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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that any axioms, understood in that sense, exist.
In mathematics, axioms are not self-evident truths. They are of two different kinds: logical axioms and non-logical axioms . Axiomatic reasoning is today most widely used in mathematics.
The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the philosophers of the ancient Greeks an axiom was a claim which could be seen to be true without any need for proof.
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms.
These are formulas which are valid, i.e., formulas that are satisfied by every model (a.k.a. structure) under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values.
Now, in order to claim that something is a logical axiom, we must know that it is indeed valid. That is, it might be necessary to offer a proof of its validity (truth) in every model. This might challenge the very classical notion of axiom; this is at least one of the reasons why axioms are not regarded as obviously true or self-evident statements.
Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in any universe, they will always hold no matter what values are assigned to the variables. Thus, this notion of axiom is perhaps the closest to the intended meaning of the word: that axioms are true, no matter what.
An example, used in virtually every deductive system , is the:
Axiom of equality.
In this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by "=" (or for all what matters, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol "=" has to be enforced - and mathematical logic does indeed that, properly delegating the meaning of "=" to axiomatic set theory.
Another, more interesting example, is that of:
Axiom of universal instantiation. Given a formula in a first order language , a variable and a term that is substitutable for in , the formulais valid.
This axiom simply states that if we know for some property , and is particular term in the language (i.e., it stands for a particular object in our structure), then we should be able to claim .
Likewise, we have the:
Axiom of existential generalization. Given a formula in a first order language , a variable and a term that is substitutable for in , the formulais valid.
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as algebraic groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.
Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and fomalized down to the bare language of logical formulas. This turns out to be impossible and proved to be quite a story.
This is the role of non-logical axioms, they simply constitute a starting point in a logical system. Since they are so fundamental in the development of a theory, it is often the case that they are simply referred to as axioms in the mathematical discourse, but again, not in the sense that they are true propositions nor as if they were assumptions claimed to be true. For example, in some groups, the operation of multiplication is commutative; in others it is not.
Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.
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