Important Vocab

atomic sentence

most basic form of a sentence; cannot be further broke up by logical connectives

logical connectives

connect sentences logically to form other sentences

premises

first two sentences of a logical argument

conclusion

third sentence of a logical argument

valid argument

when the conclusion follows from the premises

Invalid argument

Argument where the conclusion does not follow from the premises (true premises and false conclusion)

Counterexample

Case where an argument is invalid

Case

hypothetical scenario where each sentence in an argument is either true or false

object language

language we study

meta-language

language that is used to study the object language

main connective

the last logical connective used in the construction of a sentence

Negation

¬, ‘it is not the case that’

conjunction

∧, and/but

disjunction

∨, either/or

conditional

→, if…then

biconditional

↔, if and only if / iff

Inclusive ‘or’

what disjunction connective represents; either A or B or both

exclusive ‘or’ (XOR)

cannot be used with disjunction symbol; means A or B but not both

truth value

determines if a sentence is true (T) or false (F)

antecedent

beginning of a conditional statement (after ‘if’)

consequent

end of a conditional statement (after ‘then’)

Valuation

assigns truth values (T or F) to sentence letters

Entailment

another way of saying that an argument is valid; A and B entail C = A and B therefore C is valid; works if every valuation makes at least one premise false or makes conclusion true

sound argument

when all the premises of an argument are true and the argument is valid

How to write ‘neither…nor’

(-A /\ -B) or -(A \/ B)

How to write unles A,B

(-A —> B) or (A \/ B)

exclusive or symbolization

(A∨B) ∧ ¬(A∧B)

Tautology

iff A is true on every valuation (column)

Contradiction

is false on every valuation

Equivalence

Iff for every valuation, the truth values of A and B agree, and there is no valuation in which they have opposite truth values

Joint satisfiability

A1, A2,…An are this iff there is some valuation that makes them all true (row)