Chapters 1-4 Test Review!

Validity and Tautologies

100

What do we call a formula that has every entry in its truth table’s final column as true?

tautology

200

If formula A is a tautology, what can be said about the negation ~A?

~A is a contradiction

300

What is the definition of validity?

an argument is valid if and only if there is no case where all premises are true and the conclusion is false

400

Explain why an argument with all true premises and a false conclusion is always invalid.

it violates the definition of validity, allowing the conclusion to be false despite true premises

500

What type of formula has a truth table with a mixture of true and false entries in the final column?

a contingent formula

Implication and Equivalence

100

How do you test if formula A logically implies formula B using a conditional?

check if the conditional A → B is a tautology

200

Are formulas A and B logically equivalent if A ↔ B is a tautology?

yes

300

Does ~P logically imply ~ (P & Q)?

no

400

does ~ (P & Q) logically imply ~P?

no

500

A and B are logically equivalent... Can we conclude that A --> B, B-->A, A <-> B, All of these, or None of these?

All of these

Sentential Logic Formulas

100

What is the truth value of the conjunction P & Q when P is true and Q is false?

false

200

Write the symbolic formula for the negation of the conjunction of P and Q.

~ (P & Q)

300

Which is logically equivalent to ~ (P & Q): ~P & ~Q or ~P ∨ ~Q

~P ∨ ~Q

400

Is ~ (P → Q) logically equivalent to P & ~Q? Explain.

yes, this can be shown with truth tables

500

What is the truth table for the biconditional P ↔ Q?

P ↔ Q is true if and only if P and Q have the same truth value, otherwise false

Translations and Paraphrases

100

Translate “Jay and Kay are cousins” into sentential logic. Explain why it cannot be symbolized as J & K.

It must be represented as a single atomic formula C because it expresses a relation, not independent statements about Jay and Kay

200

Translate “Jay and Kay are not both Republicans.”

~ (J & K)

300

Translate “If Jones is a Republican, then Smith is a Democrat only if Lee is a Republican.”

J → (~L → ~S)

400

Translate “I will pass unless I goof off, provided that I am intelligent.” 

I → (~G → P)

500

“If the exam is easy, I will get a hundred only if I study.”

E → (~S → ~H)

Non-Standard Connectives

100

How do you symbolize “A unless B”?

~B → A

200

How do you symbolize “P only if Q”?

~Q → ~P

300

How would you symbolize "Neither A Nor B"?

~A & ~B

400

Translate “D unless E, in which case F”

( ~E → D) & (E → F)

500

How would you translate "A is not necessary for B"?

~(~A → ~B)