Satisfiability & Logical Truth PHIL 012 - 2/16/2001
Outline • Test Scores • Homework Reminder • Satisfiability • Logical Truth • Complex Truth Tables • Sample Problems
Satisfiability • A sentence is said to be satisfiable IFF under some circumstances it could be true, on logical grounds. • In other words, a sentence is satisfiable if it doesn’t involve a logical contradiction. • A sentences does not have to be actually true or even physically possible in order to be satisfiable.
Satisfiability • A set of sentences is satisfiable IFF all of the sentences could be true at the same time, on logical grounds. • In other words, a set of sentences is unsatisfiable if the truth of one or more sentences in the set precludes the truth of one or more of the others on logical grounds.
Satisfiable Sentences and Sets of Sentences • George Bush is President of the United States. • Al Gore is President of the United States. • My cat is President of the United States. • x is a tetrahedron. • Happy(Max) Happy(Claire)
Unsatisfiable Sentences and Sets of Sentences • x is a round square. • Happy(Max) Happy(Max)
Logical Truth • A sentence is said to be logically true if there is no possible world in which the sentence could be false. Home(Claire) Home(Claire) will always be true because of the definition of .
Logical Truth Home(Claire) Home(Claire) Home(Claire) Home(Claire) T F T F T T
Relationship • P is logically true just in case P is unsatisfiable. • Q is satisfiable just in case Q is not logically true.
More Truth Tables • We can determine whether a sentence is logically true, unsatisfiable, or neither by using truth tables. • In order to build a complex truth table, we must create a column for each atomic sentence and another for the complex sentence we want to analyze.
Truth Tables • Suppose we have the complex sentence, (P Q) P • We will need a column for each atomic sentence, P & Q, and one for the complex sentence. So… P Q (P Q) P
Truth Tables • We will need to show every possible combination of truth values for our atomic sentences. • Since there are two possible truth values for each sentence, the number of rows we will have for n sentences will be 2n. • So, for (P Q) P we will have 4 rows.
Truth Tables • To make sure we assign all possible truth values, it is best to use a systematic method. • Assign all truth values for a single atomic sentence all at once. • Make the first half of your rows TRUE for your first sentence, and the second half FALSE...
Truth Tables P Q (P Q) P T T F F
Truth Tables • For the next column, we want to split the truth values for the rows of the first column… P Q (P Q) P T T T F F T F F
Truth Tables • Once we have assigned truth values for our reference columns, we can begin assigning truth values to the components of the complex sentence. • The first step is to copy our truth values from our reference columns over to the complex sentence. • The truth value for each atomic sentence will be the same in any given row.
Truth Tables P Q ( P Q ) P T T T T T T F T F T F T F T F F F F T F
Truth Tables • In a complex sentence, the truth values of the connectives for more complex expressions depend upon the truth values of their component expressions. • Thus, we need to assign truth values from the “inside out”. • The last truth value we assign (the truth value for the whole complex sentence) will be that of the major connective.
Major Connective Truth Tables P Q ( P Q ) P T T T T T T F T F T F T F T F F F F F F T T F F F F F T T T F T The truth value for the complex sentence