I has wondering what these mean in Logic:
$\displaystyle
p \gets q~
p \to q~
p \leftrightarrow q
$
I've got an article on it but the explanation is too abstract.
Hello, Grich!
. . $\displaystyle \begin{array}{ccc}p \gets q & \quad & \text{If }q\text{, then }p \\ \\[-3mm]
p \to q & \quad & \text{If }p\text{, then }q \\ \\[-3mm]
p \leftrightarrow q & \quad & p\text{ if and only if }q \end{array}$
. . . I hope that's clear enough.
So,
$\displaystyle P \gets Q$
If Q is true then P is true, if Q is false then P is false
$\displaystyle P \to Q$
If P is true then Q is true, if P is false then Q is false
$\displaystyle P\leftrightarrow Q$
P is true when Q is true, P is false when Q is false
Is this correct?
Hello,
Actually, the 1st and 2nd are the same relation.
$\displaystyle P \gets Q$ is like $\displaystyle Q \to P$
Then, there's something you should learn : if $\displaystyle P \to Q$ and if P is false, then Q is true. This is weird but it's logic
Also, see the truth tables : Truth table - Wikipedia, the free encyclopedia
Okay, second attempt. So:
$\displaystyle
P \gets Q
$
so if the value that the Implication arrow is pointing to is False and the value that the arrow is pointing from is true, then the answer would be false. Else it would be true.
$\displaystyle
P \leftrightarrow Q
$
And if the values in P and Q are equal then it's true. Else it is false.
COOL
Did I get it right?
The definition of the conditional ( p--->q) from everyday life is the following:
A true event cannot imply a false event
whilst a false event can imply everything i.e. true or false
The following example it may help a Little:
we say:
if i go to London i will buy a car.
Let , p= i go to London and q= i buy a car,then we have:
case 1: i went to London and i bought a car.
in this case p is true and q is true and i will be called a true man
But what is..............(p---->q).......in this case true or false ???
I say true, what do you say???
case 2: i went to London and did not buy a car .
in this case p is true and q is false and i Will be called a lier.
Again what is .................(p----->q)..........in this case???
I say false, what do you say???
case 3: i did not go to London and did not buy a car.
in this case p is false and q is false also. Will i be called a lier ??
what is ........................(p--->q)....................in this case???
case 4: i did not go to London but i bought a car.
in this case p is false and q is true.Will i be called a lier???
...................................NO................................................
BECAUSE i could have gone to a local agency and buy a car although
i did not go to London.
and in this case .................(p--->q) ....................is true
So we see that the conditional : if i go to London i will buy a car can be broken down to 4 cases ( TT,TF,FT,FF) AND the out come can be only T or F.
THAT loosely explains the definition of the conditional in symbolic logic
Now the expression : if......then..... can be used in arguments as well,but
there the implication becomes a logical implication
......................look for relevant threads........................................... .
Finaly,what is : if 2+3=5 then 4+4=8??? true or false?????