This statement is false

Last weekend my brother and I were discussing the statement “This statement is false.” Today a colleague mentioned a similar statement, “The following statement is true. The previous statement is false.” Another colleague likened this kind of pure self-contradiction to the Cretan paradox, also known as the Epimenidean paradox: the statement “All Cretans are liars” said by a Cretan, which would seem to be a false if it’s true and true if it’s false.

But the difference between the Cretan paradox and pure self-contradiction is that the Cretan paradox has a real-world referent. It makes a statement about something external to the assertion. Pure self-contradiction has no real-world referent. It makes an assertion about nothing other than itself and thus has no truth value ascertainable.

As it happens, the source of the Cretan paradox is something Epimenides wrote in support of the immortality of Zeus:

They fashioned a tomb for thee, O holy and high one
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.

Epimenides was himself a Cretan. Thus we know through simple pragmatics that he must have been excluding himself without saying so. To treat it as a paradox is to be disingenuous. It’s fun sport, but in the end it just shows one of the things you can’t do in logical reasoning.

Statements such as the Cretan paradox are an illusion caused by conflation of one level of analysis with a higher level of analysis: an evaluation of the members of a set cannot itself be a member of the set evaluated; evaluation is a comparison of something against one or more criteria from an external perspective – what is being analysed is subsumed within its perspective. Once we acknowledge that the statement “All Cretans are liars” cannot be part of the set of statements evaluated (making it thus a simple problem in pragmatics rather than a trick of logic), we identify an unstated assumption that makes it function, without which we get a sort of Escher staircase illusion, something that can’t exist in the real world.

But with mutually evaluative statements such as the pure self-contradictions, each must be on an evaluative level above the other – each must subsume the other within its perspective. And at the same time each has no further reference; it has no claim to truth or falsehood as the set of all other statements by Cretans does (and as that set’s members individually do).

Analyzing an utterance or set of utterances is like weighing an object. In order to weigh an object, you have to lift it (or anyway support it) and you have to be resting on something that is not part of what you are weighing. In the Cretan paradox, we see that the statement that pretends to be part of the set of Cretan statements is actually weighing them and so cannot be part of them; it is evaluating them against their real-world references – that’s what it’s resting on. In the mutual contradiction case we’re looking at, each is weighing the other, and neither rests on anything else, because neither is being evaluated against anything external to itself. It’s like two dudes trying to lift each other simultaneously. In empty space.

Meaning in human communication, ultimately, is not a question first of all of logic; it is a question first of all of pragmatics. All communication is behaviour; when you utter something, you are doing something with the aim of producing a certain effect. The person hearing you will be conjecturing what effect you are trying to produce and responding accordingly. Logic helps serve this function, but pragmatics is the true basis. And the pragmatic value of things such as paradoxes is sport – mental play, fun. And a demonstration of the invalidity of certain kinds of reasoning.

6 responses to “This statement is false

  1. Further expansion on the topic from further dialogue on it:

    The real-world referent for Epimenides was the set of criteria and realities against which the set of Cretans’ statements could be evaluated. E.g., Cretan dude: “What’s for dinner?” Cretan chick: “Lasagna.” Reality: It’s boiled hamster. For each Cretan statement there is a reality against which it can be evaluated. But an evaluative statement can’t evaluate itself. And a statement about itself has nothing other than itself against which it can be evaluated.

    Evaluation is like using a device to test power outlets that has to be plugged into an outlet other than the one it’s testing. Pure self-reference is an extension cord plugged into itself. Epimenides was saying “The plugs in this house don’t work” but was plugged into one that does and that thus was not being tested.

    Epimenides made a statement which can be interpreted sensibly according to pragmatics, which are the basis of communicated meaning: Epimenides was not including himself in the statement. Pure self-contradiction has nothing like that.

    They’re similar, pure self-contradiction and the Epimenidean paradox, but even if you take the Epimenidean paradox at face value, it still has something external to itself involved (all other statements by Cretans, plus their real-world referents), which pure self-contradiction does not.

  2. I think the thing that makes Epimenides’s paradox more fun than simple self-contradiction is that it appears to be part of a valid assertion because it makes a statement with reference to a real-world set that has many members. That gives it a bit of a red herring, a misdirection. In that regard it’s a bit like those emailed number puzzles that make you go through a whole bunch of steps to discover something that could actually be simplified to 10x – x = 9x.

  3. Is there an intermediate stage between pure self-contradiction and the Cretan paradox? Here are two photographs of real objects, and I am not sure where to place either of them in your classification.

    1. A T-shirt with the slogan ‘Ceci n’est pas un T-shirt’:
    http://picturepush.com/public/6604289
    The words refer to an object other than the slogan itself, but do not make a Cretan-type generalisation, and the object itself visibly contradicts the assertion.

    2. A notice saying ‘Sorry, this sign is not in use’:
    http://picturepush.com/public/6604291
    The words refer neither to the existence nor to the veracity of the sign — but you are reading the sign, and therefore using it.

    • The T-shirt is actually a reference to René Magritte’s “Ceci n’est pas une pipe” on depiction of a pipe, which is a sort of aestheticization of the object by making strange, as well as a statement of the difference between image and what it represents – and of course a bit of fun. The T-shirt is in fact a T-shirt, so it’s a false statement, or an attempt to commandeer the designation T-shirt, and it’s a reference to Magritte.

      The sign is more amusing. One one level it’s false because if you’re reading it it’s in use, but on another level it’s true in that the sign it’s referring to – not itself but the larger one it’s posted on – is not in use for its standard purpose of displaying directions or advertising or whatever. It’s closer to the Cretan paradox in that regard, but it’s not really a paradox because if it’s false it’s false. (And if you’re not reading it, then it’s true.) Still, it’s amusing – I’m not sure if there’s a name for that particular twist. Give it one if you want!

  4. Here’s a nice self-affirming one to pass to your friends:

    1. Start with your age (e.g., 43).
    2. Add the reciprocal (e.g., 34).
    3. Divide by 11 (e.g., 34+43=77, ÷11=7).
    4. Subtract that number from your age (e.g., 43-7=36).
    5. Add together the digits of the resulting number (e.g., 3+6).
    6. Add 1 for good luck.
    7. The result is how gorgeous you are, on a scale from 1 to 10. 🙂

  5. Allan Jackson has sent me a link to another article on an example of this kind of paradox: http://maverickphilosopher.typepad.com/maverick_philosopher/2012/01/neologisms-paleologisms-and-grellings-paradox.html

    Grelling’s paradox is another example of the illusion of paradox created by closed-loop analysis. (I would add that it is a false assumption that the autologous/heterologous classification is exhaustive. There are many instances where the quality named by a word is subjective, dependent on context, unverifiable, or irrelevant to the word itself.)

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