Wednesday 2/5

Wednesday 2/5

Truth Tropic Language II: Validity by Truth Table

Readings

Notes

Synopsis

Today we defined a standard truth table to make correctly filling out all the truth values for the individual sentence letters as mechanical and straightforward as possible. If you struggle to grasp the pattern, see me for clarification.

We went on today to use standard truth tables to show how to determine the truth value of a complex PC WFF and discovered a special class of PC WFFs called 'PC Theorems' which turn out to be true on every line of a standard truth table. Since the lines of such a truth table represent all of the possible combinations of truth values for individual sentence letters, PC Theorems are special because it is impossible for them to ever be false! They are sometimes called necessary truths, laws of logic, or tautologies.

Our goal, however, was not PC Theorems but to define validity in terms of truth tables. 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. Equivalently, a PC argument is valid iff on every row of standard truth table where the premises are all true, the conclusion is also true.

We worked through some examples of valid PC arguments so as to demonstrate (show) their validity. Make a special note of the pdf handout PC Entailments and Theorems: I will draw directly from this handout in composing our first examination, which is coming up in short-order, Monday, 2/17! It would be a very good strategy in studying for the exam to practice constructing truth tables for the valid arguments and demonstrating the PC Theorems on the second page. Please note that I have office hours 11-2 TR should you run into any confusions. I also have an old-school blackboard in my office we can use in ironing out puzzles.

Next time I'll demonstrate one of the shortcomings of truth tables: Sure, they are mechanical and straightforward, but they grow enormously for even small increases in the number of sentence letters. I'll introduce a second way of defining validity for PC arguments that is not nearly so clunky or time-consuming (albeit, far less mechanical) than the method of truth tables&emdash;the elegant method of Analytic Tableaux.