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

[solakis1]

Is there to every problem in Algebra a corresponding problem in propositional calculus??

Give an example

 

[Evgeny.Makarov]

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.

 

[solakis1]

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

 

[Evgeny.Makarov]

You mean logicians?

No, I mean supporters of logicism.

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.

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.

 

[solakis1]

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