PHIL 216

Neither...nor....

Both it is not the case that....and it is not the case that... In symbols: ~(A ∨ B) = ~A & ~B Hence "not either A or B" means "both not A or not B" = each one of the two things in question does not have the property.

Truth-functional

Certain compound statements are truth-functional. A compound statement is truth-functional if its truth value is completely determined by the truth value of the atomic statements.

Since a contradiction by itself is an inconsistent set, a contradictory premise will automatically render an argument technically valid.

Consider: ◦ Godzilla is bigger than Mothra ◦ Mothra is bigger than Ultraman ◦ Ultraman is bigger than Godzilla ◦ I like pie. Since this argument's premises are inconsistent, there will never be a case in which all of the premises are true while the conclusion is false (because there will never be a case in which all of the premises are true). That means that this argument too is technically valid

Truth-functionally Equivalent

P and Q are truth-functionally equivalent iff P ≡ Q. P and Q are truth-functionally equivalent iff {~ (P ≡ Q)} is truth-functionally inconsistent.

p unless q (i.e. Car will break down unless oil is changed regularly)

~p ⊃ q: If the car does not break down, then oil was changed regularly. ~q ⊃ p: If oil was not changed regularly, then the car will break down. p ∨ q: Either the car will break down or oil was changed regularly.

If and only if (≡) Paraphrase the sentence "John will get an A in the course if and only if he does well on the final exam."

(A ⊃ W) & (W ⊃ A) Equivalently, (A & W) v (~A & ~W), which says "Either (both John will get an A and John does well on the final exam) or (both it is not the case that John get an A and it is not the case that John does well on the final exam)."

B if A is a common variant on what?

1. "If A, then B," A ⊃ B. 2. Emphasizes that A is a sufficient condition (or enough) for B. 3. Being at least seven feet tall is sufficient for being at least six feet tall.

B only if A is a common variant on what?

1. "If B, then A", B ⊃ A. 2. Emphasizes that A is a necessary condition (or requirement) for B 3. Registering for the class is a necessary condition for passing that class. (Register for the class: A; Passing the class: B)

An argument can be valid even when

1. All of its premises are false. 2. Its conclusion is false. Ex. Every Western student has four legs. (False) Everything that has four legs has three legs. (True) So, every Western student has three legs. (False) 3. There is falsehood all the way through it. Ex. All electrons are elephants. (False) Obama is an electron. (False) So, Obama is an elephant. (False)

Truth-functionally consistent

1. Apply to a set of sentences while validity applies to an argument or a sequent. 2. A set of sentences is consistent if and only if there is at least one assignment on which all members of the set are true.

Truth functionally equivalent

1. Pertain to two sentences, i.e "A & A" and "A ∨ A". 2. Sentence P is truth-functionally equivalent to Q iff there is no truth-value assignment on which P and Q have different values.

An argument can have all true premises and a true conclusion and yet not be valid. Ask: If we have a truth table with all truth value assignments for both premises and conclusion --> what can we conclude about validity? Is it not invalid, so is it valid then? but wouldn't that contradict with the above statement?

1. Some Washingtonians have hiked the Hoh River Trail. (True) 2. Some Washingtonians have hiked the PCT. (True) 3. So, some Washingtonians have hiked both the Hoh River Trail and the PCT. (True) ASK: Why is this argument not valid?

Truth-functionally false vs. Inconsistency

A sentence P is truth-functionally false iff {P} is truth-functionally inconsistent.

Truth-functionally true vs. Inconsistency

A sentence P is truth-functionally false iff {~P} is truth-functionally inconsistent.

Truth-functionally indeterminate vs. consistency

A sentence P is truth-functionally indeterminate iff both {~P} and {P} are truth-functionally consistent.

Logical truth

A sentence is logically true if and only if it is not (logically) possible for the sentence to be false.

Material conditional: "If A then B" can be alternatively paraphrased in either...or... in what way?

Either not A or B (~A ∨ B)

If an argument is valid and has a true conclusion, then it must have all true premises.

False

If an argument is valid and has at least one false premise then its conclusion must be false.

False

True or False: Some arguments are false.

False

To say that an argument is sound ASK: An argument is sound just in case it is valid and every premise is a well-known truth. (False)

Is just to say that it is valid and all the premises are true.

An argument is valid when

It is impossible for the conclusion to be false while the premises are true, i.e "truth-preserving." An argument is valid just in case, for any assignment, if the premises are true in this assignment, then the conclusion is true as well. Validity is not about the actual truth or falsity of the premises and the conclusion. Validity is about the relationship between the premises and the conclusion.

Not both...and...

It is not the case that both....and.... ~(A & B) = ~A ∨ ~B Hence "not both have" means "either one does not or the other does not have" = at least one of the two things does not have the property

An argument is invalid when

It is possible for the conclusion to be false while the premises are true.

Logical falsity

This is the opposite of logical truth. Sentences that cannot (logically) be true are called logically false.

A conditional statement is an if-then statement

True

In a sound argument the conclusion is true.

True