# Logical implication in an empty theory

How does implication work if we have an empty theory?

For example, given an empty theory Γ = { } and two propositional atoms A and B, can we say that Γ⊨A→B? If no, which logical conclusions (involving A and B) can we get from Γ?

• The theorems of an empty theory are the tautologies. Sep 26, 2022 at 17:02
• But no formula "p to q" can be taut if p and q are distinct atoms. Sep 26, 2022 at 17:03
• If the theory is empty what are "A", "B" and "→"? You need at least some language defined to write formulas and derivation rules and/or semantics to use connectives and ⊨. Sep 26, 2022 at 17:57
• @Conifold, A, B, and -> would be parts of the language. My reading of the question is that he is asking about a theory with no axioms, not a theory over an empty language. Sep 26, 2022 at 23:25
• @DavidGudeman An empty theory cannot depend on language. As soon as the theory is dependent on language, for example `A`, the theory is not empty: it holds the proposition `A`. Sep 27, 2022 at 11:24

In formal logic, a theory is a set of sentences. (Sentence here is a term of art meaning a well-formed formula with no free variables.) Usually a theory is closed under the relation of logical consequence of some underlying logic. So, if the theory is axiomatizable, the theory contains all the axioms together with all their logical consequences.

A theory can be 'empty' in the sense of having no axioms, but as soon as you close it under logical consequence, it contains all the sentences that are the tautologies or logical truths of the logic itself.

If Γ is a theory with no axioms, and A and B are specified to be distinct atomic sentences, then the sentence A → B is not a theorem of Γ, since it is not a tautology. This does not mean that A → B is false, it just means that it is independent of the theory. Its truth depends on the truth values of A and B, and the theory has nothing to say about those.

Sentences that are theorems include:

1. ¬(A ∧ B ∧ ¬A)

2. A ∨ B ∨ ¬A

3. (A → B) ∨ (B → A)

In other words, any tautology of the underlying logic is a theorem. I'm assuming here that we are using classical logic, and → is the material conditional.

How does implication work if we have an empty theory? For example, given an empty theory Γ = { } and two propositional atoms A and B, can we say that Γ⊨A→B? If no, which logical conclusions (involving A and B) can we get from Γ?

You cannot infer anything from nothing. No premise, no conclusion.

You have all the usual logical truths involving two propositions A and B, such as the Modus ponens (A → B) ∧ A ⊢ B etc., but if A and B are not assumed, then A → B and B → A are both false.

• The OP says: "two atoms". Maybe this is relevant... You say "two propositions A and B [...] then A → B and B → A are both false." Except in case A and B are the same. Sep 27, 2022 at 10:29
• @MauroALLEGRANZA "Except in case A and B are the same" If you assume that then this is one premise and then your Γ is not empty. Sep 27, 2022 at 11:32