You want a proof. I guess that means that you want to be more convinced that the Propositional Calculus is consistent than you are convinced of your own sanity. Any proof I could think of would involve mental operations of a greater complexity than anything in the Propositional Calculus itself. So what would it prove? Your desire for a proof of consistency of the Propositional Calculus makes me think of someone who is learning English and insists on being given a dictionary which defines all the simple words in terms of complicated ones…
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The character of Imprudence in the dialogue between Prudence and Imprudence in Chapter Seven of Douglas Hofstadter’s Gödel, Escher, Bach.
This criticism is very spot-on.
via molodoichelovek, maryclare
Unexpected hanging paradox - Wikipedia →
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on a Friday, as if he hasn’t been hanged by Thursday, there is only one day left - and so it won’t be a surprise if he’s hanged on a Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.
re Raven paradox →
This does not seem to me on the face to be a paradox. For why shouldn’t the whole world assert its coherence—logic—to support what is the case in logical space?
This is correct. The only paradoxic element is how a few logical operations and substitutions reveal truth-conditions that stretch far beyond the intuitive meaning of the first claim (“all ravens are black”). But if you think about mathematical statements and such, those weird truth-conditions can be quite insightful and crucial!
Raven paradox →
dailymeh quotes:
Hempel describes the paradox in terms of the hypothesis: (1) All ravens are black.
In strict logical terms, via the Law of Implication, this statement is equivalent to: (2) Everything that is not black is not a raven.
It should be clear that in all circumstances where (2) is true, (1) is also true; and likewise, in all circumstances where (2) is false (i.e. if we imagine a world in which something that was not black, yet was a raven, existed), (1) is also false. This establishes logical equivalence.
Given a general statement such as all ravens are black, we would generally consider a form of the same statement that refers to a specific observable instance of the general class to constitute evidence for that general statement. For example, (3) Nevermore, my pet raven, is black. is clearly evidence supporting the hypothesis that all ravens are black.
The paradox arises when this same process is applied to statement (2). On sighting a green apple, we can observe: (4) This green (and thus not black) thing is an apple (and thus not a raven).
By the same reasoning, this statement is evidence that (2) everything that is not black is not a raven. But since (as above) this statement is logically equivalent to (1) all ravens are black, it follows that the sight of a green apple offers evidence that all ravens are black.
No more Advanced Logic this year. No more Godel, no more Turing, no more axiomatizability, no more consistency sentences, no more reductio ad absurdum, no more sequent calculus, and no more true-false questions.
Well, good riddance to that last one.
