Logic FINAL Flashcards
PASS
Validity
iff it is impossible to make all of the
premises true and the conclusion false.
Jointly Contrary
where a sentence can be proven to be a contradiction from assuming all the sentences given
To show that some sentences A1, A2, … An are jointly contrary, construct a proof that takes A1, A2, … An as assumptions and concludes with ⊥
Modal Logical Truth
An ML sentence A is a MODAL LOGICAL TRUTH iff A is true at every world in every interpretation
Provably Equivalent
Two sentences are provably equivalent where each can be proved from the other
- provide a pair of proofs: one starts with A and concludes with B, the other starts with B and concludes with A
formal proof
a sequence of sentences some of which are
marked as assumptions (premises) and the last line of which is a conclusion, which shows how to derive conclusion from premises using good inference forms
Theorem
A is a THEOREM if and only if there is a proof of A from no assumptions (⊢ A)
extensionally equivalent
where two statements have identical truth values under all possible interpretations in a truth table.
logically equivalent
Two sentences are logically equivalent iff they are provably equivalent
Expression of ML
Any string of symbols in ML
Modal contingent truth
is a statement that could be true and could be false
Modal necessarily false
is a statement that is false in every possible world
Valuation Function
a function that assigns truth-values to atomic sentences at a world
Modally valid
iff there is no world in any interpretation in which all the premises are true and the conclusion is false
MODAL CONTRADICTION
An ML sentence A is a modal contradiction iff A is false at every world in every interpretation
Modally consistent
An ML sentence A is MODALLY CONSISTENT iff A is TRUE at some world in some interpretation
