Meaning, Presuppositions, Truth-relevance, Gödel's Theorem and the Liar Paradox
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From: Newberry on 8 Aug 2010 12:04 http://www.scribd.com/doc/35519023/Meaning-Presuppositions-Truth-relevance-Godel-s-Theorem-and-the-Liar-Paradox Section 1 reviews Strawsonâs logic of presuppositions. Strawsonâs justification is critiqued and a new justification proposed. Section 2 extends the logic of presuppositions to cases when the subject class is necessarily empty, such as (x) ((Px & ~Px) âQx) . The strong similarity of the resulting logic with Richard Diazâs truth-relevant logic is pointed out. Section 3 further extends the logic of presuppositions to sentences with many variables, and a certain valuation is proposed. It is noted that, given this valuation, Gödelâs sentence becomes neither true nor false. The similarity of this outcome with Goldstein and Gaifmanâs solution of the Liar paradox, which is discussed in section 4, is emphasized. Section 5 returns to the definition of meaningfulness; the meaninglessness of certain sentences with empty subjects and of the Liar sentence is discussed. The objective of this paper is to show how all of the above-mentioned concepts are interrelated. Feedback appreciated.
From: Daryl McCullough on 8 Aug 2010 12:43 Newberry says... > >http://www.scribd.com/doc/35519023/Meaning-Presuppositions-Truth-relevance-= >Godel-s-Theorem-and-the-Liar-Paradox > >Section 1 reviews Strawson=E2=80=99s logic of presuppositions. Strawson=E2= >=80=99s >justification is critiqued and a new justification proposed. Section 2 >extends the logic of presuppositions to cases when the subject class >is necessarily empty, such as (x) ((Px & ~Px) =E2=86=92Qx) . The strong >similarity of the resulting logic with Richard Diaz=E2=80=99s truth-relevan= >t >logic is pointed out. Section 3 further extends the logic of >presuppositions to sentences with many variables, and a certain >valuation is proposed. It is noted that, given this valuation, Godel= >=E2=80=99s >sentence becomes neither true nor false. That's nonsense. If PA is inconsistent, then the corresponding Godel sentence is *false*. If PA is consistent, then the corresponding Godel sentence is true. So to say that the Godel sentence is neither true nor false means that it is neither true nor false that PA is consistent. That is nonsensical. -- Daryl McCullough Ithaca, NY
From: Newberry on 8 Aug 2010 13:05 On Aug 8, 9:43 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >http://www.scribd.com/doc/35519023/Meaning-Presuppositions-Truth-rele... > >Godel-s-Theorem-and-the-Liar-Paradox > > >Section 1 reviews Strawson=E2=80=99s logic of presuppositions. Strawson=E2= > >=80=99s > >justification is critiqued and a new justification proposed. Section 2 > >extends the logic of presuppositions to cases when the subject class > >is necessarily empty, such as (x) ((Px & ~Px) =E2=86=92Qx) . The strong > >similarity of the resulting logic with Richard Diaz=E2=80=99s truth-relevan= > >t > >logic is pointed out. Section 3 further extends the logic of > >presuppositions to sentences with many variables, and a certain > >valuation is proposed. It is noted that, given this valuation, Godel= > >=E2=80=99s > >sentence becomes neither true nor false. > > That's nonsense. If PA is inconsistent, then the corresponding Godel > sentence is *false*. If PA is consistent, then the corresponding Godel > sentence is true. > > So to say that the Godel sentence is neither true nor false means > that it is neither true nor false that PA is consistent. That is > nonsensical. The paper is NOT about PA.
From: Aatu Koskensilta on 8 Aug 2010 16:20 Newberry <newberryxy(a)gmail.com> writes: > It is noted that, given this valuation, Gödel's sentence becomes > neither true nor false. What sentence? Any Pi-1 statement -- Fermat's last theorem, Goldbach's conjecture, etc. -- is classically equivalent to the Gödel sentence of some theory. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, darüber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 8 Aug 2010 16:47
On Aug 8, 1:20 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > It is noted that, given this valuation, Gödels sentence becomes > > neither true nor false. > > What sentence? Any Pi-1 statement -- Fermat's last theorem, Goldbach's > conjecture, etc. -- is classically equivalent to the Gödel sentence of > some theory. This sentence: ~(Ex)(Ey)(Pxy & Qy). (3.3.1) Pxy means that x is the proof of y, where x and y are Gödel numbers of wffs or sequences of wffs. Q has been constructed such that only one y = m satisfies it, and m is the Gödel number of (3.3.1). |