on 26-Dec-2018 (Wed)

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#chapter-1 #jaynes_probability_theory

Question

A -> B. Does a logical statement imply that A physically cause B?

Answer

This example shows also that the major premise, ‘if A then B’ expresses B only as a logical consequence of A; and not necessarily a causal physical consequence, which could be effective only at a later time.
The rain at 10 am is not the physical cause of the clouds at 9:45 am. Nevertheless, the proper logical connection is not in the uncertain causal direction (clouds =⇒ rain), but rather (rain =⇒ clouds), which is certain, although noncausal

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#chapter-1 #jaynes_probability_theory

Question

one would take ‘A implies B’ to mean that B is logically deducible from A.
What's wrong?

Answer

"A implies B" really does not mean that there is a chain of arguments so you can derive B from what's inside the statement A. "A implies B" indeed only means "whenever A, then B". You can say "the fact that the number 3 is odd implies that the number 100 is bigger than the number 50". "Implies" here really doesn't mean that you can derive size relations of numbers from the oddness of other numbers by looking at the concepts of "size" and "oddness". These statements haven't anything special to do with each other, they just happen to be both true.

And every true statement "follows from" / "is implied by" every other true statement in the sense of formal logic.
every statement (be it true or false) "is implied by" / "follows from" any false statement.

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#chapter-1 #has-images #jaynes_probability_theory

Question

This is just an example of a function. C = f(A, B)
How large is the class of new propositions thus generated?

Answer

two propositions, although they may appear entirely different when written out in the manner, are not different propositions from the standpoint of logic if they have the same truth value.
For A and B there 4 different inputs. TT, TF, etc. Size of the function: 2 -> 4, therefore there are 4^2 functions. A fixed number.
This easily generalizes to greater dimensions.

Tags

#chapter-1 #jaynes_probability_theory

Question

Example of "reduction to disjunctive normal" method in logic
Hint: size

Answer

Proving that there is a finite number of propositions (functions)
E.g. C = B ^ A is a proposition, that depends on 2 variables. There is a finite number of such propositions.