Propostion: an abstract math statement that is either true or false
P = Proposition
$ \neg $ = Negation / Not (the statement $ \neg P $ is true when P is False)
iff = “if and only if”
$ \wedge $ = Conjugation / And / Wedge ( $ P\wedge Q $ is true when P is True and Q is True)
$ \vee $ = Disjunction / Or ( $ P\vee Q $ is True when P or Q or both are True)
P(x) = Proposition that depends on X
$ \exists $ = " There exists" (the statement “$ \exists x \in \mathbb{R} \text{ } | \text{ } 0 < x < 1 $” would mean: there exits, at least, a number between 0 and 1)
$ \exists $! = “A unique number exists”
$ \Rightarrow $ = Implication
Implications make new propositions. The First statement is called the antecedent and the Second is called the consequent; Antecedent $ \Rightarrow $ consequent. This statement implies that if the Antecedent is true, then the consequent is true as well. So if P and Q are true, then $ P \Rightarrow Q $ is true. if P and Q are false, then $ P \Rightarrow Q $ is also True. However, if P is False and Q is True, then $ P \Rightarrow Q $ is True because all the statement is promising is if P is True, then Q is True. So as you would expect, if P is false and Q is true, then $ P \Rightarrow Q $ is false.
Here is an example of a logical Proposition:
$$ \forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \exists z \in \mathbb{R}, ( x < y ) \Rightarrow ( x < z < y ) $$
This statement would mean: for every two real numbers that are not the same, there is a real number between them.
Consider this on Negation:
$ \neg ( \exists x, P(x)) = \forall x, \neg P(x) $
$ \neg (\forall x, P(x)) = \exists x, \neg P(x) $
The statements mean the same thing with different syntax so ill explain the first. The first half of the former statements means: there exits an x value that dons’t work with P(x), and the latter half means: not all values of x work with P(x).