How do I prove, :((A ⊃ B) ⊃ C) ⊃ (B ⊃ C), using symbolic logic derivations where ⊃ represents a conditional i.e. A ⊃ B = A implies B?
The first line of my derivations is the assumption, (A ⊃ B) ⊃ C. The second line is a second sub derivation, B. However I don't know how to get C out of my second sub derivation so I can return to the original assumption ...
first of all, @Mauro ALLEGRANZA, ⊨A is a tautology. as has been explained by @N. Bar, ⊨A means A is a consequence of the empty set, so ⊨A would be a tautology.
second, here is my proof.
the first statement is equivalent to,
concerning the second statement, I have to point out that the symbol "⊨" is not a logical connective, it means "...
A formal logical system does not ever tell you what it means to prove something. It tells you what works in the model. The point is that you need to have faith that the model models your real thinking, and not something else. Otherwise, you can't use it to discuss the nature of thinking.
Without proof of a more casual sort that the proofs you are going ...
Deductive proofs in first-order logic are essentially transformations of one statement of the language into another. You start with some statement (or several or none at all) and then produce from it an ordered sequence of new statements derived by successive applications of the established rules of the system. You can end this sequence at any time and any ...
The assumption "A ⊨ B" does not mean that A is a tautology.
You have assumed f(A)=1, that means that A is true for an interpretation f.
The same for the second case : "When ⊨A → B, B must be true since ⊨A is a tautology" is wrong.
A → B is a tautology: this means that, whenever A is true also B is, and this is enough to conclude with A⊨B.
These Insights are not new, those who are meditative enough are able to perceive them. The ancient texts spoke about them;
So deep, so pure, so still It has been this way forever You may ask, “Whose child is it?”- but I cannot say This child was here before the Great Ancestor - Dao De Ching
That which is nonexistent can never come into being, that which is ...
Gödel’s Incompleteness Theorem is a result about formal systems.
Its proof requires certain assumptions about the properties of specific formal system F: basically, about its "expressive capabilities".
In a sense that can be specified rigorously, system F must have the capabilities to manufacture the provability predicate for F, i.e. a suitable formula PrF(...
(1) is a special case of the general principle that if you accept a statement, then you accept that the statement is true. If you believe snow is white, then you believe "Snow is white" is true. It remains handwavy until you give a precise explanation of what you mean by "X is true." And that turns out to be a tricky business.