MHB Propositional Calculus: Is Algebra's Problem-Solver? Example

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The discussion centers on the relationship between algebra and propositional calculus, questioning whether every algebraic problem has a corresponding propositional logic problem. Participants express skepticism about the direct correspondence, emphasizing the need for precise definitions of "algebra" and "corresponding." They note that while first-order logic encompasses propositional and predicate calculus, translating first-order statements into propositional logic is complex due to its limited expressiveness. An example is provided to illustrate how a proof in ordered fields could be represented in propositional logic, but the clarity of such translations remains debated. Overall, the conversation highlights the challenges in establishing a clear link between algebraic proofs and propositional calculus.
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Is there to every problem in Algebra a corresponding problem in propositional calculus??

Give an example
 
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I don't think so, but one has to define "algebra" and "corresponding" more carefully. For example, I don't know what propositional formula would correspond to the fact that a polynomial of degree $n$ in a field has at most $n$ roots.

However, I heard that logicists tried to reduce all mathematics to logic, and in particular they thought they reduced true arithmetic identities on natural numbers without variables (like 7 + 5 = 12) to propositional tautologies.
 
Evgeny.Makarov said:
I don't think so, but one has to define "algebra" and "corresponding" more carefully. For example, I don't know what propositional formula would correspond to the fact that a polynomial of degree $n$ in a field has at most $n$ roots.

However, I heard that logicists tried to reduce all mathematics to logic, and in particular they thought they reduced true arithmetic identities on natural numbers without variables (like 7 + 5 = 12) to propositional tautologies.

You mean logicians?
lLet us be more precise. Let us say ordered fields(without the axiom of continuity).Theoretically speaking since every proof in ordered fields is based on 1st order logic which concists of propositional and predicate calculus ,there should be a propositional proof corresponding to every proof.

For example in the following simple proof what is the relevant propositional proof ccorresponding to that proof??

a>1 & b>2 => a>0 &b>0 => ab>0 => ab>= 0
 
solakis said:
You mean logicians?
No, I mean supporters of logicism.

solakis said:
Let us say ordered fields(without the axiom of continuity).Theoretically speaking since every proof in ordered fields is based on 1st order logic which concists of propositional and predicate calculus
Yes.

solakis said:
there should be a propositional proof corresponding to every proof.
It is not clear how to translate first-order statements, let alone proofs, into propositional logic. First-order logic is used for a reason, because it is much more expressive.
 
Evgeny.Makarov said:
No, I mean supporters of logicism.

Yes.

It is not clear how to translate first-order statements, let alone proofs, into propositional logic. First-order logic is used for a reason, because it is much more expressive.

Why,if in the previous proof we put:

a>1=A, b>2=B, 1>0=C, 2>0=D, a>0=E,b>0=F, ab>0=G, (ab=0)=H,then the proof in the propositional logic corresponding to the above proof is it not the following??

IF,
1. A^C=>E
2. B^D=>F
3. E^F=> G
4.D
5.C
THEN A^B=>GvH
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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