How to (dis)solve the 'Liar's Paradox' redux:
(1) Proposition (1) is false.
(2) Proposition (1) merely refers- it has no first-order predicate.
(3) The second-order predicates of 'truth' and 'falsity' supervene on first-order predication.
(4) (1), having no first-order predicate, cannot have the second-order predicates of 'truth' or 'falsity' (i.e. (1) is not a proposition)
(5) Since (1) cannot be true or false, it cannot generate a paradox.
(6) Therefore, there is no "Liar's Paradox".
.:addendum:.
As the Philosopher says:
"None of the above is said just by itself in any affirmation, but by the combination of these with one another an affirmation is produced. For every affirmation, it seems, is true or false; but of things said without any combination none is either true or false (e.g. 'man', 'white', 'runs', 'wins', ['proposition (1)']."
Categories IV, 5-10.
(2) Proposition (1) merely refers- it has no first-order predicate.
(3) The second-order predicates of 'truth' and 'falsity' supervene on first-order predication.
(4) (1), having no first-order predicate, cannot have the second-order predicates of 'truth' or 'falsity' (i.e. (1) is not a proposition)
(5) Since (1) cannot be true or false, it cannot generate a paradox.
(6) Therefore, there is no "Liar's Paradox".
.:addendum:.
As the Philosopher says:
"None of the above is said just by itself in any affirmation, but by the combination of these with one another an affirmation is produced. For every affirmation, it seems, is true or false; but of things said without any combination none is either true or false (e.g. 'man', 'white', 'runs', 'wins', ['proposition (1)']."
Categories IV, 5-10.
Labels: liar's paradox
posted by Derek at 3:26 PM
9 Comments:
Are you sure (6) is true or false?
Consciousness colloquium at CSUF April 29,30 w/ Perry, Shoemaker, Dretske and Chalmers.
Do you know a good book/article where I can read about the definition of first-order/second-order predicates and the distinction between them? This language is familiar, but I'd like to dwell on a clear presentation of it.
i've been saying this since i was like 15. maybe.
Jon: No I don't. I just made it up.
It seems intuitive, though, doesn't it?
Example of a 'first-order' predication:
'The house is blue'
Because there is a first-order predicate (e.g. 'X is Y'), there can now be a 'second-order' predicate:
'It is true or false that 'The house is blue'.'
But if there was no first-order predication, like the following sentence:
'The house."
There could be no 'second-order' predicate of truth or falsity. I.e., it's non-sensical to say:
'It is true 'the house''.
Hence truth or falsity are second-order predicates that supervene on first-order predicates. No first-order predication, no second-order predication.
Proposition (1) violates this rule.
The only predicate in (1) is 'is false'.
But you can't do that. You can't have a second-order predicate of truth or falsity without there being a first-order predicate that isn't the second-order-predicates of truth and falsity.
The straight forward reading of (1) goes something like this:
(1) [Proposition (1)] is false.
SInce there's not "is" in the [...] there's only a subject in [...] and no predicate.
No predicate in [...] then no truth or falsity.
***SInce there's *no* "is" in the [...] there's only a subject in [...] and no predicate.
I agree that it seems like "truth", as a predicate, seems to stand apart from most other predicates, like "blue". It seems that to say "X is true" is simply to affirm X. In logical notation there is no "truth predicate" as such (that I know of); you don't write "Tp" where "T" = "is true", you just write "p". "Truth", I think (maybe; I'm just going off hunches) is a predicate endemic to our metalanguage, wherefrom on high we cast judgment on statements made in the object language. The Liar wants to play both roles at once. Maybe. I guess Tarski has a truth predicate as such, but the name of a sentence bears this predicate iff the sentence it names is assertable.
I wonder about your coined first/second order distinction, though. I like it, but I have some questions.
If I read you right, a "second-order predicate" is any predicate whose satisfaction conditions supervene on the satisfaction conditions of a "first-order predicate", and a "first-order predicate" is any predicate whose satisfaction conditions don't supervene on the satisfaction conditions of some other predicate, but rather obtain their value from some kind of direct word-world relation ("supervene solely on facts about the world" or something?) Or am I wrong?
You want to say that any sentence that lacks first-order predication has no truth-conditions. It is ill-formed. It's neither true nor false. But then what of:
(1) This sentence is neither true nor false.
It seems that (1) is truth-functionally equivalent to the conjunction of:
(1') (1) is not true.
&
(1'') (1) is not false.
I think you'd want to say that (1') and (1'') both fail to generate truth-conditions. And so their conjunction, (1), houses no first-order predicates itself (I think). But then, according to your beef with the traditional Liar, (1) fails to generate truth-conditions; it fails to be true or false. But this is just to say that (1) is neither true nor false, which is what (1) (seemingly) says, hence it is true, hence OH SHIT A LIAR'S PARADOX. It seems like you could go balls to the wall and say, "well, the thing is neither true or false because it is meaningless; so, though it looks like by claiming it is neither true nor false we seem to be agreeing with it, there really is nothing to agree with; it has no meaning, regardless of how much it may seem to have a meaning, and that that supposed meaning seems to be just what we say of it. Don't be deceived!" But this is more bravery than I am willing to exhibit; it brooks against my intuitions.
Do I misstep or misunderstand you somewhere above?
“I think you'd want to say that (1') and (1'') both fail to generate truth-conditions. And so their conjunction, (1), houses no first-order predicates itself (I think). But then, according to your beef with the traditional Liar, (1) fails to generate truth-conditions; it fails to be true or false. But this is just to say that (1) is neither true nor false, which is what (1) (seemingly) says, hence it is true, hence OH SHIT A LIAR'S PARADOX.”
I’d go further and say that the predicate ‘is neither true nor false’ is itself a second-order predicate. It’s a second order-predicate that can only be applied to sentences that don’t predicate. So the only way to make (1) intelligible is to construct it like this:
(1’’’) This sentence (viz., (1’’’)). [is neither true nor false]
We can combine the purely indexical sentence (1’’’) with the second-order predicate we made about it and form:
(1’’’’) Sentence (1’’’) is neither true nor false.
Since (1’’’’) predicates, we can give it a truth-value via another second-order intention.
(1’’’’’) Sentence (1’’’’) is true.
But neither (1’’’’) nor(1’’’’’) can generate a paradox because neither are self-referential.
::Happy face emoticon::
This comment has been removed by the author.
(1*) This sentence is neither true nor false.
(2) The predicate ‘is neither true nor false’ is a second-order predicate that can only be applied to objects (e.g. sentences, objects) that don’t predicate.
(3) Since sentence (1*) does predicate, its self-ascribed predicate of ‘neither true nor false’ is misapplied.
(4) Therefore, Sentence (1*) cannot generate a paradox.
Possible complaint: “Despite your assertion that the predicate ‘neither true or false’ is misapplied, it seems (1*) makes perfectly good sense.”
My response: That’s because you’re reading the predicate of ‘neither true nor false’ not as a first-order predicate within (1*), but as a second-order predicate about the subject term in (1*). I.e., you’re reading it like this:
(1**) This sentence. [is neither true nor false]
Where the predicate ‘neither true nor false’is made about (1**) and not contained within (1**).
Possible further complain: “You’re putting words in my mouth, I don’t read (1*) like (1**), so your analysis changes the subject.”
My further response: Well, your reading entails a paradox, and therefore so much for your reading.
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