Classical: The standard kinds of logic that you learn in undergraduate logic classes are the best (or right) logics, the ones that best model (ETA: idealized versions of) our inferential processes. Examples of classical logics are Boolean logic and first-order predicate calculus. Classical logics are bivalent (sentences can only be true or false), obey the principle of the excluded middle (if a proposition is not true, its negation must be true) and obey the law of non-contradiction (a proposition and its negation cannot both be true).
Non-classical: The best logic is not classical. Non-classical logics usually reject the principle of the excluded middle or the law of non-contradiction. An example of a non-classical logic is dialetheism, according to which there are true contradictions (i.e. some sentences of the form “A and not A” are true). Proponents of non-classical logics argue that many of our scientific theories, if you probe deeply, involve inconsistencies, yet we don’t regard them as trivially false. So they claim that we need to revise the way we understand logic to accurately model our inferential processes.
Classical: The standard kinds of logic that you learn in undergraduate logic classes are the best (or right) logics, the ones that best model our inferential processes
Is that the right criterion? Or should it be: the ones that best model the correct inferential processes, whether or not we humans adhere to them?
I lean towards classical, but with the proviso that we have to be careful about what counts as a statement. Sneak in a statement with ambiguous truth values, and classical logic halts and catches fire. Personally I’m OK with rejecting such statemetns.
In my AI lessons, the “non-classical logic” course including all the probabilistic theories : fuzzy logic, Bayesian, … that’s why I voted “lean : non-classical”, but I guess it’s just a matter of vocabulary.
Other: Different logics are appropriate for modeling how one should infer in different domains. Classical logics are fine for many applications but it is possible (maybe even plausible) that non-classical logics will be better models for certain applications. For instance, fuzzy logic (a many-valued logic) has been successfully employed to control subway systems and build thermostats.
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.
My interpretation of this question is metaphysical: ‘Is reality classical?’ This is shorthand for: ’Is there any fact that is fundamentally, objectively, and in principle...
… inexpressible?
… vague?
… contradictory?
… indeterminate? (I.e., neither the case nor not-the-case.)
… etc.
But this is a strange set of questions, and I suspect most professional philosophers and LessWrongers are instead answering a confused mixture of (mostly trivial) questions like ‘which logic do I find most useful?’.
Other: The objection to classical logic stated in pragmatist’s summary needs to be addressed, but I lack sufficient knowledge of the field to determine whether the best resolution of these objections is likely to take the form of (a) a solution within classical logic or (b) the adoption of a non-classical logic.
Logic: classical or non-classical?
[pollid:104]
Classical: The standard kinds of logic that you learn in undergraduate logic classes are the best (or right) logics, the ones that best model (ETA: idealized versions of) our inferential processes. Examples of classical logics are Boolean logic and first-order predicate calculus. Classical logics are bivalent (sentences can only be true or false), obey the principle of the excluded middle (if a proposition is not true, its negation must be true) and obey the law of non-contradiction (a proposition and its negation cannot both be true).
Non-classical: The best logic is not classical. Non-classical logics usually reject the principle of the excluded middle or the law of non-contradiction. An example of a non-classical logic is dialetheism, according to which there are true contradictions (i.e. some sentences of the form “A and not A” are true). Proponents of non-classical logics argue that many of our scientific theories, if you probe deeply, involve inconsistencies, yet we don’t regard them as trivially false. So they claim that we need to revise the way we understand logic to accurately model our inferential processes.
Is that the right criterion? Or should it be: the ones that best model the correct inferential processes, whether or not we humans adhere to them?
Good point. I’ve edited to reflect this.
I lean towards classical, but with the proviso that we have to be careful about what counts as a statement. Sneak in a statement with ambiguous truth values, and classical logic halts and catches fire. Personally I’m OK with rejecting such statemetns.
What does Bayesian probability theory count as?
Bayesian probability is an extension of classical logic. I don’t think philosophers consider it to be non-classical.
In my AI lessons, the “non-classical logic” course including all the probabilistic theories : fuzzy logic, Bayesian, … that’s why I voted “lean : non-classical”, but I guess it’s just a matter of vocabulary.
Okay, so “Accept: classical” be it.
Other: Different logics are appropriate for modeling how one should infer in different domains. Classical logics are fine for many applications but it is possible (maybe even plausible) that non-classical logics will be better models for certain applications. For instance, fuzzy logic (a many-valued logic) has been successfully employed to control subway systems and build thermostats.
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.
My interpretation of this question is metaphysical: ‘Is reality classical?’ This is shorthand for: ’Is there any fact that is fundamentally, objectively, and in principle...
… inexpressible? … vague? … contradictory? … indeterminate? (I.e., neither the case nor not-the-case.) … etc.
But this is a strange set of questions, and I suspect most professional philosophers and LessWrongers are instead answering a confused mixture of (mostly trivial) questions like ‘which logic do I find most useful?’.
Other: The objection to classical logic stated in pragmatist’s summary needs to be addressed, but I lack sufficient knowledge of the field to determine whether the best resolution of these objections is likely to take the form of (a) a solution within classical logic or (b) the adoption of a non-classical logic.
Other: probability distributions over world-histories. Otherwise classical.