept of choice is the ability to select between alternatives. The concept exists because we commonly observe people conforming to that pattern of behavior and we have needed a way to refer to it. But how that abstract concept is actualized, while of philosophical and scientific interest, has no bearing on the existence of that behavioral pattern and its central importance in our everyday activities.

A thorny issue

How do we know what pain means? Well, we experience an unpleasant feeling when we are pricked by a thorn, so it presumably refers to those kinds of feelings. While that seems right, "pain" is not just a word that we use to refer to ourselves, we also apply it to others. But how do we know that others feel pain? We don't feel what others feel, so we must infer it from their behavior (e.g., when they groan or say "ouch" or report that they are in pain).

But why would we associate a feeling that we have with the behavior that others have? The reason is that we also observe our own behavior when we experience that feeling. We notice that our behavior, when pricked by a thorn, is the same as other people's behavior when they are pricked by a thorn. Thus, from a young age, we learn that "pain" is the word that we use to talk about those situations.

Of course we nonetheless also experience an unpleasant sensation associated with our own pain behavior. We understand that that sensation is causing our behavior, and the specific sensation we have when pricked by a thorn is what we mean by pain in that circumstance. However it is the publicly observable pain behavior that leads to the word "pain" appearing and being used in our everyday language.

The curious thing about this is that it doesn't actually matter what the feeling is. If others have a different feeling when they exhibit pain behavior or even have no feeling at all (assuming they still exhibit the same behavior, albeit like a robot or a philosophical zombie), the meaning and use of the word "pain" in our language would be exactly the same. Ludwig Wittgenstein discusses this with his beetle-in-a-box analogy and concludes with, "That is to say: if we construe the grammar of the expression of sensation on the model of 'object and designation' the object drops out of consideration as irrelevant." [1]

Later Wittgenstein says (in response to his interlocutor), "'Are you not really a behaviorist in disguise? Aren't you at bottom really saying that everything except human behavior is a fiction?' — If I do speak of a fiction, then it is of a grammatical fiction." [2]

What Wittgenstein is saying is that even though the sensation of pain (and other sensations) is real, our language does not, and cannot, depend on the subjective and private nature of those sensations. Rather our language depends solely on the publicly observable behavior that those sensations produce.

The concept of pain, then, is a publicly communicable abstraction whereas our painful sensations are particular instances of pain. The only way to meaningfully talk with other people about our sensations, to the extent that we can, is via an agreed language grounded in publicly observable phenomena.

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[1] Philosophical Investigations, §293.
[2] Philosophical Investigations, §307.