Our experience of the world we inhabit leaves us supposing certain things to
be true and aware of certain relationships among these. We have learned to
formalise some of those relationships as inference
, a process whereby,
given some statements
we accept, we can construct new statements and a
case for accepting these also. The givens are then said to imply
the
new. Logic formalises this process of inference: Reason is the tool we use to
decide whether a formalism matches up with our experience (and our intuition, if
you view that as separate).
Consider a context which gives meaning to some texts as statements
,
about which the context recognises is it true ?
as meaningful. To
formalise reason it must also admit some processes of inference and gives
meaning to some texts as statements about inference. Ideally, these last will
suffice to describe all the modes of inference that the context admits: once the
text has introduced these axioms of inference
it can contribute their
meanings to other contexts; thereby providing a common, typically more formal,
idiom for inference.
For example, in this page I shall discuss enough
of logic to do the
reasoning I want: I shall presume little about the context in which you read it,
so my explanations will circle round the subject giving perspective on such
formalism as I introduce. In pages which presume this one as context, on the
other hand, I shall get on and use the logical methods introduced here with
comparative directness - presuming that you have learned, from this one, enough
of what I mean by imply
, and
, or
and not
to be able
to determine whether my proofs and explanations are formally valid (for what
that's worth).
[A good expression of formal logic gives some simple primitive
model of meaning in a preamble, introduces some axioms to which that model gives
meaning, organises these meanings into a context, provides that context as one
in which to read texts and, with the preamble left off, can itself be read in
that context, yielding the same meaning
for the text as was provided by
the primitive model.]
So we have a context which accepts some statements, some of which express accepted modes of inference, and rejects others. Given these, the accepted modes of inference enable us to accept some further statements and reject others. There are then various ways we can formalise this, and these yield descriptions of one another.
The notion of implication expresses if I accept that as true, I'll
also accept this as true
(or I infer this from
that
) in the form that implies this
. [The word
implies
may be replaced by a rightward-pointing arrow (typically
double-shafted, similar to => but neater) for which some browsers recognise
&implies; as an HTML character entity: others will (perfectly reasonably)
display it as &implies;, which I find adequately readable - also,
A&implies;B won't get split across a line, where A implies B
might be:
and some browsers don't honour the non-breaking space
, .]
Implication is quite a good idiom for use in a textual discourse, since it fits
in well with the sequential nature of text.
The notions and
and or
are so deeply wired into the English
language as to make formalisation largely fatuous. I shall only pause to remark
that I use or
in its inclusive
sense: if I accept A I accept A
or B
whether or not I accept B; and if I accept B I accept A or B
regardless of A; hence A and B
does imply A or B
.
I accept the following statements about or
, and
and
implies
(with A, B, C and D arbitrary statements):
Note, by contrast, that I shall not take for granted that A implies (B or
C)
implies (A implies B) or (A implies C)
, among other things.
Note that
(B and C) implies (B or C)
(B or C) implies D(the conclusion of 6) leads us, via 4 and 7, to
B implies D; likewise, via 5 and 7,
C implies D, so the conclusion of 6 implies
(B implies D) and (C implies D), which we now see implies the premis of 6.
A implies (B and C)gives us, via 7 and 1,
A implies Band, via 7 and 2,
A implies Cso 4, like 6, is an equivalence: its conclusion also implies its premis.
If we consider all the statements some context allows us to express, we can look at which of them imply which others.
We can describe the process of inference in terms of rules of logic
which give circumstances under which a body of givens implies some statement.