Could you give the rules of inference (transformation) that are allowed? I’m guessing double negation introduction and elimination, conjunction introduction and elimination, and hypothetical syllogism all work. Is that right? Are there other valid rules? Does the logic have a standard axiomatization? (Another way to ask this is, “In what sense are the sentences you identified theorems of the logic?”)
One worry. I’m not sure that what you say about explosion here is quite right. You say:
Isn’t the second line a statement of the principle of explosion—the fact that we can derive anything from a contradiction? Indeed it is. LP can state the principle of explosion as a (true) theorem—but it can’t actually use it as a rule of deduction.
The theorem is about a conditional sentence. The principle of explosion is about logical consequence. (Basically, this comes down to the difference between the conditional “->” and the turnstile “|-”.)That is, insofar as the logic does not allow the sentence to function as a rule of inference, the sentence is not a statement of the principle of explosion in that logic. We might recognize it as explosive in classical logic. But in classical logic, we have modus ponens!
The theorem is about a conditional sentence. The principle of explosion is about logical consequence.
Yes, I’m not going to be making that confusion again :-)
The book I was reading refered to sentences like “A ∧ (A → B) → B” as modus ponens theorems; I’ve copied that usage, but I can change it if it’s confusing.
Could you give the rules of inference (transformation) that are allowed? I’m guessing double negation introduction and elimination, conjunction introduction and elimination, and hypothetical syllogism all work. Is that right? Are there other valid rules? Does the logic have a standard axiomatization? (Another way to ask this is, “In what sense are the sentences you identified theorems of the logic?”)
One worry. I’m not sure that what you say about explosion here is quite right. You say:
The theorem is about a conditional sentence. The principle of explosion is about logical consequence. (Basically, this comes down to the difference between the conditional “->” and the turnstile “|-”.)That is, insofar as the logic does not allow the sentence to function as a rule of inference, the sentence is not a statement of the principle of explosion in that logic. We might recognize it as explosive in classical logic. But in classical logic, we have modus ponens!
Yes, I’m not going to be making that confusion again :-)
The book I was reading refered to sentences like “A ∧ (A → B) → B” as modus ponens theorems; I’ve copied that usage, but I can change it if it’s confusing.