Invitation to Critical Thinking Chapter 7 Lecture Notes Chapter 7

Invitation to Critical Thinking Chapter 7 Truth Functional Logic Known as symbolic logic Uses logical operators Uses truth tables Useful in translating claims into categorical statements Useful in determining validity

Invitation to Critical Thinking Chapter 7 Truth Tables Sample truth table Modus Ponens PQ P  Q TTT TFF FTT FFT all possible combinations of truth value for all components of a truth functional compound statement corresponding truth value of the compound statement

Invitation to Critical Thinking Chapter 7 Logical Operators Represent the relationships between the "truth values" of the compound sentences we make with them Truth functional analysis is simply a way of keeping track of this. Symbols ~ = not (negation) & = and (conjunction)  if, then (conditional) v = either or (disjunction)

Invitation to Critical Thinking Chapter 7 Logical Operators Negation Negation simply reverses the truth value of the component statement to which it is applied The symbol "~" will be used to represent the logical operator negation Thus "~P" or “not P” represents the negation of P Negation operates on a single component statement, P. Since P is either true or false (not true), our truth table for negation required only two lines. P = The weather is great. ~P = The weather is not great. P~P TF FT

Invitation to Critical Thinking Chapter 7 Logical Operators Conjunction The components of a conjunction are called "conjuncts". "P" stands for the first conjunct and the letter "Q" stand for the second, and the symbol "&" represents the logical operator conjunction. Conjunction operates with a compound statement. Since P and Q may be either true or false, our truth table for conjunction will require four lines. PQ P & Q TTTTTT TFFTFF FTFFTF FFFFFF The weather is great and I wish you were here.

Invitation to Critical Thinking Chapter 7 Logical Operators Disjunctions Component statements called “disjuncts” At least one of the disjuncts is true (possibly both) The letters "P" and "Q" represent the two disjuncts and the symbol "v" represent the operator disjunction Thus "P v Q" represents the statement "Either P or Q" o PQ P v Q TTTTTT TFTTFT FTTFTT FFFFFF Either you party or you study.

Invitation to Critical Thinking Chapter 7 Logical Operators Conditionals Compound statements in which the “if” statement is the antecedent and the “then” statement is the consequent The consequent depends on the truth of the antecedent and follows the antecedent "P" represents the antecedent and "Q" represents the consequent. The symbol "  " represents the logical operator implication. Thus "P  Q" represents the conditional "If P then Q". o -- “only if” has the effect of reversing the conditional relationship between the antecedent and consequent (see Figure 7.5 for examples) o PQ P  Q TTTTTT TFFTFF FTTFTT FFTFFT If you study, then you are likely to do better on the quiz.

Invitation to Critical Thinking Chapter 7 Argument Forms Deductively Valid modus ponens based on one hypothetical statement and the affirmation of its antecedent (1)P  Q (2)P (3) Q If there is good water on Earth, then there is adequate support for life on Earth. There is good water on Earth. So, there is adequate support for life on Earth.

Invitation to Critical Thinking Chapter 7 Argument Forms Deductively Valid modus tollens based on one hypothetical statement and the denial of its consequent (1)P  Q (2b)~Q (3b) ~P If there’s good water on Earth, then there is adequate support for life on Earth. There’s not good water on Earth. (The water on Earth is polluted.) So, there is not adequate support for life on Earth.

Invitation to Critical Thinking Chapter 7 Argument Forms Deductively Valid hypothetical syllogism based on two hypothetical statements as premises, where the consequent of the first is the antecedent of the second. (1)P  Q (2d)Q  R (3d) P  R If there is life on Mars, then there is adequate life support on Mars. If there is adequate life support on Mars, then a manned mission to Mars is feasible. If there is life on Mars, then a manned mission to Mars is feasible.

Invitation to Critical Thinking Chapter 7 Argument Forms Deductively Valid disjunctive syllogism based on a disjunction and the denial of one of its disjuncts (1)P v Q (2)~P (3) Q Either the battery is dead or there is a short in the ignition switch. The battery is not dead. There is a short in the ignition switch.

Invitation to Critical Thinking Chapter 7 Argument Forms Deductively Invalid fallacy of denying the antecedent based on a hypothetical statement and the denial of its antecedent (1)P  Q (2c)~P (3c) ~Q If a figure is square then it has four sides. This figure (a rhombus) is not a square. This figure (a rhombus) does not have four sides.

Invitation to Critical Thinking Chapter 7 Argument Forms Deductively Invalid fallacy of asserting the consequent based on a hypothetical statement and the affirmation of its consequent. (1)P  Q (2a)Q (3a) P If a figure is square, then it has four sides. This rhombus has four sides. This rhombus is square.

Invitation to Critical Thinking Chapter 7 Dilemma An argument form or strategy combining hypothetical and disjunctive premises which seeks to prove its point by showing that it is implied by each of two alternatives, at least one of which must be true