I am going to make this blogpost a bit different. I am going to make it a bit “fun”, and less proof/theorem.
Any propositional variables can be assigned a truth () or falsehood () value through a mapping . Where is a set of all propositional variables. We can show that it’s more general than this, i.e. can be a set that contains all eff
The values can be retrieved through , where we have
- for some propositional variable
- for some wff .
- for some wffs .
- For some wff , we write and say that is a tautology if for any assignment () I make to the internal statements. That is to say that evaluates to true probably because of its structure and not its content.
- If a wff always evaluates to F, then we say is a contradiction.
- implies iff
- is equivalent to iff is a tautology.
- If an assignment means that is true (need not be always), we say satisfies and we write .
- Every assignment satisfies the empty set of wff) (vacous truth)
- A set of wff is satisfiable iff there exists an assignment that satisfies, otherwise it’s contradictory.
For propositional variable, we’ll have assignments. That is to say we form a set of relations where the size of this set is .
A wff is valid with respect to an interpretation in a logistic system if under interpretation. Furthermore, an interpretation of a logistic system is sound iff, under the interpretation the axioms are valid and the rules of inference preserve the validity.
Soundness Theorem (ST)
Every theorem of our logistic system is a tautology.
Each axiom is a tautology. Futher-on, if , and then . Hence if and are tautologies, then is a tautology.
It makes sense that a system is inconsistent if every wff is a theorem. Because we could have and . Which highlights an inconsistency. Therefore we define an absolutely consistent system as a system where there exists a wff which is not its theorem. We also say that a logistic system is consistent with respect to negation iff there is no wff such that and .
Absolute consistency theorem
Let be a logistic system where is an axiomatic schema and modus ponens is a rule of inference (primitve or derived). Then is absolutely consistent iff is consistent with respect to negation.
I won’t proceed to prove this theorem, but instead, I will just briefly discuss it. I mean it does make sense that if we have a system that is absolutely consistent that is to say that there exists a wff that is not a theorem, I can easily go the other way around for this theorem and say that if I have consistency with respect to negation then a theorem where a proposition implies its negation is not a theorem in the system.
One can then also show that our system in question is absolutely consistent and is thus consistent with respect to negation
All tautologies in are theorems in .
This is a desired result for any logistic system, if I define a statement where states that if n is a natural number, then is an even natural number. To apply this theorem, I’d have to use modus ponens on it given that it holds. We know that it holds because that statement along with its converse is equivalent to the definition of the natural even numbers. So we know it’s a tautology because it’s independent of whatever natural number I choose, I can still apply M.P to it. And all such are theorems in whatever logistic system I’m working in.