Necessity and Contingency

From Logic

Current revision as of 19:13, 14 April 2010

Necessary and Contingent Truth

For every argument there corresponds a conditional statement whose antecedent is the conjunction of the argument's premise and whose consequent is the argument's conclusion. For example, an argument using the form of modus ponens:

p ⊃ q p therefore q

...could be expressed as a conditional statement thusly: [(p ⊃ q) & p] ⊃ q. We can read this conditional statement this way: "If it is true that 'p is true, then q is true' AND p is in fact true, then in this case, q is true."

Now why go through all this? Because something very interesting happens to a valid deductive argument when stated in a conditional statement: it becomes a tautology. We learn then that a deductive argument is valid if and only if its expression in the form of a conditional statement is a tautology.

This all goes back to my original expostulation on the nature of deductive arguments: deductive arguments deal in equivalencies, in equalities, i.e. with a priori truths.

So how do we deal with inductive matters, with matters that concern real world phenomena?

Here, we can only deal in contingent truths. If we think back to the deductively invalid forms of affirming the antecedent and denying the consequent, we will recall while there were possible permutations where all true premises led to a false conclusion, we also can recall that there were permutations where this was not the case. Therefore, along with necessarily contradictory conclusions, these deductively invalid forms also led to contingent truths: claims that may in fact be true. As we will see in the section on inductive logic, we can rely on such forms to give us probable or possible truths.