And yet the metalanguage is always classical logic. Even the most enthusiastic proponents of other systems never use them to talk about those systems. So I go firmly with “classical”.
That seems consistent with my view. For the specific application you mention—talking about logical systems—classical logics are our best models. It could still be the case that other logics are better for other applications. What makes this particular application the trump card, so that the fact that classical logic is best for doing metalogic means that it is the best simpliciter?
First, I shall ask the question “what is logic?” And I shall answer it. In the context of the present poll, “logic” means those methods of reasoning that are guaranteed to produce, from true premises, only true conclusions. And the poll is asking whether classical logic is it.
Particular formalisms used to model particular things are not, in this sense, logic, although they may be expressed in logic. For example, number theory is not logic. Neither is geometry, or physics, or probability theory. Neither, I claim, is fuzzy logic, despite the word “logic” in its name. You can say, “here is a set of functions (which I shall call fuzzy logic truth tables), and here are some theorems about how they behave (which I shall call fuzzy reasoning), and here are some physical systems whose description uses these functions.” That does not mean that those functions are actually a form of logic, as I just defined it. Bang-bang controllers like the room thermostat were invented (in 1883) long before fuzzy control theory (about which I’ve heard anecdotally that the term was invented only to avoid someone’s patent claims).
The closest anyone has come to promulgating an alternative system is intuitionistic logic, which is a pessimistic version of classical logic, in which the axiom of the excluded middle is dropped. In intuitionistic logic, you cannot infer P from not-not-P, or carry out proof by contradiction. However, I think intuitionism is simply a mistake, a historical accident which would never have happened if there had not been a half century between the codification of mathematical logic and the invention of the computer. Everything that is useful in intuitionism is given by computability theory and classical logic.
I agree with you that a logic is an account of truth-preserving inference. But, by this definition, fuzzy logic absolutely qualifies as a logic. The rules of inference in fuzzy logic are truth-preserving, provided we’re talking about “full” truth, i.e. we’re not in the realm of fuzziness. There are other non-classical logics, besides intuitionism, that also provide accounts of valid inference that are truth-preserving. Relevance logic, for example.
I still see those as mathematics, rather than logic, and the same goes for all other non-classical systems, such as all the modal logics. All of these are more like group theory than they are like logic, in the fundamentalist sense of “logic” I read the poll as talking about. They axiomatise certain mathematical objects, but not the general process of valid reasoning itself. That, I claim, is a problem completely solved by the classical first-order predicate calculus.
In the context of the present poll, “logic” means those methods of reasoning that are guaranteed to produce, from true premises, only true conclusions. And the poll is asking whether classical logic is it.
I think you’re begging the question. I think you’ve given a definition of “classical logic” rather than “logic”.
I think you’re begging the question. I think you’ve given a definition of “classical logic” rather than “logic”.
I think that it seems that way only because classical logic has so definitively answered the question. The question is “how shall we reason?”, and it was not obvious beforehand that the first-order predicate calculus was the answer. It took two thousand years to get there from the ancient Greeks’ understanding.
And yet the metalanguage is always classical logic. Even the most enthusiastic proponents of other systems never use them to talk about those systems. So I go firmly with “classical”.
That seems consistent with my view. For the specific application you mention—talking about logical systems—classical logics are our best models. It could still be the case that other logics are better for other applications. What makes this particular application the trump card, so that the fact that classical logic is best for doing metalogic means that it is the best simpliciter?
First, I shall ask the question “what is logic?” And I shall answer it. In the context of the present poll, “logic” means those methods of reasoning that are guaranteed to produce, from true premises, only true conclusions. And the poll is asking whether classical logic is it.
Particular formalisms used to model particular things are not, in this sense, logic, although they may be expressed in logic. For example, number theory is not logic. Neither is geometry, or physics, or probability theory. Neither, I claim, is fuzzy logic, despite the word “logic” in its name. You can say, “here is a set of functions (which I shall call fuzzy logic truth tables), and here are some theorems about how they behave (which I shall call fuzzy reasoning), and here are some physical systems whose description uses these functions.” That does not mean that those functions are actually a form of logic, as I just defined it. Bang-bang controllers like the room thermostat were invented (in 1883) long before fuzzy control theory (about which I’ve heard anecdotally that the term was invented only to avoid someone’s patent claims).
The closest anyone has come to promulgating an alternative system is intuitionistic logic, which is a pessimistic version of classical logic, in which the axiom of the excluded middle is dropped. In intuitionistic logic, you cannot infer P from not-not-P, or carry out proof by contradiction. However, I think intuitionism is simply a mistake, a historical accident which would never have happened if there had not been a half century between the codification of mathematical logic and the invention of the computer. Everything that is useful in intuitionism is given by computability theory and classical logic.
I agree with you that a logic is an account of truth-preserving inference. But, by this definition, fuzzy logic absolutely qualifies as a logic. The rules of inference in fuzzy logic are truth-preserving, provided we’re talking about “full” truth, i.e. we’re not in the realm of fuzziness. There are other non-classical logics, besides intuitionism, that also provide accounts of valid inference that are truth-preserving. Relevance logic, for example.
I still see those as mathematics, rather than logic, and the same goes for all other non-classical systems, such as all the modal logics. All of these are more like group theory than they are like logic, in the fundamentalist sense of “logic” I read the poll as talking about. They axiomatise certain mathematical objects, but not the general process of valid reasoning itself. That, I claim, is a problem completely solved by the classical first-order predicate calculus.
I think you’re begging the question. I think you’ve given a definition of “classical logic” rather than “logic”.
I think that it seems that way only because classical logic has so definitively answered the question. The question is “how shall we reason?”, and it was not obvious beforehand that the first-order predicate calculus was the answer. It took two thousand years to get there from the ancient Greeks’ understanding.