Well, to me, modal logic is part and parcel of propositional logic — Agent Smith
proof of consistency — Banno
inconsistent language - or theory, if you prefer — Banno
[with an inconsistent theory] every theorem can be deduced; on in which everything is true. — Banno
if a contradiction is true in our system, then anything is derivable. — Banno
Some simplified detail might be fun.Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations. — TonesInDeepFreeze
this thread is not intended to be so formal but to get on with outlining what is going on. — Banno
rain on the parade — Banno
Some simplified detail might be fun. — Banno
If every proposition is true, then truth is trivial. It does nothing. — Banno
if contradictions (p & ~p) are allowed, "every proposition is true" — Agent Smith
So the argument from the principle of explosion ( ex contradictione sequitur quodlibet) is, in fact, the circular argument: — Agent Smith
contradictions are unacceptable because contradictions are unacceptable. — Agent Smith
...but every refers not to logically independent propositions like "some swans are not white" and "Socrates was bald" but to logically dependent propositions like "all swans are white" and "some swans are not white" — Agent Smith
Not sure what "logically dependent" is doing here. — Banno
the final row — Banno
I explained in detail why. — TonesInDeepFreeze
...bivalence... — Srap Tasmaner
Well, you are better informed than I. If you did it might be interesting.Are you suggesting that I provide more detail... — TonesInDeepFreeze
Rejecting bivalence — Banno
But perhaps not all that clearly. — Banno
If one sets (ϕ &~ϕ) as true — Banno
then since (ϕ &~ϕ)⊃ρ where ρ is any wff, — Banno
You can go back to the posts. — TonesInDeepFreeze
If a theory has a statement of the form P & ~P, then the theory has no model. — TonesInDeepFreeze
Get a good book on mathematical logic to learn the notions of provability, truth in a model, entailment, etc. — TonesInDeepFreeze
Your quibbles are doubtless correct. But not helpful. — Banno
Your quibbles are doubtless correct. But not helpful. — Banno
The traditional early chapters of a logic textbook try to show how the logical constants capture some of what we mean by familiar idioms. (The exception might be Kalish and Montague, because they're not kidding.) — Srap Tasmaner
I would be wary of thinking that the book suggests that natural languages lie down so easily that we can just read off its sentences always unambiguously into formal sentences. — TonesInDeepFreeze
For example, I would be surprised if the authors held that "if then" is always in English the material conditional. — TonesInDeepFreeze
I agree that the book is very careful indeed in how it states things and formulates things. I always recommend the book. — TonesInDeepFreeze
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