Truth Tropic Language II
- The Propositional Calculus (from last time)
- PC Entailments and Theorems (from last time)
- PC Translations(from last time)
Today we defined validity in terms of truth tables and worked some problems to see how the definition plays out in practice.
Recall the general definition of validity: An argument is valid iff it is impossible for the conclusion to be false when all the premises are true.
Since the rows of a standard truth table represent all the possibilities, this is equivalent to saying: A PC argument is valid iff on no row of a standard truth table are all the premises true but the conclusion false.
Note that the handout above, PC Entailments and Theorems, gives lots of practice examples of valid arguments (entailments) for you to practice. Note also that apart from correctly filling out the truth values for the individual sentence letters--the 'standard' in standard truth table--there is no one right way to proceed. You can use temporary columns to keep track of your work, as I do, or you can break expressions apart and put them side by side to be evaluated each in turn, as Gabby neatly showed you how to do. What does matter is that clearly indicate those rows on which all the premises are true so as to verify (display) that the conclusion is also true on those rows--therewith demonstrating that it is impossible for the conclusion to be false when all the premises are true, going back to the general definition of validity.
As useful as truth tables are for defining (in a transparent way) validity (and theoremhood!), they quickly become unwieldy, perhaps even unworkable. After all, by the formula for filling out a standard truth table, the number of rows for formulas with four distinct sentence letters is 16--complicated, but not awfully so. The number of rows for formulas with 5 distinct sentence letters is 32, 6 gives you 64 rows, and a mere 7 distinct sentence letters requires a whooping 128 rows!
Surely there is a better, more economical approach. And, indeed, there is. We take up this new way of defining validity next time.