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I will try to explain in other words. Given a formal theory of absolute geometry, we are able to write the expression whose meaning is the paralel axiom.

The question is, that expression is a "proposition"? AFAIK a "proposition" in propositional logic is a "thing" that is true or false. It cannot be undecided.

On the other side, we can add this expression as an axiom and we will have euclidean geometry.

Now I guess there is no issue to consider that these expression is a proposition.

Probably I need some precise definitions of what a proposition is, what a predicate is, what an axiom is, how a language is defined, ...

Eduard
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  • As you pointed out i suggest you to learn some logic before dealing with such questions. You will find that your stated question is reduced to find alternative (but equivalent) set of axioms for a theory. – No Signals Apr 24 '22 at 10:25
  • Any link where it is clearly expressed the diference between expresions and propositions? – Eduard Apr 24 '22 at 12:51
  • You have to consider the difference between a proposition in prop logic and a sntence in natural language. – Mauro ALLEGRANZA Apr 24 '22 at 13:53
  • In the formal language of predicate logic we call sentence a formula with no free variables, like $\forall x (x=x)$ – Mauro ALLEGRANZA Apr 24 '22 at 13:57
  • If we formalize elementary geometry with the language of predicate calculus, the axioms are sentences. – Mauro ALLEGRANZA Apr 24 '22 at 13:58
  • In formalized mathematical theories we may have undecidable sentences. – Mauro ALLEGRANZA Apr 24 '22 at 14:00
  • "If we formalize elementary geometry with the language of predicate calculus, the axioms are sentences." Given this particular elementary geometry formalized with the language of predicate calculus, is any axiom a proposition too? "A1 v A2" (meaning axiom1 or axioms2) is a sentence? Is it true? Is a proposition? – Eduard Apr 24 '22 at 14:18
  • A sentence of a formalized theory has a definite truth value in each interpretation; changing interpretation, the truth value may change. Consider the sentence expressibg "there ate two objects": it will be false in some interpretation and true in other. – Mauro ALLEGRANZA Apr 24 '22 at 14:27
  • And yes, axioms are true in the models of the theory. – Mauro ALLEGRANZA Apr 24 '22 at 14:28
  • "A sentence of a formalized theory has a definite truth value in each interpretation". Then the sentence "paralell axiom" (there is one single line that goes throught this point and doesn't cross this given line) in a formalized theory of absolute geometry has a definite truth value? Sorry to insist... I don't see what I am doing wrong... – Eduard Apr 24 '22 at 14:33
  • If we consider the parallel postulate and we interpret it in the physical space of everyday experience, it holds. If we interpret in the cosmic space of general relativity, it does not. – Mauro ALLEGRANZA Apr 24 '22 at 14:35
  • An interpretation must decide the truth value of any sentence (including the undecidable ones)? – Eduard Apr 24 '22 at 14:37
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    Truth is about the relationship between language and world: we expect that a statement has a definite truth value in the world (the model of the theory). But this does not imply that the axioms and the underlying logic are sufficient to formally prove (or disprove) it. – Mauro ALLEGRANZA Apr 26 '22 at 07:47
  • The parallel postulate is independent of Absolute geometry (i.e. undecidable) because we have models of absolute geometry where the parallel postulate holds (Euclidean gem) and others where it does not (non-Euclidean geom) [we have already discussed it above]. – Mauro ALLEGRANZA Apr 26 '22 at 12:33
  • See Formal definition of proposition and Sentence vs proposition as well as Defining what a proposition is is in propositional logic – Mauro ALLEGRANZA Apr 26 '22 at 12:38
  • See also What is the difference between a statement and proposition? as well as Is "This sentence is true" a proposition? – Mauro ALLEGRANZA Apr 26 '22 at 12:39
  • "Truth is about the relationship between language and world: we expect that a statement has a definite truth value in the world (the model of the theory)." If I believe that in my world a statement could remain unverificable forever (neither true nor false) should I ban clasical propositional calculus and replace it with Kleene three valued logic for example? And then think about a predicate calculus that follows Kleene logic etc etc... If I do that, provability and thruth could become equivalent again? (like my oldself before reading about Godel, Cantor and so on...) – Eduard Apr 26 '22 at 14:46
  • In fact in quantum mechanics, the position and the momentum of a particle cannot be fully interpreted. There is some kind of built-in undecidability in the quantum world... – Eduard Apr 26 '22 at 14:57
  • After some more thoughts, it is not that easy to build some kind of deductive systems with a three valued logic. If (U=>U is U) and (T and U is U) , modus ponens is not a tautology anymore... Which rules of inference can we use? – Eduard Apr 26 '22 at 20:41

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