Possibly this is GD, although the fact that I’m looking for a yes/no is making me put this in GQ. Anyway, I am attempting to resolve the liar paradox. If this has already been done, than apologies for wasting time and hampster points.
The liar paradox goes that the statement “This statement is false” cannot be true, because it states that it is false, and cannot be false, because if it were false, than it would be true, and it cannot be true. However, not every statement is either true or false. The statement “Ooga”, for instance, is also neither true nor false. If the statement “This statement is false” is neither true nor false, than it tells us nothing about anything, and can be therefore treated like “Ooga”. (I.e., ignored.)
The distinction between sentences and propositions is certainly not original. You will find a discussion in the early chapters of any textbook of logic. The basic ideas of the discussion are
(i) Not all sentences express propositions. To take more orthodox examples than yours, neither “Hello!” nor “What time is it?” do so.
(ii) Different sentences can express the same proposition. Thus the sentences
“Tolstoy is dead.”
“The author of War and Peace is dead.”
“Tolstoy est mort.”
all make the same claim.
Having made this distinction, we revise our definition and say that every proposition ( but not every sentence) is either true or false.
The problem with this as a resolution of the Liar Paradox is that the sentence “This sentence is false.” clearly seems to be expressing a proposition. The usual approach is to establish a hierarchy of propositions. Firstly, we conceive of a realm of objects. Propositions about these objects we call ‘first-order propositions’. Propositions about first-order propositions are called second-order propositions, and so on. We place restrictions on the way propositions can be formed by ( for example) imagining the formation of propositions as a tempopral process. At each stage, we can only form propositions about objects and propositions created at an earlier stage. The sentence “This sentence is false” clearly breaks this rule, as it would be a proposition about itself. Thus the sentence is not a proposition and so the question of its truth or falsity does not arise.
If any analysis provides insufficient information to answer the question asked, it is necessary to conduct further investigations in order to obtain the answer.
Any investigation has the potential to interfere with its own analysis.
For example I could investigate whether eating broken glass cured the sore throat caused by colds. I could do this by feeding people broken glass. The trouble is that the act of swallowing glass will almost certainly mask any changes in the cold symptoms. These sorts of situations are fairly common in the real world. To overcome such problems it’s necessary to eliminate the ‘interference’. Perhaps we could grind the glass up finely, perhaps we could place the glass directly into the stomach via an incision.
The liars paradox is no different and no more paradoxical. The means of investigation we have adopted interferes with the analysis we are attempting. We need to find other ways of investigating the truth of the statement. The simplest solution would simply be to interrogate the speaker and ask “was the last statement you made correct?”.
There is no paradox here that I can see, it’s just that the investigator is restricted to using invalid or insufficient investigative methods. It only remains paradoxical if all other means of investigation are prohibited. That being the case it’s not really any more paradoxical than asking someone to open a box with the key locked inside. That’s not a paradox, it’s just a really dumb place to put the key.
Jabba, your distinction between first-order and second-order propositions, and the restrictions you place upon them, seems equivalent to the Zermelo-Fraenkel set theory proposed after Russel’s Paradox: It is constrained specifically to prevent sets from encompassing themselves, or to at least prevent paradoxes arising from sets encompassing themselves. Is that the case?
(Russel’s Paradox was proposed by Bertrand Russel, and it runs like this: A set may be normal or self-encompassing. Normal sets do not list themselves as members, whereas self-encompassing sets do. Is the set of all normal sets itself normal? (If yes, it must be self-encompassing, but then it cannot contain itself, so it cannot be. If no, it must be normal, in which case see previous sentence.))
Discussing the truth or falsity of your sentence does not seem to lead us into any major problems, as it appears to be obviously true. I would mention two points, though.
(i) We do not want the truth of our propositions to depend on the precise wording we use ( unless the change of wording results in a different propostion, of course). For example, the sentences
Tolstoy is dead
and
Tolstoy is definitely dead
express the same proposition.
On the other hand, there is an obvious problem in saying that the sentences
This sentence has five words
and
This sentence definitely has five words
express the same proposition, and so have the same truth value.
(ii) The aim of the step-by-step approach outlined above is to rule out propositions which lead to contradictory conclusions. Almost all the propositions we consider in maths, science, history and philosophy ( for example) are first- or second-order. Declaring improper a few sentences which appear to express true propositions is a small price to pay for ensuring that the resulting theory is consistent.
Derleth wondered:
It is a parallel process, yes. For example, Michael D. Potter, in his Sets: An Introduction ( a book he says he was narrowly dissuaded from calling The Joy of Sets) constructs sets in a step-by step process:
“The idea is that on the first day we start with the objects that already exist, i.e. the individuals; on each subsequent day we can collect together into a new collection any collections which were created on earlier days.” ( Potter, p. 16)
This is obviously strongly analogous to the procedure I described in my first post.
True enough, but
What real number x satisfies x[sup]2[/sup] + 1 = 0?
is a sentence but not a proposition.
There is a real number x such that x[sup]2[/sup] + 1 = 0
is a proposition, and is false.
No, and yes. The proposition expressed by that sentence would be ‘correctly’ expressed as “The sentence ‘This sentence has five words,’ has five words.” Using this ‘quotation’ method, one can arbitrarily define the truth-value of indeterminate propositions. Eg. “The sentence ‘This sentence is false,’ is false,” can be defined as expressing a true proposition. There’s no paradoxical self-reference there, because no sentence can actually refer to itself.
W.V.O. Quine killed that one, of course. True or false: “Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
The OP has the same solution that I do. Jabba’s objection is preceded by “The problem with this as a resolution of the Liar Paradox is that the sentence “This sentence is false.” clearly seems to be expressing a proposition.” My response is to note that many things seem to be what they aren’t, and IMO, this is one of them.
By the way, if “This sentence has five words.” is OK, what about “This sentence does not have seven words.”? Which of it or its negation is true?
Dogface: I am positing that rather than being both true and false, “This statement is false” is neither. Is there a way to show it true or false that does not depend on showing it non-false or non-true?
Truth or falsehood are irrelevant to evaluating “Ooga”. “This statement is false.” is a paradox because it is self-contradictory. “Ooga” is not self-contradictory.
And in any case “This statement is false.” is not actually a paradox. It is a semantically true self-referential statement. By being nearly perfectly self-contradictory, it has rendered itself meaningless, thus being semantically false. Things that are semantically true have some sort of meaning.
Read “Godel, Escher, Bach: An Eternal Golden Braid” by Douglass Hofstadter, for more insight into “Yields falsehood when preceeded by it’s quotation” yields falsehood when preceeded by it’s quotation, aka a “Quined” sentence. Hofstadter talks about the origin of self-reference and recursion in mathematics, music, and art. Another book which covers the subject is “A New Kind of Science” by Stephen Wolfram.
