a. ~p : if p is T, then ~p is F, and vice versa

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Dominick Neal
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1 Lecture 10: Propositional Logic II Philosophy & 8 November 2016 O Rourke & Gibson I. Administrative A. Group papers back to you on November 3. B. Questions? II. The Meaning of the Conditional III. More Propositional Logic A. Computing Truth Values 1. Complex sentences in propositional logic are formed by combining other sentences with propositional connectives. a. Syntactically, you need only follow the formal combination rules associated with the connectives to ensure that you get a well-formed sentence, assuming you had as inputs wellformed sentences. b. Semantically, it is important to recognize that propositional connectives are truth functions they take truth values as inputs (i.e., domain values) and produce a truth value as output (i.e., range value). 2. For the ones we are working with: a. ~p : if p is T, then ~p is F, and vice versa b. p v q : this is T if one or both of p and q are T; otherwise, it is F c. p & q : this is T if both p and q are T; otherwise, it is F 1

2 d. p q : this is T if either p is F or q is T; otherwise, it is F specifically, if p is T and q is F, then it is F 3. These can be composed, or combined, to enable us to determine truth values for more complex sentences. Two examples, where A and B are T and X and Y are F: a. ~(A & X) v (A Y) ~(T & F) v (T F) ~F v F T v F T b. ~(~(A (B & (X v Y)))) ~(~(T (T & (F v F)))) ~(~(T (T & F ))) ~(~(T F )) ~(~( F )) ~( T ) F B. Truth Tables 1. Given that connectives are functions, their impact on the meaning of complex sentences can be summarized in a table known as a truth table. a. Truth tables can tell us how the meaning of the whole sentence is composed out of the meanings of its parts, where this is understood in terms of truth conditions. b. They can also be used to answer questions about the validity of arguments or the logical equivalence of sentences. 2. One can put a truth table together following a mechanical procedure: a. Put the sentence into symbolic form by swapping out the propositional expressions for either propositional variables or capital letters. b. Put the symbolic sentence whose truth condition is to be determined to the right of a double line and the individual atomic sentences to the left. 2

3 c. The individual atomic sentences will be at the top of the columns the reference columns that are to be filled in with truth values. d. All possible combinations of "T"s and "F"s should appear in the rows of the reference column part of the truth tables. e. Beginning with the connectives that apply to atomic sentences, use their meanings to fill in the column under the sentence with the truth values generated by the assignment of truth values to the constituent sentences. (To determine the order, see Algorithm handout.) f. Continue assigning truth values to connectives based on the sentences they connect and their meanings until there is a column of truth values under the principal connective. This completes the truth table and gives the meaning of the sentence. 3. Put together truth tables for the four connectives from the last lecture: p & q, p v q, ~p, p q. a. Conjunction: b. Disjunction: c. Negation: p q p & q T F F F T F F F F p q p v q T F T F T T F F F p ~ p T F F T 3

4 d. Conditional: p q p q T F F F T T F F T 4. For more complex sentences, we must fill out the truth tables in a way that respects how they were constructed. This means that we must often work from the inside-out. a. There are various ways to determine how to do this, such as the construction technique or the algorithm technique (see handout). b. Start with the smallest constituents and work out to the largest. i. You can fill the table out by computing columns of Ts and Fs under each of the connectives. ii. The last one you compute is the only one that matters it is the truth condition. All the rest are scaffolding. 5. Example: ~(p ~(q r)) (10.1, 10.2) C. Evaluating Arguments for Validity and Sentences for Equivalence 1. One can use truth tables to assess arguments for validity or sentences for logical equivalence. a. In both cases, you will take the sentences in question either those that constitute the argument or those whose equivalence is being interrogated and put them above their own columns on the right-hand side of the truth table. b. Fill out the table as above, treating each of the sentences on the right as independent sentences whose truth conditions are to be computed. c. Evaluate the rows that result to determine if validity or equivalence are manifest. 4

5 2. Validity a. Look only at those rows where all of the premises are assigned the value T. The other rows are irrelevant. (Why?) b. In any of those rows, is the conclusion assigned the value F? If so, then the argument is invalid; otherwise, it is valid. c. Example 3. Equivalence a. Look at the rows of the truth condition for each sentence. b. Are they all the same? If so, the sentences are logically equivalent; if not, then they are not. c. Example D. Relationships with Natural Language 1. Conversational implicatures and the connectives 2. Does if then really mean that? 3. Interesting and complex connective interactions: unless, neither nor 5

Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)

Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook) Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or