Exam 2: Propositional Logic: Syntax and Semantic

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -P ≡ ∼Q ∼(Q $\lor$ ∼P) Q ⊃ ∼Q / P

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(Y $\lor$ P) ⊃ (B • P)

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -(∼A • B) ≡ (B ⊃ A)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(H ⊃∼I) • ∼J ∼I ⊃ [(J $\lor$ K) • (J $\lor$ H)] ∼H $\lor$ (K ⊃ L) / L ⊃∼H

use the following key to determine which English sentence best represents the given formula of PL. A: Peirce studied logic. B: James was a pluralist. C: Dewey wrote about thirdness. D: Dewey denigrated the quest for certainty. E: Peirce emphasized education. -∼C ⊃ (A ≡ B)

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator. -∼A $\lor$ (B • D)

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator. -[(W ⊃ X) • (Y $\lor$ ∼X)] ≡ [∼(Z $\lor$ Y) ⊃ ∼W]

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -{X $\lor$ [C • (Y ⊃ B)]} ⊃ {Z ⊃ [Z ⊃ (Z ⊃ Z)]}

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(∼P ⊃ P) $\lor$ (P ⊃ A)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -Q • (∼A ≡ Q)

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the α assignment; the second variable in the formula read left to right (if any) gets the β assignment; the third variable in the formula read left to right (if any) gets the γ assignment; and the fourth variable in the formula read left to right (if any) gets the δ assignment. For exercises with only one propositional variable, the standard assignment is: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \beta \gamma \delta 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 Complete truth tables for each of the following propositions. -(V ? W) • (V ? ?W)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -J ≡ (∼K • L) L ⊃ J / L ⊃ K

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼(Q ⊃ C) $\lor$ (Z • ∼X)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -S ∼T • U / ∼(T • U) • S

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -∼J ⊃ K ∼K $\lor$ (J • K) ∼K ⊃ K / ∼J

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -(W $\lor$ X) ⊃ ∼(Y • ∼Z) ∼(Y • W) ⊃ ∼Z ∼(W $\lor$ X) $\lor$ Z / (Z $\lor$ W) ⊃ (X • Y)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -G ⊃ (J • ∼K) (I ⊃ H) • G H ⊃ (K $\lor$ I) J • ∼I / I $\lor$ K

Instructions: For 11-20, use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, provide a consistent valuation. -D ≡ (A • B) D $\lor$ (∼E • F) ∼E ⊃ A F ⊃ B A $\lor$ B ∼A $\lor$ ∼B

Instructions: For questions 11-20, construct complete a truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers. -(D ⊃ E) • (E ⊃ D) and (E $\lor$ ∼D) • (D $\lor$ ∼E)

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. Justify your answers by appeal to the meanings of those terms. -(A ? ?B) ? ?(B • A)