Timeline for Adding a truth-like predicate to PA
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 27, 2016 at 19:03 | history | edited | Nik Weaver | CC BY-SA 3.0 |
added 283 characters in body
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| Jan 27, 2016 at 19:00 | comment | added | Nik Weaver | I'll edit my answer to clarify that. | |
| Jan 27, 2016 at 19:00 | comment | added | Nik Weaver | Oh, now I understand what was bothering you. Yes, these are rules of inference for Hilbert-style reasoning. Of course if you allowed them in natural deduction then you could use $\to$-introduction to turn them into implications. | |
| Jan 27, 2016 at 17:03 | vote | accept | Cecilia Burrow | ||
| Jan 27, 2016 at 17:03 | vote | accept | Cecilia Burrow | ||
| Jan 27, 2016 at 17:03 | |||||
| Jan 27, 2016 at 17:02 | comment | added | Cecilia Burrow | (cont'd) If T-intro, etc., are taken to be rules of inference, then a contradiction does indeed follow, but if they are only taken to be proof transformation rules, then the Friedman Sheard paper shows that we do not get a contradiction. In that sense, the Friedman Sheard result answers my question. | |
| Jan 27, 2016 at 17:02 | comment | added | Cecilia Burrow | Just to clear up some confusions here and make it easy for the reader: we need to distinguish rules of inference that apply anywhere (even within hypothetical reasoning) from rules of inference of the form R1-R4, which only take complete proofs to complete proofs. I don't know that there is accepted terminology here, but let's call call the former a 'rule of inference' and the later a 'proof transformation rule.' | |
| Jan 27, 2016 at 11:15 | comment | added | Lucas K. | @CeciliaBurrow To get a Liar paradox, you need to make T and the system where you add these axioms in such way that they are the same. | |
| Jan 27, 2016 at 5:02 | comment | added | Cecilia Burrow | Yes, looking at their paper, T-intro etc. do seem to be just the rules R1-R4, rather than the more general sort of deduction rule that applies even in hypothetical reasoning. | |
| Jan 27, 2016 at 4:02 | comment | added | Nik Weaver | ... $T$-intro etc. are deduction rules, not axiom schemes. | |
| Jan 27, 2016 at 4:02 | comment | added | Nik Weaver | Well, if you can prove $\phi$ then you can infer $T(\ulcorner\phi\urcorner)$, and then $\phi \leftrightarrow T(\ulcorner\phi\urcorner)$ follows trivially, but there's no technique for proving biconditionals generally. You explained why yourself: we can't reason hypothetically about truth, we can only talk about the truth or falsity of something when we've actually established its truth or falsity. | |
| Jan 27, 2016 at 3:19 | comment | added | Cecilia Burrow | Thanks, I'll look at all those materials, they sound fascinating. Still, I'm puzzled about the Friedman and Sheard result - if the rest of the logic is standard, and T-intro etc. mean what one would think, wouldn't one then able to prove every Tarksi biconditional, and thus get into trouble by considering Liar type sentences defined by the usual diagonalization method? | |
| Jan 27, 2016 at 3:08 | history | answered | Nik Weaver | CC BY-SA 3.0 |