How do we know what we claim to know? We wish to begin here by assuming as little as
possible. However, we cannot begin by assuming nothing. As soon as a writer sets down his
first word he is already assuming a great deal, whether he acknowledges it or not.
The ability to communicate through **language**, written or spoken, is a very complex achievement.

So how does language work? An **argument** proceeds by a series of **statements**,
which are expressed in **sentences** made up of words, which may be in spoken or written form.

The use of language in a manner seeking to gain agreement or at least tolerance of our
point of view, without the use of force or shouting down the opposition or any more subtle
forms of compulsion, is **reason**. The way we do this is by **reasoned** (or
*rational*) argument in which we set out our arguments in simple stages, seeking to
get agreement that each step is acceptable to our audience. If all the steps are accepted,
then the whole argument should be accepted, or else our opponent is being
**unreasonable** (or *irrational*).

By **truth** we mean suitable to be accepted by our audience. The nature of truth
may thus depend on our audience, some audiences will be prepared to accept our assumptions,
or presumptions, for their beauty, ingenuity, common sense, self-evidence or other qualities, while
others may simply be prepared to suspend judgment, to see what our assumptions lead to,
and some may refuse to accept our assumptions, or may wish to subject our assumptions to
tests of some sort designed to assess their ‘correspondence with reality’.
Statements which have passed the particularly strict tests of ‘scientific method’
may be termed **scientific truth**.

Any audience usually has an accepted body of truths, constituting their collective
**knowledge** (or *belief*) against which the truth or falseness of new propositions
is judged. Historical studies show that what is accepted as true by the scientific
community, or by a religious community, evolves over time. A collective world-view,
against which new propositions are assessed, is known as a **paradigm**. Major new
ideas may require deep changes in the paradigm, that may amount to **paradigm-shifts**
(or *revolutions*) in ideas, often accompanied by revolutions in social
organisation.

The attempt to make reasoning more reliable by paying close attention to the way statements
are formulated and the steps by which arguments are constructed is **logic**.
The most basic kind of logic deals only in **propositions**, that is precisely
defined statements, that are either **true** or **false**. This is **two-valued
logic**. There have been attempts to develop a three-valued logic, but the two-valued
form is so much simpler that it in effect chooses itself for study.

A proposition has a **truth value**, that is it is either true (1) or false (0).
We use these number-symbols for truth values (rather than say the letters T and F which
are often used) since we will eventually develop a more comprehensive system in which
intermediate values can be assigned that measure the **probability** (or *degree of
truth* or *certainty*) that may be assigned to a proposition. It is perhaps
surprising that by using two-valued logic we can develop this more complex system that
can cope with multiple or fuzzy valuations.

Reasoned arguments, at least when tidied up, consist of sequences of propositions
whose truth we assert. An argument to gain acceptance of one proposition from the accepted
truth of others is a **proof**, and a proposition together with a proof of it is a **theorem**.
(A minor theorem that is used in the proof of a more important one is sometimes called a *lemma*,
while a minor theorem whose truth follows in a simple way from that of a major theorem is
called a *corollary* or *rider*.) The propositions with which we start a proof are
termed **assumptions** (or *premises*) and those we finish with are **conclusions**.
The conclusions are said to be **inferred** (or *deduced*) from the assumptions, and
the process is one of **inference** (or *deduction*). A conclusion of one argument
may be an assumption of another.

So what are the simplest steps by which rational arguments proceed? An inference from
one statement to another is an **implication** and takes the form ‘If X then Y’.
It seems apparent that Y must either say the same as X or say less than X, which doesn't
seem to be progress, but in fact the implications of an assumption may not be obvious and
can often be overlooked, particularly special cases. An inference from two statements to
one is a **syllogism** and takes the form ‘If X and Y then Z’. If Z follows
by implication from X or Y alone then this is not a genuine syllogism, since Y or X is not
needed. So it is clear that Z must combine part or all of X with part or all of Y.
A series of syllogisms is called a **sorites** (pronounced sore-eye-tease). Most
arguments however are far more complex than these simple forms.

A long series of connected arguments constitutes a **theory**. An **axiomatic**
development of a subject proceeds by linear or branching chains of arguments that can be
traced back to certain initial assumptions, which are called **axioms** (or *postulates*).
There may however be many different ways of developing a subject in this manner, and
there may not be any one way that is best or right. A non-axiomatic development of a
subject, in which concepts may be circularly related or form a complex network, is also
possible and can be equally logical and correct, so long as the circularity is recognised.

A theory whose axioms contain undefined terms, or whose terms can be interpreted in
different ways, is an **abstract** (or *mathematical*) theory, while one whose
terms get their meaning from outside the system itself, i.e. in the ‘real world’
is a **concrete** (or *scientific*) theory. As a matter of history, the first
abstract theories were developed from concrete examples, but as mathematics developed,
it has often happened that abstract theories have been developed before any applications
of them have been found.

In applying an abstract theory to concrete uses, or in other words using a theory
to **model** a real situation, our success or failure depends on the accuracy with
which the terms of the theory correspond in their theoretical behaviour to the observed
phenomena. One can say all sorts of things in theory, but whether they work in practice
depends on the fit. The fitting of theories more and more closely and economically to
the facts is what **science** is about. Abstract theories can often be applied to
other abstract theories, or indeed to themselves. In fact this process is the principal
way in which we develop complex abstract theories from initially simple beginnings.