A forum on philosophy ought have threads on the basics of logic. — Banno
Some string of symbols are well-formed, others are not.&pv~~⊃
orp
and so on.q
So since "p", we can now also write "~p". And since we can write "~p", we can write "~~p", and so "~~~p" and so on. The process is iterative.If ϕ is a wff, then ~ϕ is a wff
I understand why you might want to brush up, but why should we watch? — Srap Tasmaner
+---+ | p | +---+ | T | +---+ | F | +---+
+---+----+ | p | ~p | +---+----+ | T | F | +---+----+ | F | T | +---+----+
+---+----+ | ϕ | ~ϕ | +---+----+ | T | F | +---+----+ | F | T | +---+----+
Best of luck with your new blog. — Srap Tasmaner
+---+---+-------+ | ϕ | ψ | ϕ & ψ | +---+---+-------+ | T | T | T | +---+---+-------+ | T | F | F | +---+---+-------+ | F | T | F | +---+---+-------+ | F | F | F | +---+---+-------+
+---+---+-------+ | ϕ | ψ | ϕ v ψ | +---+---+-------+ | T | T | T | +---+---+-------+ | T | F | T | +---+---+-------+ | F | T | T | +---+---+-------+ | F | F | F | +---+---+-------+
+---+---+-------+ | ϕ | ψ | ϕ ⊃ ψ | +---+---+-------+ | T | T | T | +---+---+-------+ | T | F | F | +---+---+-------+ | F | T | T | +---+---+-------+ | F | F | T | +---+---+-------+
1. Identity A = A — Agent Smith
+---+----+--------+ | p | ~p | p v ~p | +---+----+--------+ | T | F | T | +---+----+--------+ | F | T | T | +---+----+--------+
+---+----+----------+-----------+ | p | ~p | (p & ~p) | ~(p & ~p) | +---+----+----------+-----------+ | T | F | F | T | +---+----+----------+-----------+ | F | T | F | T | +---+----+----------+-----------+
The law of identify holds between individuals, and as mentioned earlier propositional calculus deals in whole propositions. SO strictly the law of identity is a part of predicate clacualus rather then propositional calculus — Banno
Notice that it is the negation of (p & ~p), a contradiction?
The contradiction has "F in each row in the truth table. The tautology has T.
So the first way we have of proving a theorem is looking at its truth table. — Banno
But natural deduction is more common and more direct, so we might start there. — Banno
The point to truth tables is that given a set of rules on how logical connectives function (quite like mathematical operations), what will be — Agent Smith
Evaluate for validity of an argument: Is there a possible world in which, for a given argument form, all the premises are true and the conclusion false? If there is, the argument is invalid and if there's none, the argument is valid. — Agent Smith
+---+---+-------+ | p | q | p & q | +---+---+-------+ | T | T | T | +---+---+-------+ | T | F | F | +---+---+-------+ | F | T | F | +---+---+-------+ | F | F | F | +---+---+-------+
+---+----+----------+ | p | ~p | (p & ~p) | +---+----+----------+ | T | F | F | +---+----+----------+ | F | T | F | +---+----+----------+
+---+----+----------+ | p | ~p | (p v ~p) | +---+----+----------+ | T | F | T | +---+----+----------+ | F | T | T | +---+----+----------+
Nothing to do with possible worlds. — Banno
will always be true; (p v ~p).the cow needs milking or the cow does not need milking
will never be true; (p & ~p).the cow needs milking and the cow does not need milking
might be either; (p & q)."the cow needs milking and the cat is having kittens"
1. p → q
2. p
Ergo
3. q
+----+---+----+---+----+---+---+ | (p | & | (p | ⊃ | q) | ⊃ | q | +----+---+----+---+----+---+---+ | T | T | T | T | T | T | T | +----+---+----+---+----+---+---+ | T | F | T | F | F | T | F | +----+---+----+---+----+---+---+ | F | F | F | T | T | T | T | +----+---+----+---+----+---+---+ | F | F | F | T | F | T | F | +----+---+----+---+----+---+---+
...you don't sound convinced... — Banno
I need feedback.I don't wanna derail a good thread. — Agent Smith
I need feedback. — Banno
Yet natural deduction is as powerful and valid as an axiomatic system. — Banno
My responses are limited. You'll have to ask the right questions. — Dr. Lanning (I Robot)
A theory [natural deduction] is said to be consistent if falsehood is not provable (from no assumptions)^{1} and is complete if every theorem or its negation is provable using the inference rules of the logic^{2}. — Wikipedia
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