Conjunctions, Disjunctions, and Negations

Conjunctions, Disjunctions, and Negations Symbolic Logic 2/12/2001 Outline

Test Homework & Schedule Truth Tables Negation Conjunction Disjunction Ambiguity & Parentheses Test

Handed back Wednesday Solutions posted after all have taken it. See Sara if you have questions. Note on Caps & letters: Caps (A, B, C) are reserved for Predicates. Letters like (a, b, c) are usually for constants.

Letters like (x, y, z) are usually for variables. Homeworks & Schedule New Procedures for homeworks. Homework will be Due in class every Monday. You will hand in a disk and/or paper.

What is due is the problems for the sections on the schedule for the preceding week. An updated schedule with tests & assignments is online. More on Homework Problems will be sampled. That is, not every problem of every assignment will be graded. You are expected to do all the assignments. You will need two disks for your homework so that you can keep working while the current assignment is being graded.

Your second disk will also serve as a backup. Truth Tables We use a truth table to define logical constants. A truth table lists all possible combinations of truth values for a particular constant. The left side of the table contains the truth values of the components of a sentence. The right side contains the truth value of the sentence.

Negation Heres the truth table for negation (~) P ~P TRUE FALSE FALSE TRUE This table lists all possible combinations of truth values for the sentence P. Negation Continued

Negation simply reverses the truth value of whatever it is negating. For example, if we have the true sentence, Home(John) the negation of this sentence is ~Home(John) and its truth value will be false. Truth table for Home(John) Home(John) TRUE FALSE

~Home(John) FALSE TRUE Conjunction The ordinary language definition of conjunction is AND. Words like but and moreover also do much the same thing in English as ^ in FOL. ^ implies that a compound sentence will be

truth IFF both sides are true, false otherwise. Truth table for ^ P T T F F Q T F

T F P^Q T F F F Example of ^ Happy(John) T

T F F Sad(Mary) T F T F Happy(John)^Sad(Mary) T

F F F Disjunction Logical OR (v) The resulting sentence is true if either or both sentences are true. This differs from the ordinary English meaning of OR Disjunction

P T T F F Q T F T F

PvQ T T T F v vs. XOR The ordinary English use of or ordinarily means XOR, which is true IFF only one of the sentences is true. P Q P xor Q

T T F T F T F T T F F F Exercises