Re: A is subst of B


gaya.patel@xxxxxxxxx wrote:
Frederick Williams wrote:
gaya.patel@xxxxxxxxx wrote:


I read in a Logic Tutorial (very very basic, small section of Discrete
Math book) that -> is the logic connective and that => is similar, but
we are asserting that the implication is true always (something like an
equivalence in one direction).

Hmm... sounds like a bit of a confusion to me. P -> Q has it's
well-known truth functional meaning. Did your tutorial define => by

P => Q is defined to be |= P -> Q ?

First it defines <=> to mean logically equivalent:

"P and Q are logically equivalent, denoted P <=> Q, whenever P <-> Q is
a tautology."

Later defines => to mean logically implies:

"We say P logically implies Q whenever P -> Q is a tautology. We write
this as P => Q.
So, P => Q means that P->Q is a tautology."

And in math courses I always encountered => and <=> used for
implications which I had mentally tried to link back to the definitions
above. I always thought the double lined versions were just
asserting that the implications were ALWAYS true (as it is for Math
Theorems).

After reading the thread answers, I'm beginning to believe that there
are different forms of logic out there. Is this true? If so, is the
basic logic something like "naive" logic ( like naive set theory is
related to the other axiomatic set theories that I don't know anything
about)?

There's basic propositional logic, with first-order and second-order
quantifier logic as one extension of that, and the modal logics as
another. Modal logics model non-logical connectives like necessity and
possibility, axiomatically. Some modal logics use P => Q to mean
[](P-Q) ("Necessarily, if P then Q") or strict implication, which
basically means what your book says: P => Q is true if P->Q is a
propositional tautology.

I'd just stick with -> if I was you.

IIf one's just talking about the propositional calculus, that's wise.

.

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