A Priori vs. A Posteriori Truth

An argument is valid if its conclusion is necessarily true on the assumption that all of its premises are true without regard to whether or not those premises actually are true in the real world. As described Sentential Calculus, a sentence denoted by a primitive term in the sentential calculus may be either true or false. If P is defined to correspond to some contingent fact about the world that can only be determined by empirical observation – “It is raining,” “The moon is full” etc. – its truth value is said to be known a posteriori (i.e. “after observation”) and symbolic logic by itself says nothing about its actual truth value at any given point in spacetime.

A “fact” that can be known with certainty without knowing anything about the current state of affairs is said to be an a priori (“prior to observation”) truth. An argument does not need to be a premise-less tautology in order to be valid. The following argument is valid. Whether or not its conclusion is actually true depends on the current of the world in regards to its premises:

$$ \begin{aligned} &P \rightarrow Q \\ &P \\ \hline \\ \therefore &Q \end{aligned} $$

Symbolic logic is useful in providing tools for verifying the validity of some piece of a priori reasoning while not even attempting to verify any bit of a posteriori truth.