A statement asserts that a statement is a statement that is false.
The statement violates an axiom of logic, so it doesn’t really matter whether the ostensible statement the so-called liar, really is a statement or has a truth value.
When I endeavored some years ago to examine the so-called “liar paradox” from what I take to be a pragmatic, semiotic, sign relational standpoint, I arrived at a way of understanding it that dispelled, for me, every air of paradox about it. I wrote out an outline of that analysis under the same title I’m using here and shared it in several discussion groups. The couplet above is a bare bones rendering of that analysis.
The more rambling version can be found at the following locations.
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Moulton is the name of my avatar in a text-only virtual community called MicroMuse.
> Look at Moulton
You see a slow moving red-headed scientist wearing a wrinkled lab coat over a Science Museum Polo Shirt. He is wearing a button which reads “Reach Out and Teach Someone”.
Moulton is carrying:
A Button reading: Incorrigible Punster. Please do not incorrige.
Button A: The sentence on Button B is True.
Button B: The sentence on Button A is False.
“The statement S_0 violates an axiom of logic”
An interesting solution, and one that I think closely approximates how real humans think: we take one look at a contradiction, go ‘that’s wrong’, file it as False, and then because it’s False we know not to evaluate it further. It’s only logicians who go on to try to derive valid information from a recursively contradictory statement.
But while S0 (if we can find one) can be given False, I think that the logical judgement ‘it doesn’t really matter’ for S1 is perhaps the heart of the issue: it’s a judgement which is neither True nor False. In computing we’d simply flag ‘x = not x’ as the non-terminating series ‘not not not not…..’ and assign it a value like _|_ for Nonterminating or Bottom Type. So there’s a case to be made that logic could use a value like _|_, where ‘not _|_’ remains ‘_|_’.
Right now I’m interested in four-valued logic which does perhaps have such a value (eg Both and Unknown as well as True and False), and seems to arise naturally from allowing ‘Not X’ as well as ‘X’ statements. (Eg Both is simply what you get when you have both X and Not X statements in the same data file, which can happen when you merge data from inconsistent sources, as happens all the time on the Web). So the question I have is what the correct way of dealing with Both if it’s detected. My feeling is it should be treated like a kind of error-state (like Undefined or Not-A-Number in maths).
Belnap Logic seems initially to go in this direction, ( https://en.wikipedia.org/wiki/Many-valued_logic#Belnap_logic_.28B4.29 ) but looking at his truth-tables I don’t think he was thinking quite of ‘Both’ as being an error-state as I do.
Also, even in a four-valued logic, I’m not sure the Liar would actually evaluate as Both – I think Unknown from three-valued logic would be sufficient. Because it can neither be evaluated as True or as False, so Unknown (or ‘Empty’ perhaps) might be the only sensible answer.
And again, evaluating S1 as ‘Unknown’ seems basically the same as ‘removing S1 from the database because S0 was false’. It is neither a true nor a false statement; it simply does not exist for the purposes of any proof.
“Some expressions can evaluate as Unknown” also seems to me to be a restatement of “Not all signs denote”.
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