0

From various Wikipedia pages and Stack Exchange questions, I have come to understand that "A is stronger than B" means that A can prove everything B proves, and then some. I am aware that this is vague, and the only page I could find that specifically discussed the strength of theories was stronger than Presburger Arithmetic but weaker than Peano Arithmetic. The only solid piece of information I could glean from that page was that, when it comes to theories, there are a lot of different and conflicting ways you could say one theory is stronger than another.

But my question pertains to this part of Wikipedia's page on elementary theories...

an elementary theory is one that involves axioms using only finitary first-order logic

...and this part of Wikipedia's page on algebraic theories:

Sentential logic is the subset of first-order logic involving only algebraic sentences

The page on elementary theories explicitly states that,

Saying that a theory is elementary is a weaker condition than saying it is algebraic.

I'm very uneducated, but it seems like Wikipedia is telling me that propositional logic is stronger than predicate logic somehow. Is this incorrect, or is there a meaning of "stronger" in this context that I'm missing?

Community
  • 1
sdleihssirhc
  • 121
  • In this setting, that a condition is weaker means that it is less restrictive. An algebraic sentence is a particular kind of first-order sentence, so any algebraic theory is automatically elementary. However, many elementary theories are not algebraic and many first-order statements cannot be derived solely from algebraic "axioms". I am not sure why you mention propositional logic here, it seems extraneous to the discussion. – Andrés E. Caicedo Mar 15 '17 at 05:06
  • Ah yes, I see my mistake – sdleihssirhc Mar 15 '17 at 05:33

1 Answers1

0

Although "sentential logic" is another term for the propositional calculus... and even though the "sentential logic" link goes to the page for the propositional calculus... it refers here to equations involving free variables; i.e., the variables refer to objects, not propositions

sdleihssirhc
  • 121