Logical validity
Inconsistent arguments are automatically deductively valid.
A is B, so x is y or x is not y – valid.
Three connectives – and, or, not
Transitional chapter to propositional logic (PL).
“Logic is easy, because everything that is difficult, we will ignore” – Laurence.
And – used to connect two expressions of the same grammatical category (nouns, verbs, adjectives, adverbs). Must importantly it can connect two sentences. – Propositional connective. – binary (connects exactly two propositions).
A and B – A^B. A but B – A^B.
‘But’ and ‘and’ have the same meaning to logicians. But = and + contrast. ‘But’ implies only rhetorical nuance, so from a logical perspective we don’t care.
‘And’ nuances
Temporal succession – implies that propositions follow each other in an order – logicians don’t care.
Simultaneity – implies that propositions are happening simultaneously – logicians don’t care.
Causality – implies that A implies B – a vase fell and it broke – logicians don’t care.
We only care about the truth-conditional aspect – meaning that we don’t care about the implied nuances. Nuances require encyclopaedic/world knowledge, we want to avoid that in logic.
Or - used to connect two expressions of the same grammatical category (nouns, verbs, adjectives, adverbs). Most importantly connects propositions.
A or B – the disjunction of A and B.
‘Or’ is a binary connective; ‘or’ is a propositional connective.
If at least one of A and B is true, then A or B is true as well.
Otherwise, i.e., when A and B are both false, then A or B is false.
A or B = A v B (inclusive disjunction)
Inclusive disjunction – true even when both A and B is true.
Exclusive disjunction – true when exactly one proposition is true.
Not – use to deny or negate a sentence.
Careful when using with ‘some’.
Some students are smart.
It is not the case that some students are smart.
Not A = ¬A
Truth conditions:
If A is true, then A is not true is false. Not not a is A.