Predicate logic is a development of propositional logic,
which should be familiar to you. In proposition logic a
fact such as ``Alison likes waffles'' would be represented
as a simple atomic proposition. Lets call it P.
We can build up more complex expressions (sentences) by combining
atomic propositions with the logical connectives
and
. So if we had the proposition
Q representing the fact ``Alison eats waffles'' we could
have the facts:
P Q : ``Alison likes waffles or Alison eats waffles''
P Q : ``Alison likes waffles and Alison eats waffles''
Q : ``Alison doesn't eat waffles''
P Q : ``If Alison likes waffles then Alison eats waffles''.
In general, if X and Y are sentences in propositional logic, then
so are X Y, X
Y,
X, X
Y, and
X
Y. So the following are valid sentences in the logic:
P Q
P (P
Q)
(Q R)
P
Propositions can be true or false in the world. An intepretation function assigns, to each proposition, a truth value (ie, true or false). This interpretation function says what is true in the world. We can determine the truth value of arbitrary sentences using truth tables which define the truth values of sentences with logical connectives in terms of the truth values of their component sentences. The truth tables provide a simple semantics for expressions in propositional logic. As sentences can only be true or false, truth tables are very simple, for example:
In order to infer new facts in a logic we need to apply inference rules. The semantics of the logic will define which inference rules are universally valid. One useful inference rule is the following (called modus ponens) but many others are possible:
a, a b
---
b
this rule just says that if a b is true,
and a is true, then b is necessarily true. we could
prove that this rule is valid using truth tables.