Kirk Rader  1.0-SNAPSHOT
Validity

A Priori Validity vs. A Posteriori Truth.

Overview

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.

As described elsewhere in this document, 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 the sentential calculus by itself says nothing about its actual truth value at any given point in spacetime.

Also as alluded to at several points in this document. some formulas have truth values that can be computed without regard to the truth values of the terms from which they are composed. For example, looking at the truth tables for $\lognot \phi$ and $\phi \vee \psi$, above, it is easy to see that the following is always true without regard to the a posteriori truth value of $P$:

\[ \begin{array}{cc|c} P & \lognot P & P \vee \lognot P \\ \hline \\ T & F & T\\ F & T & T\\ \end{array} \]

The preceding is an example of a tautology and is, as the saying goes, "self-evidently" true. Where a formula's truth value can be computed this way without knowing anything about the actual values of its terms, its truth value is said to be known a priori (i.e. "prior to observation").

Not all a priori truth values correspond to tautologies. Given the preceding examples, it is easy to deduce a priori that the following formula is "self-contradictory" and, as such, false:

\[ \begin{array}{cc|c} P & \lognot P & P \wedge \lognot P \\ \hline \\ T & F & F\\ F & T & F\\ \end{array} \]

Symbolic logic is not the only formal language concerned with a priori validity. The arithmetic formula $1 + 2 = 3$ represents an a priori truth. In fact, every branch of mathematics is simply a system for manipulating some particular set of symbols according to some set of agreed-upon rules to demonstrate tautologies and, so, is simultaneously: