When does p implies q noted as p---->q,
and when does p logically implies q noted as p====>q
solid examples would help.
in a mathematical proof we use logical implication or just simple implications
When does p implies q noted as p---->q,
and when does p logically implies q noted as p====>q
solid examples would help.
in a mathematical proof we use logical implication or just simple implications
In my view there is absolutely no difference is the two expressions.
Now, be warned that I am also sure the there is out there somewhere an author and textbook that will take great exception with my thinking that. For example, I recall reviewing a logic textbook in the 70’s in which the author took some five pages to argue that there are no such things as propositions. He maintained there are only sentences that are true or false. I think I recall his going so far as to say those who talk about propositions are just idealists and Platonists.
ok i was just browsing through some notes and came across the following. hope it helps ....
The truth of $\displaystyle p \rightarrow q$ is sometimes described by the expression "p is a sufficient condition for q" or "q is a necessary condition for p"
example:
p = "it is raining"
q = "there are clouds in the sky"
$\displaystyle p \rightarrow q$ is true
but $\displaystyle q \rightarrow p$ is false.
Given two compound propositions P and Q, we say that P logically implies Q, if Q has the truth value true whenever P has truth value true. we write $\displaystyle P \Rightarrow Q$ when this occurs.
Note that $\displaystyle P \Rightarrow Q$ iff ( $\displaystyle p \rightarrow q$ ) is a tautology.
That is, $\displaystyle P \Rightarrow Q \longleftrightarrow (p \rightarrow q)$