Truth Tropic Language I
By way of investigating one of the more important truth-tropic properties of language, today I introduced a formal logic.
As we see in specifying its vocabulary, syntax, and semantics, the Propositional Calculus (PC) is a lean, some might say simplistic, logic. Nevertheless, it provides us with lots of room to develop a more sophisticated understanding of validity.
Today we defined PC by specifying its syntax and semantics. We used the semantics to develop a method of displaying in a table precisely how the truth of a Well Formed Formula (WFF) of PC--that is, a syntactically correct formula or sentence of PC--depends on the truth of its constitutive sentence letters.
You should in this regard be familiar with the mechanical method of filling out a truth table so as to systematically exhaust all the possible combinations of truth values of constitutive sentence letters. (We call the result a 'standard truth table'.)
We went on today to use standard truth tables to demonstrate (show) validity and invalidity for a number of arguments from the pdf handout PC Entailments and Theorems. Please note that I will draw directly from this handout in composing our first examination, which is coming up in short-order, Tuesday, 9/17! It would be a very good strategy in studying for the exam to practice constructing truth tables for the valid arguments. 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 explain how we can use the method of truth tables to prove theorems of PC and 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.