Axioms Part 4: Commentary on Logical Axioms
(this page can easily be skipped by noting that there are axiomatic assumptions at the basis of logic)

In this part of the discussion, I do not seek do make any inferences about mathematical logic. I am only concerned with logically constructed declarations in natural languages such as English.

Logical systems are akin to other axiomatically defined systems in that they can be modified and indeed they have been many times and for different purposes. Of necessity, the potential starting points in logic vary between systems and are quite abstract.

Since logic is merely concerned with the relationship of entities the axioms for logical systems are mostly to do with the nature of the logical connections (connectives), logical conclusions (or outcomes) and qualifiers of those conclusions. Possibility and necessity, in the context of logic, are examples of qualifiers that can vary what outcome of a logical argument that we are prepared to accept. Nevertheless, within a given system of logic, we need some primitive declarations of existence.

A very informal description concerning the nature of axioms (or core beliefs) that might with respect to some form of classical logic could be a follows:

a) Conceptually separate entities exist. (Entities, in this context, are things, abstract things, or properties of both types of things.)

b) The entities have an identity or distinctness. (the law of identity).
Entities need only be represented in a non-descriptive or symbolic way that preserves distinctness.

c) Assertions are constructed by including the types of connections that are allowed between conceptual entities in a particular system of logic. (Relevance, for example, is not usually defined as a logical operator or modifier in classical logic).

d) It is possible to specify a complete list of axiomatic bifunctional connectives and the unitary operator ‘not’ for a given deductive system.

e) Compound connectives may be defined which result from the combination of simpler connections and the unitary operator. Compound statements involving strings of connectives can be constructed.

f) There is an implicit or implied outcome value associated with a logical expression created through the connection of entities.
(In some systems of logic, this is the bivalent property of truth and falsity. In binary arithmetic, this idea is expressed as 0 or 1, where 0 is the logical equivalent of false and 1 is true.

g) The implicitness of a logical outcome (or conclusion) is said to be what we agree on when we define the terms ‘logical implication’ and ‘logical entailment’ within a particular system.

h) A contradiction of outcomes is not permissible (the law of non-contradiction).

i) No inconsistency is allowed. This rule is at least partly be expressed as ‘the law of excluded middle‘, which says that we cannot, hold that an outcome can have opposite binary values at the same time. The conclusion is either one outcome or the other but not both at the same time.

j) There exist qualifiers of logical statements (know as modals) that concern the conditions under which logical conclusions can be accepted.

In some defined forms of betting, like card games or roulette, we are not primarily concerned with what the false or true value of an assumption might be. Instead, if we are determined to win by ‘playing the odds’, we would want to understand or compute (using binary logic) what the probability of a particular outcome or range of outcomes could be.

In contrast to the situation described in the list above, which applies to classical logic, there are some situations in life where we can accept that holding contradictory binary or apparently opposing beliefs, seems to make sense. For example, in chemotherapy of cancer, we can accept that a mutagenic treatment agent can kill already mutated aberrant cells with some dose-related probability and simultaneously also create new carcinogenic mutations in otherwise healthy cells. It can, therefore, be seen that the binary treatment properties of bad or good, beneficial or harmful, life-preserving or life-threatening can be held simultaneously as valid. 

There is an obvious way out of this apparent dilemma.  We believe that one good outcome is possible with one probability (for example 70%) and that the bad outcome is associated with another (for example 2%). Clearly, both probabilities need not add up to 100% as they refer to different outcomes, which are not truly independent.

As the above example demonstrates in a non-classical alternative, we can also choose an outcome value to be a point on a continuum, such as a variable probability that is expressed as a decimal number anywhere between 0 and 1 or as a percentage.) More generally, one interpretation of probabilistic belief is that we are to some variable extent holding true and false values simultaneously.

Steve Campbell
Glasgow, Scotland

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