Reason

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.

Imply And/Or Not

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):

1. (B and C) implies B
2. (B and C) implies C
3. ((A implies B) and (A implies C)) implies (A implies (B and C))
4. B implies (B or C)
5. C implies (B or C)
6. ((B implies D) or (C implies D)) implies ((B or C) implies D)
7. ((A implies B) and (B implies C)) implies (A implies C)
8. (A and (A implies B)) implies B

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

• applying 7 to 1 and 4, or equally to 2 and 5, we get (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.
• more directly, A implies (B and C) gives us, via 7 and 1, A implies B and, via 7 and 2, A implies C so 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.