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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 ar

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).
yorwba 9y ago
Logic is mostly not about the semantics of natural languages, because even very simple sentences depend on a large amount of implicit information. Consider the case of computer vision, and how no one gives a formal definition of objects that should be recognized. Instead, they throw lots of data at a neural network and hope it learns the right thing. If you really want to apply formal reasoning to philosophical issues, you will need an understanding not only of elementary propositional logic, but also of higher order quantification, formal language semantics and probably modal logics and probability theory. If that doesn't discourage you, let me try to help you over the bump that is material implication. Consider the statement "If it's raining, the sky is cloudy." (And ignore any counterexamples that come to mind.) Everyone seems to get stuck on the case where it's not raining, and the statement is true regardless of the presence of clouds. Now treat the statement as a general rule instead, and make it's time-dependence explicit: "If it's raining on a certain day, the sky is cloudy on that day." On Monday, it rains and the sky is cloudy. You look at the rule and see that it agrees with reality. The opposite rule "If it's raining on a given day, the sky is not cloudy on that day." is wrong. (Hopefully that's obvious.) On Tuesday, it doesn't rain and the sky is still cloudy. You look at the rule and realize that it doesn't have anything to say about your situation. That means it can't be wrong, so the rule must be true in this instance. The opposite rule is also right in this instance, because it doesn't say anything either (but it has been wrong before). On Wednesday, it doesn't rain and the sky is not cloudy. Both rules are also right about this day. Now, when we state a rule, we usually mean that it always holds. (This is called universal quantification and you will learn about it when you continue studying.) Since the first rule has been correct in every instance so far, it is simply called true. The second rule has been wrong once, so it is disqualified forever, since a rule that's sometimes right and sometimes wrong doesn't give you any certainty. And what about rules like "If [something impossible], then [something ridiculous]."? Since their conditions are impossible, they never tell you anything. They are the same as saying "". That's useless, but not wrong. EDIT: If something I wrote is incorrect, or you think it's not helpful, or not expressed clearly, please tell me! I really want to get better at writing explanations.
dvt 9y ago
First of all, logic is a game. A game in the same sense that algebra is a game. Allow me to give you a very simple analogy. AC circuit theory and quantum mechanics require the use of complex numbers -- that is, the square root of negative one. In a very real sense, taking the square root of negative one sounds incredibly stupid. In fact, imaginary numbers were a very divisive issue when introduced (not unlike negative numbers before them). What does it really mean to use i? What does i map to in the real world? We're trying to calculate voltages here.. does it make any sense? Well, probably not. I could start talking about how i sort of extends the number line in the second dimension, and so forth, but that would only move the goal post (what about hypercomplex units j or k or l ... ? ). So that's why the best way to see sqrt(-1) = i is as a rule. Second of all, material implication is also a rule. There's nothing to connect to the real world. The problem with logic students is that they often don't understand the differences between abductive, deductive, and inductive logic. You mention atheism vs. religion (I assume you mean theism vs. atheism, as you can presumably have some kind of religion without a deity). When trying to win this kind of debate, you first need to pinpoint what kind of argument you're dealing with. If, for example, you're dealing with an inductive argument, the rules of deduction will not hold -- especially a rule like material implication[1]. I think that your problem is trying to apply formalisms built for one kind of game to another kind of game. Natural language is not logic. Just how Chess is not Checkers. Going back to material implication for a second, what really made material implication "click" for me was studying the Deduction Theorem[2]. This is, of course, a bit more advanced, but understanding MI from a "provability" perspective really clarifies why it's so obviously true. [1] http://ac.els-cdn.com/0888613X94900027/1-s2.0-0888613X94900027-main.pdf?_tid=e265747a-8bc7-11e7-8b1c-00000aacb35d&acdnat=1503907807_40d920a8cae68359ea5e98f028c1a537 http://ac.els-cdn.com/0888613X94900027/1-s2.0-0888613X949000... [2] https://www3.nd.edu/~cfranks/dt.pdf https://www3.nd.edu/~cfranks/dt.pdf
- fizixer 9y ago
  Okay, here's a concrete example, if you or anyone reading this is up for thinking about it: John's wife Jennifer says the following to her friend, "If John left office by 4:55 pm, he'd be home in a minute" (e.g., she claims that if John's office leaving time is 4:55 pm, he'd be home at 4:56 pm). Now, John never leaves office at 4:55 pm. It is always 5:00 pm or later. If he leaves at 5:00 pm sharp, it takes about 90 minutes on average for him to get home. If there is no traffic, it will still take at least 30 minutes for John to get home. So we know, as a behind-the-scenes knowledge, that if John left at 4:55 pm, there is no way he'd reach home before 5:25 pm. But, since John has never left office before 5:00 pm, and let's assume he never will, according to the rules of logic, Jennifer's claim is True: It is True that if "John left office by 4:55 pm" then "it'll take 1 minute for him to get home". Now, I don't read "A => B" as "A implies B". I find it more convincing to read it as "A => B" being True means "Information about A and B is insufficient to rule out implication". If "A => B" is False, that means it is not true that information about A and B is insufficient to rule out implication. In other words, information about A and B is sufficient to rule out implication. And the only way that's possible is when A is True and B is False. Now, given our problem, if we only look at Jennifer's claim, we would agree with her, because A is False, and B is False. (A: John's office leaving time is 4:55 pm or less, B: John 's travel time is 1 minute). If all we know is that A is False and B is False, we can claim that this is insufficient information to rule out Jennifer's claim. But, we do have extra information, call it C: that it takes John at least 30 minutes to get home, no matter what. The question is, how do we incorporate this extra information, C, in our problem, consisting of A and B, if we wish to refute Jennifer's claim of 'A => B' being True?
mbrock 9y ago
Logicians have been extremely useful to fields like formal software methods, linguistics and semantics, programming language design, etc. The latest rumblings are that linear logic may be perfectly applicable to reasoning about blockchains; see https://arxiv.org/abs/1506.01001 https://arxiv.org/abs/1506.01001 Wanting to treat religion vs atheism using tools from logic strikes me as... kind of old-fashioned or eccentric! You seem to have chosen the most contentious, culturally complex, ambiguous topic of all time. Maybe you'd be amused by this -- https://github.com/FormalTheology/GoedelGod https://github.com/FormalTheology/GoedelGod -- an open source effort to formalize Gödel's proof of God's necessary existence with the framework of higher-order modal logic.

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).

yorwba 9y ago

Logic is mostly not about the semantics of natural languages, because even very simple sentences depend on a large amount of implicit information. Consider the case of computer vision, and how no one gives a formal definition of objects that should be recognized. Instead, they throw lots of data at a neural network and hope it learns the right thing. If you really want to apply formal reasoning to philosophical issues, you will need an understanding not only of elementary propositional logic, but also of higher order quantification, formal language semantics and probably modal logics and probability theory. If that doesn't discourage you, let me try to help you over the bump that is material implication. Consider the statement "If it's raining, the sky is cloudy." (And ignore any counterexamples that come to mind.) Everyone seems to get stuck on the case where it's not raining, and the statement is true regardless of the presence of clouds. Now treat the statement as a general rule instead, and make it's time-dependence explicit: "If it's raining on a certain day, the sky is cloudy on that day." On Monday, it rains and the sky is cloudy. You look at the rule and see that it agrees with reality. The opposite rule "If it's raining on a given day, the sky is not cloudy on that day." is wrong. (Hopefully that's obvious.) On Tuesday, it doesn't rain and the sky is still cloudy. You look at the rule and realize that it doesn't have anything to say about your situation. That means it can't be wrong, so the rule must be true in this instance. The opposite rule is also right in this instance, because it doesn't say anything either (but it has been wrong before). On Wednesday, it doesn't rain and the sky is not cloudy. Both rules are also right about this day. Now, when we state a rule, we usually mean that it always holds. (This is called universal quantification and you will learn about it when you continue studying.) Since the first rule has been correct in every instance so far, it is simply called true. The second rule has been wrong once, so it is disqualified forever, since a rule that's sometimes right and sometimes wrong doesn't give you any certainty. And what about rules like "If [something impossible], then [something ridiculous]."? Since their conditions are impossible, they never tell you anything. They are the same as saying "". That's useless, but not wrong. EDIT: If something I wrote is incorrect, or you think it's not helpful, or not expressed clearly, please tell me! I really want to get better at writing explanations.

dvt 9y ago

First of all, logic is a game. A game in the same sense that algebra is a game. Allow me to give you a very simple analogy. AC circuit theory and quantum mechanics require the use of complex numbers -- that is, the square root of negative one. In a very real sense, taking the square root of negative one sounds incredibly stupid. In fact, imaginary numbers were a very divisive issue when introduced (not unlike negative numbers before them). What does it really mean to use i? What does i map to in the real world? We're trying to calculate voltages here.. does it make any sense? Well, probably not. I could start talking about how i sort of extends the number line in the second dimension, and so forth, but that would only move the goal post (what about hypercomplex units j or k or l ... ? ). So that's why the best way to see sqrt(-1) = i is as a rule. Second of all, material implication is also a rule. There's nothing to connect to the real world. The problem with logic students is that they often don't understand the differences between abductive, deductive, and inductive logic. You mention atheism vs. religion (I assume you mean theism vs. atheism, as you can presumably have some kind of religion without a deity). When trying to win this kind of debate, you first need to pinpoint what kind of argument you're dealing with. If, for example, you're dealing with an inductive argument, the rules of deduction will not hold -- especially a rule like material implication[1]. I think that your problem is trying to apply formalisms built for one kind of game to another kind of game. Natural language is not logic. Just how Chess is not Checkers. Going back to material implication for a second, what really made material implication "click" for me was studying the Deduction Theorem[2]. This is, of course, a bit more advanced, but understanding MI from a "provability" perspective really clarifies why it's so obviously true. [1] http://ac.els-cdn.com/0888613X94900027/1-s2.0-0888613X94900027-main.pdf?_tid=e265747a-8bc7-11e7-8b1c-00000aacb35d&acdnat=1503907807_40d920a8cae68359ea5e98f028c1a537 http://ac.els-cdn.com/0888613X94900027/1-s2.0-0888613X949000... [2] https://www3.nd.edu/~cfranks/dt.pdf https://www3.nd.edu/~cfranks/dt.pdf

fizixer 9y ago
Okay, here's a concrete example, if you or anyone reading this is up for thinking about it: John's wife Jennifer says the following to her friend, "If John left office by 4:55 pm, he'd be home in a minute" (e.g., she claims that if John's office leaving time is 4:55 pm, he'd be home at 4:56 pm). Now, John never leaves office at 4:55 pm. It is always 5:00 pm or later. If he leaves at 5:00 pm sharp, it takes about 90 minutes on average for him to get home. If there is no traffic, it will still take at least 30 minutes for John to get home. So we know, as a behind-the-scenes knowledge, that if John left at 4:55 pm, there is no way he'd reach home before 5:25 pm. But, since John has never left office before 5:00 pm, and let's assume he never will, according to the rules of logic, Jennifer's claim is True: It is True that if "John left office by 4:55 pm" then "it'll take 1 minute for him to get home". Now, I don't read "A => B" as "A implies B". I find it more convincing to read it as "A => B" being True means "Information about A and B is insufficient to rule out implication". If "A => B" is False, that means it is not true that information about A and B is insufficient to rule out implication. In other words, information about A and B is sufficient to rule out implication. And the only way that's possible is when A is True and B is False. Now, given our problem, if we only look at Jennifer's claim, we would agree with her, because A is False, and B is False. (A: John's office leaving time is 4:55 pm or less, B: John 's travel time is 1 minute). If all we know is that A is False and B is False, we can claim that this is insufficient information to rule out Jennifer's claim. But, we do have extra information, call it C: that it takes John at least 30 minutes to get home, no matter what. The question is, how do we incorporate this extra information, C, in our problem, consisting of A and B, if we wish to refute Jennifer's claim of 'A => B' being True?

mbrock 9y ago

Logicians have been extremely useful to fields like formal software methods, linguistics and semantics, programming language design, etc. The latest rumblings are that linear logic may be perfectly applicable to reasoning about blockchains; see https://arxiv.org/abs/1506.01001 https://arxiv.org/abs/1506.01001 Wanting to treat religion vs atheism using tools from logic strikes me as... kind of old-fashioned or eccentric! You seem to have chosen the most contentious, culturally complex, ambiguous topic of all time. Maybe you'd be amused by this -- https://github.com/FormalTheology/GoedelGod https://github.com/FormalTheology/GoedelGod -- an open source effort to formalize Gödel's proof of God's necessary existence with the framework of higher-order modal logic.