-If this is true and correct then only the 'elements' belonging to this set (i.e. the 'inner environment', in Herb Simon's terminology for a system) are true, and all 'elements' outside that set (i.e. the 'outer environment', in Herb Simon's terminology for a system) is untrue, if not false.
-What is your opinion? Thanks for the answers! Marc.
Philadelphia, PA
Dear Carvallo,
The pair of concepts "set" and "elements of a set" are mathematical concepts. So far as I am aware, there are very few restrictions on what can count as an element of a set. What is perhaps more familiar is the idea that truths can be elements of sets, or more specifically true (declarative) sentences of some specified language. One may debate, I suppose, the usefulness of such an approach to the concept of truth. But it is not unknown. Still, taking up such an approach is not, in general, a way of finding out what is true and what is not.
But consider, for instance Godel's theorem. What it tells us is that there is no consistent set of axioms for mathematics, including arithmetic, capable of generating all mathematical truths. No consistent set of mathematical truths defined by a set of axioms is all encompassing. For, given any consistent set of axioms for mathematics, it is always possible to prove that there is some additional mathematical truth not included in the set. Interesting result and claim, though very abstract.
This would seem to show that it makes sense to talk of truths (in the plural) as elements of sets. Regarding the specifics of your question, it may be important to distinguish between what is simply true or false, and what is true or false, according to particular systems of axioms or assumptions. Such systems may differ and conflict with one another.
H.G. Callaway
Dear dr. Callaway,
-Thank you for the clear answer! Let me however take only the first and the last (third) paragraph of your answer.
(i) By talking of truths (in the plural) you 'a-priori'ly say that: there are many truths, so truth is a relative concept; but that is not my question! My question regards truth (in the singular) in general, that is the essence of my question.
(ii) By taking of truths (in the plural), viz. as elements of sets, you already say that truths (not my question!) have the feature of 'elementariness'! Which? The essence of my question is truth (in the singular, see (i)), not what is true or false according to particular systems of axioms or assumptions; this latter would engender the relativism of truth (see again (i)).
(iii) Is there not truth (not 'a' truth!!!) that is universally valid, at least scientificly? That is the essence of my question (see again (i)).
-My next questions (for everyone) would then look as follows:
[1] Is 'truth' severable or at least distinguishable?
[2] Is 'truth' (or 'falsehood') according to the established mainstream science (viz. the s.c. 'normal' or 'standard' science) itself not grounded on 'assumed' and 'axiomatized' systems?
Thanks for your time to answer! Marc.
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