|Axiom: principle or rule for linking elements of a theory that is not proven within the theory. It is assumed that axioms are true and evident. Adding or eliminating axioms turns a system into another system. Accordingly, more or less statements can be constructed or derived in the new system. > System.|
Books on Amazon
Axioms/Intuition/Bigelow/Pargetter: nevertheless, intuitions should not be allowed to throw over entire axiom systems. E.g. the principle of distribution of the disjunction can be explained as follows: Suppose that in natural languages a conditional "If A, then B" is equivalent to a quantification over situations:
"In all situations where A applies, B also applies."
Then you could read the distribution of the disjunction like this:
(x)((Ax v Bx) would > would Cx) (x) (Ax would > would Cx) u (x)(Bx would > would Cx)).
This is indisputably logical!
Bigelow/Pargetter: therefore the quantified form seems to capture the everyday language better than the unquantified. E.g. "In any situation where you would eat..." This is then a logical truth.
This again shows the interplay of language and ontology.
Axioms/Realism/Bigelow/Pargetter: our axioms are strengthened by a robust realistic correspondence theory. And this is an argument for a conservative, classic logic.
Theorems/Bigelow/Pargetter: Need a semantic justification because they are derived. This is the foundation (soundness).
Question: Will the theorems also be provable? Then it is about completeness.
Axioms/Axiom/Axiom system/Axiomatic/Bigelow/Pargetter: can be understood as a method of presenting an interpretation of the logical symbols without using a meta-language (MS). That is, we have here implicit definitions of the logical symbols. This means that the truth of the axioms can be seen directly. And everyone who understands it can manifest it by simply repeating it without paraphrasing it.
Language/Bigelow/Pargetter: ultimately we need a language which we speak and understand without first establishing semantic rules.
In this language, however, we can later formulate axioms for a theory: that is what we call
Definition "extroverted axiomatics"/terminology/Bigelow/Pargetter: an axiomatics that is developed in an already existing language.
Definition introverted axiomatics/terminology/Bigelow/Pargetter: an axiomatics with which the work begins.
Extrovert Axiomatics/Bigelow/Pargetter: has no problems with "metatheorems" and no problems with the mathematical properties of the symbols used. We already know what they mean.
Understanding and accepting the axioms is one thing here.
That is, the implicit definition precedes the explicit definition. We must understand what we are working with.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990