Is the argument valid or not? Let’s assume the conclusion P is false, but that
all the premises are true. If we can show this forces a contradiction, it means the
argument must be valid. It will then be impossible to have the premises true but
the conclusion false. If P is false, then from the first premise C → P, C must be
false. As C VS is true, the fact that C is false means that S is true. From the third
premise S → H this means that H is true. That is, ¬H is false. This contradicts
the fact that ¬H, the last premise, was assumed to be true. The content of the
statements in the newspaper leader may still be disputed, but the structure of the
argument is valid.
V or
⋀ and
¬not
→implies
for all
there exists
Other logics
Gottlob Frege, C.S. Peirce, and Ernst Schröder introduced quantification to
propositional logic and constructed a ‘first-order predicate logic’ (because it is
predicated on variables). This uses the universal quantifier, ∀, to mean ‘for all’,
and the existential quantifier, ∃, to mean ‘there exists’.
Another new development in logic is the idea of fuzzy logic. This suggests
confused thinking, but it is really about a widening of the traditional boundaries
of logic. Traditional logic is based on collections or sets. So we had the set of
spaniels, the set of dogs, and the set of brown objects. We are sure what is
included in the set and what is not in the set. If we meet a pure bred ‘Rhodesian
ridgeback’ in the park we are pretty sure it is not a member of the set of spaniels.
Fuzzy set theory deals with what appear to be imprecisely defined sets. What if
we had the set of heavy spaniels. How heavy does a spaniel have to be to be
included in the set? With fuzzy sets there is a gradation of membership and the