Chapter 8

Question 1

Q: What is propositional logic?

Answer 1

A: That part of logic that deals with the relationships holding between simple propositions or statements and their compounds. In propositional logic, the basic logical terms are not, or, and, and if then.

Question 2

Q: What is a conjunction (of statements)?

Answer 2

A: A compound statement in which all the statements are asserted, linked by and or an equivalent term. For the conjunction to be true, each component statement or conjunct must be true. The conjunction of statements P and Q is written as P. Q.

Question 3

Q: What is a Conditional Statement?

Answer 3

A: A statement of the form “If P then Q.” As such, it does not assert either P or Q. Rather, it asserts a connection between them in the sense that provided P is the case, Q will be also. Example: “If the population of Vancouver increases, the cost of housing in Vancouver will increase” does not say that the population of Vancouver increases or that the cost of housing in Vancouver will increase. It says that if the first happens, the second will happen. In propositional logic, the conditional is symbolized as P ⸧Q.

Question 4

Q: What is an Antecedent (of a conditional)?

Answer 4

A: Statement that follows if in a conditional of the form “If P then Q.” For example, in “If the population increases, the price of housing will increase,” the antecedent is “the population increases.”

Question 5

Q: What is a consequent (of a conditional?)

Answer 5

A: Statement that follows then in a conditional form “If P then Q.” For example, in “If the population increases, the price of housing will increase,” the consequent is “the price of housing will increase.”

Question 6

Q: What is a horseshoe?

Answer 6

A: A connective written as “⸧”, used in propositional logic to represent basic conditional relationships. A statement of the form “P ⸧ Q” is defined as false if P is true and Q is false, and true otherwise.

Question 7

Q: What is disjunction (of a statement)?

Answer 7

A: A compound statement in which the statements are asserted as alternatives; the connective is or. For the disjunction to be true, at least one of the disjoined statements must be true. The disjunction of statement P and statement Q is written disjoined statements must be true. The disjunction of statement P and statement Q is written as P v Q.

Question 8

Q: What is Exclusive disjunction?

Answer 8

A: A disjunction that is true if and only if one and only one of the disjuncts is true. An exclusive disjunction of statements P and Q is represented as (P v Q) · - (P · Q).

Question 9

Q: What is an inclusive disjunction?

Answer 9

A: A disjunction that is true if and only if one or both of the disjoined statements are true. The symbol “v” in propositional logic is used to represent inclusive disjunction.

Question 10

Q: What is a truth table?

Answer 10

A: Set of rows and columns that systematically display the truth values of basic statements and the compound statements formed from them.

Question 11

Q: What is counterfactual?

Answer 11

A: A conditional statement in which the antecedent is known to be false. Example: “If Hitler had been murdered when he was 20, world war 2 would not have occurred.” Note: Do not be misled by the term counterfactual into thinking all counterfactuals are false. All counterfactuals have antecedents that are false; however, many counterfactuals themselves are plausibly regarded as true statements.

Question 12

Q: What are contrary statements?

Answer 12

A: Statements that cannot both be true, although they can both be false. For example, “The rose is pink” and “the rose is yellow” are contrary statements.

Question 13

Q: What are Contradictory statements?

Answer 13

A: Statements that must have opposite truth values. A statement and its denial are contradictory. If the statement is true, its denial must be false. And if its denial is true, the statement must be false. For example, “he applied for the job” and “he did not apply for the job” are contradictory statements.

Question 14

Q: What is a necessary condition?

Answer 14

A: A condition that is required for another statement to be true. Using the horseshoe, if Q is a necessary condition of P, we would symbolize this as “P ⸧ Q.” To say that Q is a necessary condition of P is to say that P will be true only if Q is true.

Question 15

Q: What is a sufficient condition?

Answer 15

A: A condition that is enough to establish a further statement as true. Using the horseshoe, if Q is a sufficient condition for P, then Q horseshoe P. To say that Q is sufficient for P is to say that, if Q is true, P will be true as well.

Question 16

Q: What is biconditional?

Answer 16

A: A conjunction of a conditional and its transposition (the conditional that results from transposing the antecedent and the consequent). Example: (P ⸧ Q). (Q ⸧ P)

Question 17

Q: What is modus ponens?

Answer 17

A: A valid argument form, in which from P ⸧ Q and P, we may infer Q.

Question 18

Q: What is modus tollens?

Answer 18

A: A valid argument form, in which P ⸧ Q and -Q, we may infer -P.

Question 19

Q: What is an affirming the consequent?

Answer 19

A: An invalid form of inference of the type “P ⸧ Q; Q; therefore P.”

Question 20

Q: what is denying an antecedent?

Answer 20

A: An invalid form of inference of the type “P ⸧ Q; -P; Therefore -Q.”

Question 21

Q: Conditional proof?

Answer 21

A: Proof incorporating an assumption explicitly introduced into the argument and then represented as the antecedent of a conditional of a statement of which the consequent is derived from it together with the given premises.

Question 22

Q: What is reductio ad absurdum?

Answer 22

A: Means of proving a proposition by assuming its opposite and showing that its opposite leads to a contradiction. You then a conditional proposition “if the denial, then a contradiction.” You deny the contradiction and, using modus tollens, on the conditional, are able to infer the negation of the antecedent. That is denial of what you assumed. Being the denial of your initial denial of a proposition, it is that very proposition, proven.

Question 23

Q: What is denial (of a statement)?

Answer 23

A: A statements contradictory or negation. It must have the opposite truth value to the statement. The denial of a statement S is symbolized as -S (not-S).

Question 24

Q: What is indirect proof?

Answer 24

A: Proof of a conclusion by introducing its denial, on the rule of conditional proof, and then deriving a contradiction from the denial and stated premises. Using modus tollens, we infer from that contradiction the negotiation of the statement introduced. That is the denial of the denial of what we set out to prove. This (by double negotiation) is what we set out to prove.

Question 25

Q: What are Dilemma arguments?

Answer 25

A: Both constructive and destructive dilemmas constitute valid forms of argument. A constructive dilemma has the form P⸧ Q; R⸧ S; P v R; therefore, Q v S. A destructive dilemma has the form P⸧ Q; R⸧ S; -Q v -S; therefore, -P v -R. The disjunctive premises of dilemma arguments should be carefully scrutinized for acceptability to ensure that no false dichotomies are involved.

Question 26

Q: What is escaping through the horns of dilemma?

Answer 26

A: Showing that a dilemma argument, though valid, is not cogent because it is based on a false dichotomy. The disjunctive premise P v Q is not true because there is a third possibility. This expression is also used when a person shows a disjunctive statement to be false because there is a third alternative; the person said to have escaped through the horns of a dilemma.