If we have another proposition b such as ‘Ethel is a cat’ then we can combine
these two propositions in several ways. One combination is written a V b. The
connective V corresponds to ‘or’ but its use in logic is slightly different from ‘or’ in
everyday language. In logic, a V b is true if either ‘Freddy is a spaniel’ is true or
‘Ethel is a cat’ is true, or if both are true, and it is only false when both a and b
are false. This conjunction of propositions can be summarized in a truth table.
Implies truth table
We can also combine propositions using ‘and’, written as a⋀b, and ‘not’,
written as ¬a. The algebra of logic becomes clear when we combine these
propositions using a mixture of the connectives with a, b and c like a ⋀ (b Vc).
We can obtain an equation we call an identity:
a⋀(b V c) = (a ⋀ b) V (a V c )
The symbol ≡ means equivalence between logical statements where both sides
of the equivalence have the same truth table. There is a parallel between the
algebra of logic and ordinary algebra because the symbols Λ and V act similarly
to × and + in ordinary algebra, where we have x × (y + z) = (x × y) + (x × z).
However, the parallel is not exact and there are exceptions.
Other logical connectives may be defined in terms of these basic ones. A
useful one is the ‘implication’ connective a→b which is defined to be equivalent
to ¬ a ⋀ b and has the truth table shown.
Now if we look again at the newspaper leader, we can write it in symbolic
form to give the argument in the margin: