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Not to trivialize your issue, but I always am dismayed at how much time is spent trying to map English statements into those of sentential logic, especially in

Not to trivialize your issue, but I always am dismayed at how much time is spent trying to map English statements into those of sentential logic, especially in cases where their usage differ, like inclusive ors or material implication. In a more mathematical study of logic, you define your logical connectives using truth tables. So the material implication is nothing more and nothing less than X | Y | X -> Y ---------------- F | F | T F | T | T T | F | F T | T | T For the unfamiliar, a truth table works as follows: in the sentence "X -> Y", evaluate the truth values of X and Y (If X and Y are themselves sentences, this follows recursively. If they are atomic predicates, they should have fixed truth values). Then, refer to the appropriate row of the table to find the value of the sentence. Rather tangentially, my logic teacher would always say "to verify a material implication, we must falsify the hypothesis or verify the conclusion". I'm not sure that helped any of her students come to terms with the connective, but it is a succinct (if not jargon-heavy) way to summarize that table.

zengid 9y ago
I think time is spent trying to map Logic to English because its our more familiar framework for thinking through logical problems. We do it everyday when problem-solving and weighing options, so its only natural that we want a way to make the abstractions of formal Logic into a concrete 'cause and effect' type of explanation via language. That being said, I agree that one needs to try and resit this urge, otherwise you won't get past the barrier that material implication presents. It really just doesn't make sense at first, and not for a long time of playing with it and using it within larger statements does one start to sus-out some kind of concrete mechanical meaning for its behavior.
fizixer 9y ago
There are two ways of looking at logic, roughly speaking: - Logic as a game (as in a set of rules). E.g., in that regard pure math is a game, formal systems are a game, even programming and CS is a game. - Logic as an applied discipline used to formalize rational discourse, to arrive at a set of conclusive facts, with sound and valid reasoning, given a problem (which could be a problem in the real world) and a set of assumptions. Part of my frustration with logicians is that they almost exclusively treat logic as belonging to the first case, and hate it when you ask them about the second case. I am interested in logic partly because it's an aspect of the foundation of pure mathematics, but around the time I started learning logic, that was the least of my concerns. Instead, I was highly motivated by the need to have a formal system in place which I could rely on while arguing about philosophical issues, and especially topics related to religion vs atheism, because the arguments I would hear from the other side could safely be described as "stupid reasoning". However, when I started taking logic seriously, I realized that, not only do the fundamental rules of logic sound stupider (hint: material implication) than the logic I was hearing from my opponents, but logicians have no interest in helping the logic student connect the formal ideas to real world problems. When 99.999% of logicians, no matter how experienced they are, tell me to treat logic as a game, and nothing but a game, what I hear is this: I don't know how to apply logic to philosophical discourse and real world problems, and I am content with my state of knowledge, therefore you should be happy treating logic as just a game too and stop asking any deeper questions, or trying to connect it to philosophy and rational discourse.
- benkuykendall 9y ago
  That's an interesting dichotomy to establish. At first, it appears to mirror the theoretical/applied division we see in a lot of fields. But in addition to an applied mindset, you seem to be demanding something stronger out of logic: applicability to philosophical discourse. To me, this seems like a very lofty goal: I would be very surprised if someone treated religion (to use your example) productively with formal logic. However, just because it's not immediately applicable in one domain does not make logic useless. A great example of this is Prolog, a logical programming language which uses declarative statements reminiscent of first order logic. It can solve real world problems, though it has not yet seen wide adoption as a programming language. People do use automatic theorem provers, which have their roots in logic. The are used not only to further the "game" of mathematics, but to find also bugs in real hardware and software systems. These application have very little to do with philosophy, and nothing to do with translating between English language and logical sentences. However, I think they are quite interesting. (I'm not sure what to tell you about logic "sounding stupid". In the end, things like definitions of connectives are just conventions. Although it's good to understand their motivations, I'm not sure how much good objecting to them does. It kind of reminds me of half serious attempts to adopt tau := 2 pi as mathematical notation; maybe using a different constant makes more sense to you, but finally, it won't make a big difference, and you should probably just continue to use pi for interoperability with the rest of the world.)
  - fizixer 9y ago
    I'm wondering what would be your take on a concrete example that I just posted here: https://news.ycombinator.com/item?id=15115319 https://news.ycombinator.com/item?id=15115319
    - benkuykendall 9y ago
      Personally, I don't really like the translation "information about A and B is insufficient to rule out implication". At least when I say "A implies B" I mean exactly "A => B". But maybe I hang out with too many mathematicians. Other common translations you might be interested in are "A suffices for B" or "B is necessary for A". Though admittedly these particular words have a bit of a mathy tone, so no wonder they are so precisely tied to the definition of implication. As far as your idea of "outside information" goes, I'm afraid it kind of breaks outside the bounds of sentential logic, which cannot deal with numbers in the manner you are attempting. However, that does not mean we cannot analyze it with logic. It should be simple to put your sentences into the logical "language" of Arithmetic and to prove more or less what you would expect: A is false (because John never leaves the office before 5), B is false (because it takes him 30 minutes to get home), and that "A => B" is true (by the definition of material implication, however vacuously).

zengid 9y ago

I think time is spent trying to map Logic to English because its our more familiar framework for thinking through logical problems. We do it everyday when problem-solving and weighing options, so its only natural that we want a way to make the abstractions of formal Logic into a concrete 'cause and effect' type of explanation via language. That being said, I agree that one needs to try and resit this urge, otherwise you won't get past the barrier that material implication presents. It really just doesn't make sense at first, and not for a long time of playing with it and using it within larger statements does one start to sus-out some kind of concrete mechanical meaning for its behavior.

fizixer 9y ago

There are two ways of looking at logic, roughly speaking: - Logic as a game (as in a set of rules). E.g., in that regard pure math is a game, formal systems are a game, even programming and CS is a game. - Logic as an applied discipline used to formalize rational discourse, to arrive at a set of conclusive facts, with sound and valid reasoning, given a problem (which could be a problem in the real world) and a set of assumptions. Part of my frustration with logicians is that they almost exclusively treat logic as belonging to the first case, and hate it when you ask them about the second case. I am interested in logic partly because it's an aspect of the foundation of pure mathematics, but around the time I started learning logic, that was the least of my concerns. Instead, I was highly motivated by the need to have a formal system in place which I could rely on while arguing about philosophical issues, and especially topics related to religion vs atheism, because the arguments I would hear from the other side could safely be described as "stupid reasoning". However, when I started taking logic seriously, I realized that, not only do the fundamental rules of logic sound stupider (hint: material implication) than the logic I was hearing from my opponents, but logicians have no interest in helping the logic student connect the formal ideas to real world problems. When 99.999% of logicians, no matter how experienced they are, tell me to treat logic as a game, and nothing but a game, what I hear is this: I don't know how to apply logic to philosophical discourse and real world problems, and I am content with my state of knowledge, therefore you should be happy treating logic as just a game too and stop asking any deeper questions, or trying to connect it to philosophy and rational discourse.

benkuykendall 9y ago
That's an interesting dichotomy to establish. At first, it appears to mirror the theoretical/applied division we see in a lot of fields. But in addition to an applied mindset, you seem to be demanding something stronger out of logic: applicability to philosophical discourse. To me, this seems like a very lofty goal: I would be very surprised if someone treated religion (to use your example) productively with formal logic. However, just because it's not immediately applicable in one domain does not make logic useless. A great example of this is Prolog, a logical programming language which uses declarative statements reminiscent of first order logic. It can solve real world problems, though it has not yet seen wide adoption as a programming language. People do use automatic theorem provers, which have their roots in logic. The are used not only to further the "game" of mathematics, but to find also bugs in real hardware and software systems. These application have very little to do with philosophy, and nothing to do with translating between English language and logical sentences. However, I think they are quite interesting. (I'm not sure what to tell you about logic "sounding stupid". In the end, things like definitions of connectives are just conventions. Although it's good to understand their motivations, I'm not sure how much good objecting to them does. It kind of reminds me of half serious attempts to adopt tau := 2 pi as mathematical notation; maybe using a different constant makes more sense to you, but finally, it won't make a big difference, and you should probably just continue to use pi for interoperability with the rest of the world.)
- fizixer 9y ago
  I'm wondering what would be your take on a concrete example that I just posted here: https://news.ycombinator.com/item?id=15115319 https://news.ycombinator.com/item?id=15115319
  - benkuykendall 9y ago
    Personally, I don't really like the translation "information about A and B is insufficient to rule out implication". At least when I say "A implies B" I mean exactly "A => B". But maybe I hang out with too many mathematicians. Other common translations you might be interested in are "A suffices for B" or "B is necessary for A". Though admittedly these particular words have a bit of a mathy tone, so no wonder they are so precisely tied to the definition of implication. As far as your idea of "outside information" goes, I'm afraid it kind of breaks outside the bounds of sentential logic, which cannot deal with numbers in the manner you are attempting. However, that does not mean we cannot analyze it with logic. It should be simple to put your sentences into the logical "language" of Arithmetic and to prove more or less what you would expect: A is false (because John never leaves the office before 5), B is false (because it takes him 30 minutes to get home), and that "A => B" is true (by the definition of material implication, however vacuously).