Rule of
Inference: Statement to the
effect that if one has asserted (said, written) one or more of certain kinds
of statements, collectively called the premisses, one may, therefore, assert
(say, write) some kind of statement, called the conclusion.
Human beings use and rely on many kinds of rules
of inference in their speaking, writing and argueing, but often  when not
much concerned with logic  are not much aware of this, and anyway would find
it difficult to precisely state the rules of inference they use.
Here is an example of a rule of inference for the term "and",
when used as a term that connects two statements into a statement, that is
used and relied upon by the vast majority of English speakers: "If p and q are
two arbitrary statements, and one has asserted that p and q is true, then one
may assert that p is true".
Any rule of inference is valid or invalid, and is so with respect to
some interpretation of the
terms in the rule, that state what
the terms mean. A rule of inference is valid iff its conclusion is true in
every case that all its premisses are true. Consequentely, a
valid rule of
inference has the property that, for the given interpretation, it can conclude
only true statements from true
premisses.
A rule of inference is invalid iff it is not valid.
In general in logic and mathematics, deductively
valid inferences
correspond to valid implications, but do
not reduce to the latter, in as much as they add a permission to
infer (write, assert) a
conclusion if it is logically implied by
earlier proved or assumed formulas.
