'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 25 Mar 2010 06:35 Newberry says... >The sentence "if it rains then some roads are wet" describes a >possible state of affairs. I can picture to myself what it means. I >can even picture "if it rains then no roads are wet." It is still >conceivable although very unlikely. "If it rains and does not rain >then the roads are wet" does not describe any possible state of >affairs. I cannot picture to myself what it expresses. Well, this is something you need to work on. The "state of affairs" associated with an implication A -> B is any situation in which A is false or B is true. If A is always false, then A -> B describes every state of affairs. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 25 Mar 2010 06:37 Newberry says... >In my logic the Liar paradox can be expressed as follows. > > ~(Ex)(Ey)(Pxy & Qy) (L) > >where Pxy means that x is a proof of y, Q is satisfied by only one y = >m, and m is Goedel number of (L). That's not the Liar sentence. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 25 Mar 2010 06:38 Newberry says... >If you take the position that there are truth value gaps then the Liar >papradox is solvable in English. What does it mean to be "solvable" and why do you want it to be solvable? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 25 Mar 2010 06:49 Newberry says... >Tarski's theorem does not apply to formal systems with gaps. I think >it is preferable. If you the way you express Tarski's theorem is like this, then truth gaps don't change anything: There is no formula T(x) such that if x is a Godel code of a true sentence, then T(x) is true, and otherwise, ~T(x) is true. Anyway, *why* is it preferable to have a formal system for which Tarki's theorem does not apply? Preferable for what purpose? -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 25 Mar 2010 13:26
Newberry <newberryxy(a)gmail.com> writes: > On Mar 24, 3:32 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > Plus >> >> > (x)((x = x + 1) -> (x = x + 2)) >> >> > does not look particularly meaningful to me. >> >> I don't believe you. > > Trust me. > >> You know what it means. It's perfectly clear >> what it means. It means that whenever x = x + 1, then x = x + 2.[1] > > The sentence "if it rains then some roads are wet" describes a > possible state of affairs. I can picture to myself what it means. I > can even picture "if it rains then no roads are wet." It is still > conceivable although very unlikely. "If it rains and does not rain > then the roads are wet" does not describe any possible state of > affairs. I cannot picture to myself what it expresses. Is the statement "Honesty is a virtue" meaningful? What do you picture when you think about that statement? As usual, your claim that meaning involves picturing various states of affairs is silliness. I can understand various theorems about, say, infinite dimensional spaces. I daresay that I know those theorems are meaningful, even though I cannot picture a space with more than three dimensions. Of course, as Daryl points out, it is very easy to "picture" what the above sentence means. It means the same thing as (Ax)( ~(x = x + 1) or (x = x + 2) ). I see no problem understanding that sentence at all. > The analytic sentences are rather odd. But even then given "all > bachelors are unmarried" if you examine every bachelor you will find > that he is umarried. Given "all married bachelors are unmarried > bachelors" is just like "when it rains and does not rain ..." I cannot > picture anything. > > Similaly I cannot picture (x)(x = x+1) -> (x = x+2) any better than I > can picture anything being attributing to married bachelors. As I said previously, I understand the meaning of that sentence and can even immediately see that it is true, through the following perfectly simple reasoning. >> [1] In fact, this statement seems obviously true! Suppose >> x = x + 1. Then we may substitute x + 1 for x in the right hand side >> of the equation x = x + 1, thus: >> >> x = x + 1 >> = (x + 1) + 1 >> = x + 2. >> >> I see nothing the least bit fishy about this reasoning. -- Jesse F. Hughes "As you can see, I am unanimous in my opinion." -- Anthony A. Aiya-Oba (Poeter/Philosopher) |