Chapters 3 & 4: The Boolean Connectives and the Logic of Boolean Connectives

truth functional connectives
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Terms in this set (19)
truth functional connectives
logical connectives which connect sentences in such a way that the truth-value of the compounded sentence is a function of the truth-value of the constituent sentences
Negation
"not, it is not the case that, non, or un"
Conjunction (^)
"and, moreover, but..."

Game Rule: if at least one of P or Q is false, then P ^ Q is false.
Both P and Q must be true for the sentence to be true.
Image: Conjunction (^)
Disjunction (V)
"or"
"Either...or..."
"Neither...nor..."

Game rule: The sentence is true if at least on the the sentences is true. If both sentences are false then the whole sentence is false.
Image: Disjunction (V)
Inclusive disjunction
at least one and possibly. both of the two disjoined sentences is true.
DeMorgan's Law
The negation of a conjunction, ~(P ^ Q), is logically equivalent to the disjunction of the negation of the original conjuncts: ~P V ~Q.

The negation of a disjunction, ~(P V Q), is logically equivalent to the conjunction of a negation the original disjuncts: (~P ^ ~Q).

~(P ^ Q) = (~P V ~Q)
~(P V Q) = (~P ^ ~Q)
truth conditions
For an fol sentence to be a good translation of an English sentence, it is sufficient that the two sentences have the same truth values in all possible circumstances
Logical truth
Sentence that is a logical consequence of any set of premises.

No matter what the premises may bee it is impossible for the conclusion to be false.
(logical necessity)
logical possibility
the is some logically possible circumstance in which the situation is true
(not physically possible to go as fast as the speed of light but they do it on Star Trek)
logical necessity
if the sentence is true in every logically possible circumstance (identity: a=a)
tautologyA sentence that is logically true in virtue of its truth-functional structure. A tautology is a sentence that comes out true under every row of its truth table Any tautology is logically necessary. Not all logically necessary claims are tautologies.A sentence S is TT-necessary if and only ifthe truth table for S is T in EVERY row under its main connective.A sentence S is TT-possible if and only ifthe truth table for S is T in at least ONE row under its truth tablelogicallly equivalent sentencessentences that have the same truth value in every possible circumstance. sentences have the same truth conditions, since the conditions under which they came out true and false are identical.tautologically equivalentlogically equivalent sentences whose equivalence is due to the meaning of the truth functional connectives they contain. to check of TE, if two sentences on a joint truth table have main connective columns which are completely identical, then the sentences are TE.tautological vs logical equivalenceAll tautologically equivalent sentences are logically equivalent. However, logically equivalent sentences are NOT always tautologically equivalent:logical consequencea sentence, S, is a logical consequence of some premises if S is true in every logically possible situation in which the premises sentences are true or equivalently, if there is no possible situation in which the premises are true and the conclusion false.tautological consequenceS is true in every row of the truth table that makes the premise sentences true. "all we care about is the rows in which P is true. For each row in which P is true, reference the value of Q in that row, and if it is false, we know that Q is not a tautological consequence f P."A connective is truth-functional if and only ifthe truth value of a complex sentence-formed by using the connective - is a function of the truth values of the solo sentences.