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Good article overall, but I think I should make a comment: > Many understand implication intuitively, yet find its symbolic formulation puzzling. Yes, and for

by sid0 16y ago

Good 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.

mattmight 16y ago
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.
thebooktocome 16y ago
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.
moultano 16y ago
Is that still true constructively?
- sid0 16y ago
  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.
  - moultano 16y ago
    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. :)
- vilhelm_s 16y ago
  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
Someone 16y ago
I also think it is unfortunate that his first theorem is incorrect (n=1 cannot be written as the product of two distinct integers) That may cost the author some of the readers for whom this text is the most educational, i.e. the ones that have trouble grasping the logic.
- mattmight 16y ago
  Good catch! Will fix. ;)
- nene 16y ago
  Even if it's not meant to be true, I find it even more disturbing that the symbolic and English formulations doesn't seem to quite match up. (IMHO the symbolic formulation doesn't state that n must be integer.) It's like the source code and comments being out of sync - you don't know which one is wrong.
  - mattmight 16y ago
    Another great point. I fixed the broken theorem, and I added a blurb on the "universe of discourse."
astrofinch 16y ago
It makes perfect intuitive sense if you're used to the way the word "if" is used in logic.
- sid0 16y ago
  It makes perfect intuitive sense if you're used to the way the word "if" is used in classical logic. Remember that it's you who has bent your intuitive sense to fit how material implication works. Most people are not going to think it makes intuitive sense, and the fault doesn't lie with them -- it lies with the logic. Logic is like UI: it's Not the User's Fault he doesn't find implication intuitive.
  - astrofinch 16y ago
    Logic is not like UI. UI is a translation layer between the way humans think and the way a machine works. Logic isn't a translation layer. It's more like a theoretical machine. The English terms we use to talk about logic could be considered similar to UI. But I don't think that using the word "if" (instead of making up a new word like "garply") was obviously the wrong design decision for the inventors of English logical terminology. "If" is the word in the English language that most closely maps to the concept the terminology designers wanted to describe.
    - sid0 16y ago
      > Logic isn't a translation layer. It's more like a theoretical machine. More precisely, it is a model for human thought. The way logic is similar to UI is that logic's philosophical success is a function of how well human thought maps to it and how intuitive theorems in the logic are, just as UI's success is a function of how intuitive it is. The last hundred years of discourse on philosophical logic have been focused on one thing -- to formalize the human notion of implication better than material implication does. That's why logicians came up with non-classical logics of the modal, counterfactual (which captures the very intuitive notion of mutatis mutandis), fuzzy and intuitionistic varieties.
      - 16y ago

by sid0 16y ago

mattmight 16y ago

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.

thebooktocome 16y ago

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.

moultano 16y ago

Is that still true constructively?

sid0 16y ago
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.
- moultano 16y ago
  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. :)
vilhelm_s 16y ago
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

Someone 16y ago

I also think it is unfortunate that his first theorem is incorrect (n=1 cannot be written as the product of two distinct integers) That may cost the author some of the readers for whom this text is the most educational, i.e. the ones that have trouble grasping the logic.

mattmight 16y ago
Good catch! Will fix. ;)
nene 16y ago
Even if it's not meant to be true, I find it even more disturbing that the symbolic and English formulations doesn't seem to quite match up. (IMHO the symbolic formulation doesn't state that n must be integer.) It's like the source code and comments being out of sync - you don't know which one is wrong.
- mattmight 16y ago
  Another great point. I fixed the broken theorem, and I added a blurb on the "universe of discourse."

astrofinch 16y ago

It makes perfect intuitive sense if you're used to the way the word "if" is used in logic.

sid0 16y ago
It makes perfect intuitive sense if you're used to the way the word "if" is used in classical logic. Remember that it's you who has bent your intuitive sense to fit how material implication works. Most people are not going to think it makes intuitive sense, and the fault doesn't lie with them -- it lies with the logic. Logic is like UI: it's Not the User's Fault he doesn't find implication intuitive.
- astrofinch 16y ago
  Logic is not like UI. UI is a translation layer between the way humans think and the way a machine works. Logic isn't a translation layer. It's more like a theoretical machine. The English terms we use to talk about logic could be considered similar to UI. But I don't think that using the word "if" (instead of making up a new word like "garply") was obviously the wrong design decision for the inventors of English logical terminology. "If" is the word in the English language that most closely maps to the concept the terminology designers wanted to describe.
  - sid0 16y ago
    > Logic isn't a translation layer. It's more like a theoretical machine. More precisely, it is a model for human thought. The way logic is similar to UI is that logic's philosophical success is a function of how well human thought maps to it and how intuitive theorems in the logic are, just as UI's success is a function of how intuitive it is. The last hundred years of discourse on philosophical logic have been focused on one thing -- to formalize the human notion of implication better than material implication does. That's why logicians came up with non-classical logics of the modal, counterfactual (which captures the very intuitive notion of mutatis mutandis), fuzzy and intuitionistic varieties.
    - 16y ago