Logical Literacy
matt.might.net
Logical Literacy
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Re: Logical Literacy
#2Good article overall, but I think I should make a comment:
> Many understand implication intuitively, yet find its symbolic formulation puzzling.
Yes, and for a reason: material implication (the kind of implication discussed in the article) cannot always explain our intuitive sense of implication. My favourite example, from Priest's Introduction to Non-Classical Logic: (A ⇒ B) ∧ (C ⇒ D) ⊦ (A ⇒ D) ∨ (C ⇒ B) is valid when ⇒ is material, yet the statement:
"If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France."
makes no intuitive sense. It is very important to discard intuition when dealing with implication in mathematics -- this is why although I'm a big fan of using English connectives instead of symbolic ones, I try to not use the English "if" and "then" in proofs.
Re: Logical Literacy
#3Good article overall, but I think I should make a comment: > Many understand implication intuitively, yet find its symbolic formulation puzzling. Yes, and for a reason: material implication (the kind of implication discussed in the article) cannot always explain our intuitive sense of implication. My favourite example, from Priest's Introduction to Non-Classical Logic : (A ⇒ B) ∧ (C ⇒ D) ⊦ (A ⇒ D) ∨ (C ⇒ B) is valid…
This is a great point.
For something that seems so "common sense," the meaning of implication is difficult to grasp.
I wondered whether I should spend some time discussing the differences between material and logical implication.
In the end, I decided that was just too much for a short overview on the basics.
Re: Logical Literacy
#4Good article overall, but I think I should make a comment: > Many understand implication intuitively, yet find its symbolic formulation puzzling. Yes, and for a reason: material implication (the kind of implication discussed in the article) cannot always explain our intuitive sense of implication. My favourite example, from Priest's Introduction to Non-Classical Logic : (A ⇒ B) ∧ (C ⇒ D) ⊦ (A ⇒ D) ∨ (C ⇒ B) is valid…
Maybe because I'm a mathematician my idea of "intuitive sense" is warped, but intuitively speaking, either John is in Paris or London exclusively. (This isn't given in the statement, but it holds intuitively.) Therefore, in the conclusion, we can assume that at least one of the antecedents "If John is in Paris..." or "If John is in London..." is false, and therefore at least one of the implications in the conclusion is true.
Re: Logical Literacy
#5Upvoted especially for his list of suggested reading at the end of the article.
Re: Logical Literacy
#6It's unfortunate that this kind of logic isn't emphasized more in mathematics or science in a pure form. I'm pretty sure in high school I had one semester of this kind of logic associated with geometry. That was it until I hit discrete structures in college.
Failure to understand this logic might be part of the reason there's so much magical thinking about things relating to pseudoscience (e.g. paranormal activity, UFOs, conspiracy theories, New Age anything). A stronger base in logic would help people understand exactly why these things are impossible, stupid, or plain crazy with simple deductive and inductive reasoning techniques taught in logic courses.
Re: Logical Literacy
#7Good article overall, but I think I should make a comment: > Many understand implication intuitively, yet find its symbolic formulation puzzling. Yes, and for a reason: material implication (the kind of implication discussed in the article) cannot always explain our intuitive sense of implication. My favourite example, from Priest's Introduction to Non-Classical Logic : (A ⇒ B) ∧ (C ⇒ D) ⊦ (A ⇒ D) ∨ (C ⇒ B) is valid…
Is that still true constructively?
Re: Logical Literacy
#8Good article overall, but I think I should make a comment: > Many understand implication intuitively, yet find its symbolic formulation puzzling. Yes, and for a reason: material implication (the kind of implication discussed in the article) cannot always explain our intuitive sense of implication. My favourite example, from Priest's Introduction to Non-Classical Logic : (A ⇒ B) ∧ (C ⇒ D) ⊦ (A ⇒ D) ∨ (C ⇒ B) is valid…
Is that still true constructively?
By "constructively" do you mean in a constructivist logic like intuitionist logic? I'm too lazy to work it out right now, but I'd place my money on that statement not being valid in intuitionist logic.
Re: Logical Literacy
#9Earlier quoted context omitted.
Is that still true constructively?
By "constructively" do you mean in a constructivist logic like intuitionist logic? I'm too lazy to work it out right now, but I'd place my money on that statement not being valid in intuitionist logic.
Yes. That's what I meant. That was my feeling too, but I know myself too well than to place any bets on what is provable or not constructively unless I work it out. :)
Re: Logical Literacy
#10Good article overall, but I think I should make a comment: > Many understand implication intuitively, yet find its symbolic formulation puzzling. Yes, and for a reason: material implication (the kind of implication discussed in the article) cannot always explain our intuitive sense of implication. My favourite example, from Priest's Introduction to Non-Classical Logic : (A ⇒ B) ∧ (C ⇒ D) ⊦ (A ⇒ D) ∨ (C ⇒ B) is valid…
Is that still true constructively?
No.
Intuitionistic logic has the pleasing property that whenever G |- P \/ Q is provable, then either G |- P is provable or G |- Q is provable.
I think this gets to the heart of what is confusing about this example: (London -> England) /\ (Paris -> France) |- (London -> France) \/ (Paris -> England) is classically valid, even though neither (London -> England) /\ (Paris -> France) |- (London -> France) nor (London -> England) /\ (Paris -> France) |- (Paris -> England) is