9 ms·

What is logic?

139 points by Hooke 10y ago

kensai 10y ago
"The idea that ordinary language is expressively inadequate to account for mathematical (or even logical) reasoning became a recurring theme in the ensuing tradition of mathematical logic, so much so that the term ‘symbolic logic’ became synonymous with this tradition. Doing logic came to mean simply working with special symbols, not with ordinary words. In this respect, it is worth noting that the humanist authors had criticised the Latin of scholastic logicians precisely as ‘too artificial’, and even the Greek language that Aristotle relies on for syllogistic logic is regimented and removed from ordinary ways of speaking at the time. In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage." True that, but were/are there alternatives? Difficult to say.
- throwaway729 10y ago
  > In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage Of course, you could say the exact same thing for programming languages (after all, what do programmers do if not "simply working with special symbols, not with ordinary words"? And isn't this the primary criticism of programming brought by people who find it distasteful -- that expressing their fuzzy intuitions in a precise enough manner for the computer feels like a pointless exercise in frustration?) Really, the only difference between the proponents of mathematical logic over informal prose in the early 20th century and the proponents of programming languages over informal prose today is that computers obviate the problem with fuzzy thinking and fuzzier exposition. The philosopher or mathematician could hand-wave about their holes in their arguments, but the programmer can only hand-wave so much the uselessness of his unexecutable pseudo-code. The golden standard would be a form of speech that is natural and ordinary, but also avoids misleading, omitting, and eliding essential details. In other words, a language that is formal enough to be parsed, checked, and generated by a proof assistant program. But also natural enough to be read as if it's prose written by a native speaker of a natural language. I'm of the opinion that such a golden standard does not exist, and so we're forever doomed to incremental improvement.
  - tikhonj 10y ago
    On a slightly orthogonal note, I think that not only can you think of programming languages this way, but that you should—programming languages and formal logics are ultimately instances of the same general idea. I've found this to be a consistent and powerful view for reasoning about programs, programming languages, proofs and logics in a uniform sort of way. The Curry-Howard correspondence is an important idea that's a specific instance of this, showing a direct relationship between specific kinds of logic and specific kinds of typed programming languages, but I think it's useful to think about programming languages that don't correspond to well-studied logical systems in a similar way. This helps explain why the study of programming languages—and especially programming language theory—isn't really about programming languages: it's more about abstraction and reasoning in general. It's deeper than you'd suspect from the name. I'd go so far as saying that programming language theory is less a theory of programming languages and more a general theory of computation from a language point of view, as compared to "normal" theoretical CS.
    - mafribe 10y ago
      programming languages and formal logics are ultimately instances of the same general idea. That's deeply questionable. The CH-correspondence breaks down as soon as you your computation includes non-termination, concurrency, timing, distribution etc. Classical logic doesn't really have wholly convincing CH-correspondences either. It seems to me that constructive proofs are a special class of programs, and fall under the purview of programming language, but not the other way around.
      - tikhonj 10y ago
        The point is that they're both formal systems of reasoning, not that they have exactly the same properties. The techniques and abstractions from both PL and formal logic often generalize well to both.
        mafribe 10y ago
        Non-determinism, a key property of concurrent systems, doesn't generalise well to logic.
- SomeStupidPoint 10y ago
  The whole drive of mathematics (and logic) is ultimately that syntax is the semantics. This, of course, is annoying for almost all communication, particularly when we want to use fuzzy, hard to define terms. Languages are useful because you can pretty heavily butcher the syntax, but still be understood. (And arguably, that fuzziness is what drives new thoughts.)
pitaj 10y ago
From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic and math can lead us to useful conclusions.
- FrostyElsa 10y ago
  Didn't it take some guy like 200 pages to prove logically that 2+2=4?
- redial 10y ago
  But isn't reason is the result of logic? as in something is reasonable only when it is logically sound?
  - pitaj 10y ago
    A line of thought can only be reasonable if it is logically sound. Reason is built on top of logic. All of this IIRC.
- lgas 10y ago
  Where can I find this standard set of axioms?
  - cmrx64 10y ago
    One widely-used system is Higher Order Logic (with various extensions sometimes), one core of which you can find here: http://sourceforge.net/projects/hol/files/hol/kananaskis-10/kananaskis-10-logic.pdf/download http://sourceforge.net/projects/hol/files/hol/kananaskis-10/... This is based on type theory (and has a model in set theory), but isn't as powerful as the logics used in Coq and similar theorem provers (the Calculus of (Inductive) Constructions). You might be interested in metamath: http://us.metamath.org/ http://us.metamath.org/, which lists the axioms used in its largest body of work: http://us.metamath.org/mpegif/mmset.html#axioms http://us.metamath.org/mpegif/mmset.html#axioms
  - baddox 10y ago
lngnmn 10y ago
A set of general rules for not losing a connection to reality. Could be used in reasoning or computer programming.
- metaphorm 10y ago
  this is amusingly missing the point. logic is, by its nature, an abstraction away from reality. it's a set of specific (not general) rules for evaluating statements of language (whether that language is natural, mathematical, or artificial/computer) such that meaning can be disambiguated. "reality", whatever it is, has no such requirement that it can be expressed in unambiguous statements.
  - paganel 10y ago
    > it's a set of specific (not general) rules for evaluating statements of language Not the OP, but you could say that no "language" exists outside of "reality", and stretching it further you could also say that all types of languages (formal, mathematical, artificial/computer), including the associated "rules", are social constructs which depend heavily on the time and place of their "existence", i.e. on "reality". Until we meet some quantum-like entity which might "talk" to us from outside the boundaries of this physical Universe we're doomed to be solipsistic: we construct "abstractions" that are meant to distance us from "reality" only to find out in the end that "reality" was behind those "abstractions" all along.
  - lngnmn 10y ago
    Syllogism is not general? WTF am I reading?
s_kilk 10y ago
"We just don't know" https://www.youtube.com/watch?v=Kh0Y2hVe_bw https://www.youtube.com/watch?v=Kh0Y2hVe_bw
westoncb 10y ago
My take: it's a human means of systematically exploring implications of already accepted beliefs. There are implicit and explicit forms of it. The implicit form is something our brains do automatically and is a consequence of its structure. Perhaps the way it models things intrinsically does not allow for what we would call contradicting statements (though it contains many separate models which if unified would contain contradictions); then, the space of desirable implications is narrowed by a motivation to find a certain kind of implication (i.e. there is typically a goal when engaging in reasoning—we aren't often indifferently interested in all implications) combined with the fact that contradictions aren't allowed. It seems like implications are usually discovering that some entity belongs to a class it wasn't previously known to belong to, at which point it inherits the attributes of other things in that class. The explicit forms seem like attempts to model the implicit form in mathematical language, in order to bring logical processes more under control of conscious thought, or 'executive function' (plus reasoning on paper extends working memory). It seems possible that with the 'correct' formulation, a logic could viewed as a theoretical science describing properties of the structure of the human brain that give rise to reasoning.
- kazagistar 10y ago
  The interesting point of the given article was that pre-mathematical-formalization, logic was backwards of that purpose, and often remains so even today. The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them. A vast majority of mathematicians don't care about the axiomatic foundations of mathematics, nor mind if it gets entirely reformulated; they already know math works, and its irrelevant which particular set of axioms is used to prove that. The assuptions => theorm structure is an artificial abstraction over a very different logic process. Symbolic logic is a framework for mechanical validation, rather then anything like how people actually do logic.
  - westoncb 10y ago
    > The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them. Right, there was an issue with labeling justification 'logic'. An interesting point, but not as interesting as the title of this story—which I prefer to discuss :) I was getting at the general process of inference making in the human brain (not even conscious inference making necessarily), which I see as the real root of all the different things we call logic.
tel 10y ago
Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one another—some statements' truth depends upon what logic they're formalized within. Model theory is good here in that it studies the connection between logical language and more "physical" models. Some of these models can give a relatable account for what a logic "means". For instance, Boolean logic correspond with set operations and therefore we can use that correspondence to portray a meaning for statements in Boolean-like logics. Intuitionistic logic is an interesting one if you're not familiar with it. It can be seen as the logic of working with "demonstrations", building and analyzing them. For instance, if I have an object "X" such that witnessing it immediately convinces someone of the truth of "1+1=2", and if I have a whole collection of such items, then intuitionistic logic talks about how to compose and decompose collections of these things. Another interesting operation is "not" such that "not X" is a tool for using any X you happen to have to show that the world is inconsistent. This acts as a refutation of the existence of values like X! This differs from set-like logics because, for instance, "not (not X)" is not the same as X: being able to show that if you had (a way to prove the world inconsistent if you had an X) you could show the world as inconsistent... it's a far cry away from actually having an X! Intuitionistic logic is interesting for two reasons. First, it models communication and persuasion by being an algebra of "persuasive items" like X. Second, it's "constructive" and thus works similarly to how programming languages express things through their creation.
- hackits 10y ago
  Correct me if I'm wrong, I've all ways seen Maths to be easy part. The translation of English Language (grammar, sentence structure) that can be ambiguous to a very precise meaning such as Math's to be the hardest part and where people with good translation skills excel.
  - tel 10y ago
    I think much of maths is that translation. You learn a lot of mathematics and you recognize powerful, subtle, deep patterns. Then you spend your time looking for those patterns in the world. It's the same thing with even the most basic "translation problems" that kids in schools learn. They're armed with arithmetic and high-school algebra so they get arithmetic and high-school algebra "real world" scenarios to analyze. At higher levels its the same. The problems are just harder and the patterns aren't necessarily given to you. Sometimes, oftentimes, nobody even knows what they are.
cousin_it 10y ago
Gödel's arithmetization of logic is one hell of a beautiful mathematical idea, up there with linear algebra or probability theory. One of its big selling points for me was how easily it solves the unexpected hanging paradox. (The simplest arithmetization of the judge's self-referential statement is a statement about numbers that can be shown to be self-contradictory.) It's also cool how it turns Russell's paradox into Gödel's theorem, Curry's paradox into Löb's theorem, etc. The connections to algorithms and computability theory are also neat (Gödel's idea of "effectively axiomatized system" is any computer program that can print sentences, which has just the right amount of generality and connects to the halting problem in the obvious way). To me arithmetization is simply the right approach to logic, which easily subsumes everything that makes sense and rejects everything that doesn't.
- westoncb 10y ago
  Do you have a recommendation for a book that covers these topics that's technical but not necessarily a text book?
  - empath75 10y ago
    The classic popular treatment is: Godel Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter
    - westoncb 10y ago
      I've read about a third of it. It's too long and too focused on other topics for me personally. I also read the book 'Gödel's Proof' a long time ago, before I had much familiarity with pure math, and remember it describing a very interesting sort of 'architecture' that Gödel had put together—but I left still wondering about any applications other than producing the incompleteness theorems. Maybe I'd get more out of it now though...
      - alasdair_ 10y ago
        GEB is well worth a second attempt. It's one of the very few books that completely changed my worldview.
catnaroek 10y ago
An article titled ”What is logic?” with no reference to computing? Does the author still live in the 19th century or what?
abtinf 10y ago
> “It’s logical, but logic has nothing to do with reality.” Logic is the art or skill of non-contradictory identification. Logic has a single law, the Law of Identity, and its various corollaries. If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. If so, then: a. things are not what they are; b. things can be and not be at the same time, in the same respect, i.e., reality is made up of contradictions. If so, by what means did anyone discover it? By illogical means. (This last is for sure.) The purpose of that notion is crudely obvious. Its actual meaning is not: “Logic has nothing to do with reality,” but: “I, the speaker, have nothing to do with logic (or with reality).” When people use that catch phrase, they mean either: “It’s logical, but I don’t choose to be logical” or: “It’s logical, but people are not logical, they don’t think—and I intend to pander to their irrationality.” http://aynrandlexicon.com/lexicon/logic.html http://aynrandlexicon.com/lexicon/logic.html
- westoncb 10y ago
  > If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. That may be the case. If our brain is only capable of forming self-consistent schemas (consider how it reacts when encountering new facts that contradict existing schemas: https://en.wikipedia.org/wiki/Cognitive_dissonance https://en.wikipedia.org/wiki/Cognitive_dissonance), then the world's apparent consistency may only be a consequence of our contemplating it through the human brain. (To be clear though, it's consistency within single schemas—not between, where contradictions abound.) > If so, then: a. things are not what they are; b. things can be and not be at the same time Can someone be a hero and not be a hero at the same time? I think there are some subtleties to this issue arising from the nature of language. Certainly at our scale (i.e. not quantum scales—although even there I don't really think 'real' inconsistency is happening, but it's less clear cut) a certain kind of consistency has very strong empirical support (it's probably our best supported hypothesis!), so it would be adaptive for us to build models of the world that would reflect that property—but this doesn't mean we've hit on something deep about the nature of reality.
  - abtinf 10y ago
    To be honest with you, I just about fell out of my chair reading your comment. Seriously, this is the first time I've posted a relevant quote from Ayn Rand on the web and received a substantive and thoughtful response. I will have to think on the points you raise.
danielam 10y ago
Speaking of classical logic, I came across a book on Fred Sommers' work entitled "The Old New Logic" (David Oderberg is editor). I haven't read it yet, so I can't comment, but based on some excerpts and reviews, it sounds like a good read. I thought I'd mention it here for those interested. Link: https://mitpress.mit.edu/books/old-new-logic https://mitpress.mit.edu/books/old-new-logic Also, those interested in the fallout of Russell's paradox are encouraged to look into Lesniewski's mereology (this is a topic that occurs in my master's thesis where I attempted to formalize the structure of Aristotelian living things mereologically using many-sorted logic). In short, Lesniewski held that the axiomatic set theory of the day betrayed the intuitions of naive set theory and resulted in paradoxes. He attempted to dispense with the these paradoxes by proposing a formal system known as mereology that replaced the primitive element relation in set theory with the parthood relation (other relations can be defined in terms of parthood, e.g., overlap, disjoint, proper/improper parthood depending on which one you assume, and vice versa). Link: https://plato.stanford.edu/entries/lesniewski/ https://plato.stanford.edu/entries/lesniewski/
spacehacker 10y ago
I've been surprised to not find anything for ctrl-F "Bayes". It turns out that traditional deductive logic is simply a special case of applying Bayes rule in order to find out the state of some binary variables given the other variables in which the probabilities are only 0 or 1. See for example these examples from Barber's "Bayesian Reasoning and Machine Learning" (p. 39): https://i.imgur.com/W1b0Wem.png https://i.imgur.com/W1b0Wem.png https://i.imgur.com/JOFvd9C.png https://i.imgur.com/JOFvd9C.png
- GregBuchholz 10y ago
  I think you might enjoy E.T. Jaynes http://bayes.wustl.edu/ http://bayes.wustl.edu/ https://www.amazon.com/Probability-Theory-E-T-Jaynes/dp/0521592712 https://www.amazon.com/Probability-Theory-E-T-Jaynes/dp/0521...
- cscurmudgeon 10y ago
  Deductive logic is sort of like the machine code for reasoning. It encompasses everything, but it is hard to see in daily usage, just like machine code. Bayes Rule is just one part of it. If you look at the Mizar mathematical library you will find a formalization of Bayes Rule using plain FOL. In fact, one of my Logic II (an intermediate logic class) exercises was formalizing a version of Bayes' rule in classic first-order logic.
bluetwo 10y ago
Logic is a tool. Like all tools, they hold no value until given a purpose, then they take on the value of the task they solve. (Just my opinion. Agree or disagree. That's cool.)
samirillian 10y ago
Great summary/history lesson, but the title is a bit misleading. The author spends very little time discussing the nature of logic per se.
juskrey 10y ago
Just a tool to embrace the unknown.
woodandsteel 10y ago
Interesting article. I would add that Frege, Russell and Whitehead did not just want to logicize mathematics. They thought that reality itself followed this logic, and so were doing metaphysics. The later Wittgenstein is a decisive refutation of this idea, and this is a key reason logic has gone out of fashion.

139 points by Hooke 10y ago

"The idea that ordinary language is expressively inadequate to account for mathematical (or even logical) reasoning became a recurring theme in the ensuing tradition of mathematical logic, so much so that the term ‘symbolic logic’ became synonymous with this tradition. Doing logic came to mean simply working with special symbols, not with ordinary words. In this respect, it is worth noting that the humanist authors had criticised the Latin of scholastic logicians precisely as ‘too artificial’, and even the Greek language that Aristotle relies on for syllogistic logic is regimented and removed from ordinary ways of speaking at the time. In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage." True that, but were/are there alternatives? Difficult to say.

throwaway729 10y ago
> In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage Of course, you could say the exact same thing for programming languages (after all, what do programmers do if not "simply working with special symbols, not with ordinary words"? And isn't this the primary criticism of programming brought by people who find it distasteful -- that expressing their fuzzy intuitions in a precise enough manner for the computer feels like a pointless exercise in frustration?) Really, the only difference between the proponents of mathematical logic over informal prose in the early 20th century and the proponents of programming languages over informal prose today is that computers obviate the problem with fuzzy thinking and fuzzier exposition. The philosopher or mathematician could hand-wave about their holes in their arguments, but the programmer can only hand-wave so much the uselessness of his unexecutable pseudo-code. The golden standard would be a form of speech that is natural and ordinary, but also avoids misleading, omitting, and eliding essential details. In other words, a language that is formal enough to be parsed, checked, and generated by a proof assistant program. But also natural enough to be read as if it's prose written by a native speaker of a natural language. I'm of the opinion that such a golden standard does not exist, and so we're forever doomed to incremental improvement.
- tikhonj 10y ago
  On a slightly orthogonal note, I think that not only can you think of programming languages this way, but that you should—programming languages and formal logics are ultimately instances of the same general idea. I've found this to be a consistent and powerful view for reasoning about programs, programming languages, proofs and logics in a uniform sort of way. The Curry-Howard correspondence is an important idea that's a specific instance of this, showing a direct relationship between specific kinds of logic and specific kinds of typed programming languages, but I think it's useful to think about programming languages that don't correspond to well-studied logical systems in a similar way. This helps explain why the study of programming languages—and especially programming language theory—isn't really about programming languages: it's more about abstraction and reasoning in general. It's deeper than you'd suspect from the name. I'd go so far as saying that programming language theory is less a theory of programming languages and more a general theory of computation from a language point of view, as compared to "normal" theoretical CS.
  - mafribe 10y ago
    programming languages and formal logics are ultimately instances of the same general idea. That's deeply questionable. The CH-correspondence breaks down as soon as you your computation includes non-termination, concurrency, timing, distribution etc. Classical logic doesn't really have wholly convincing CH-correspondences either. It seems to me that constructive proofs are a special class of programs, and fall under the purview of programming language, but not the other way around.
    - tikhonj 10y ago
      The point is that they're both formal systems of reasoning, not that they have exactly the same properties. The techniques and abstractions from both PL and formal logic often generalize well to both.
      - mafribe 10y ago
        Non-determinism, a key property of concurrent systems, doesn't generalise well to logic.
SomeStupidPoint 10y ago
The whole drive of mathematics (and logic) is ultimately that syntax is the semantics. This, of course, is annoying for almost all communication, particularly when we want to use fuzzy, hard to define terms. Languages are useful because you can pretty heavily butcher the syntax, but still be understood. (And arguably, that fuzziness is what drives new thoughts.)

pitaj 10y ago

From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic and math can lead us to useful conclusions.

FrostyElsa 10y ago
Didn't it take some guy like 200 pages to prove logically that 2+2=4?
redial 10y ago
But isn't reason is the result of logic? as in something is reasonable only when it is logically sound?
- pitaj 10y ago
  A line of thought can only be reasonable if it is logically sound. Reason is built on top of logic. All of this IIRC.
lgas 10y ago
Where can I find this standard set of axioms?
- cmrx64 10y ago
  One widely-used system is Higher Order Logic (with various extensions sometimes), one core of which you can find here: http://sourceforge.net/projects/hol/files/hol/kananaskis-10/kananaskis-10-logic.pdf/download http://sourceforge.net/projects/hol/files/hol/kananaskis-10/... This is based on type theory (and has a model in set theory), but isn't as powerful as the logics used in Coq and similar theorem provers (the Calculus of (Inductive) Constructions). You might be interested in metamath: http://us.metamath.org/ http://us.metamath.org/, which lists the axioms used in its largest body of work: http://us.metamath.org/mpegif/mmset.html#axioms http://us.metamath.org/mpegif/mmset.html#axioms
- baddox 10y ago

lngnmn 10y ago

A set of general rules for not losing a connection to reality. Could be used in reasoning or computer programming.

metaphorm 10y ago
this is amusingly missing the point. logic is, by its nature, an abstraction away from reality. it's a set of specific (not general) rules for evaluating statements of language (whether that language is natural, mathematical, or artificial/computer) such that meaning can be disambiguated. "reality", whatever it is, has no such requirement that it can be expressed in unambiguous statements.
- paganel 10y ago
  > it's a set of specific (not general) rules for evaluating statements of language Not the OP, but you could say that no "language" exists outside of "reality", and stretching it further you could also say that all types of languages (formal, mathematical, artificial/computer), including the associated "rules", are social constructs which depend heavily on the time and place of their "existence", i.e. on "reality". Until we meet some quantum-like entity which might "talk" to us from outside the boundaries of this physical Universe we're doomed to be solipsistic: we construct "abstractions" that are meant to distance us from "reality" only to find out in the end that "reality" was behind those "abstractions" all along.
- lngnmn 10y ago
  Syllogism is not general? WTF am I reading?

s_kilk 10y ago

"We just don't know" https://www.youtube.com/watch?v=Kh0Y2hVe_bw https://www.youtube.com/watch?v=Kh0Y2hVe_bw

westoncb 10y ago

My take: it's a human means of systematically exploring implications of already accepted beliefs. There are implicit and explicit forms of it. The implicit form is something our brains do automatically and is a consequence of its structure. Perhaps the way it models things intrinsically does not allow for what we would call contradicting statements (though it contains many separate models which if unified would contain contradictions); then, the space of desirable implications is narrowed by a motivation to find a certain kind of implication (i.e. there is typically a goal when engaging in reasoning—we aren't often indifferently interested in all implications) combined with the fact that contradictions aren't allowed. It seems like implications are usually discovering that some entity belongs to a class it wasn't previously known to belong to, at which point it inherits the attributes of other things in that class. The explicit forms seem like attempts to model the implicit form in mathematical language, in order to bring logical processes more under control of conscious thought, or 'executive function' (plus reasoning on paper extends working memory). It seems possible that with the 'correct' formulation, a logic could viewed as a theoretical science describing properties of the structure of the human brain that give rise to reasoning.

kazagistar 10y ago
The interesting point of the given article was that pre-mathematical-formalization, logic was backwards of that purpose, and often remains so even today. The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them. A vast majority of mathematicians don't care about the axiomatic foundations of mathematics, nor mind if it gets entirely reformulated; they already know math works, and its irrelevant which particular set of axioms is used to prove that. The assuptions => theorm structure is an artificial abstraction over a very different logic process. Symbolic logic is a framework for mechanical validation, rather then anything like how people actually do logic.
- westoncb 10y ago
  > The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them. Right, there was an issue with labeling justification 'logic'. An interesting point, but not as interesting as the title of this story—which I prefer to discuss :) I was getting at the general process of inference making in the human brain (not even conscious inference making necessarily), which I see as the real root of all the different things we call logic.

tel 10y ago

Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one another—some statements' truth depends upon what logic they're formalized within. Model theory is good here in that it studies the connection between logical language and more "physical" models. Some of these models can give a relatable account for what a logic "means". For instance, Boolean logic correspond with set operations and therefore we can use that correspondence to portray a meaning for statements in Boolean-like logics. Intuitionistic logic is an interesting one if you're not familiar with it. It can be seen as the logic of working with "demonstrations", building and analyzing them. For instance, if I have an object "X" such that witnessing it immediately convinces someone of the truth of "1+1=2", and if I have a whole collection of such items, then intuitionistic logic talks about how to compose and decompose collections of these things. Another interesting operation is "not" such that "not X" is a tool for using any X you happen to have to show that the world is inconsistent. This acts as a refutation of the existence of values like X! This differs from set-like logics because, for instance, "not (not X)" is not the same as X: being able to show that if you had (a way to prove the world inconsistent if you had an X) you could show the world as inconsistent... it's a far cry away from actually having an X! Intuitionistic logic is interesting for two reasons. First, it models communication and persuasion by being an algebra of "persuasive items" like X. Second, it's "constructive" and thus works similarly to how programming languages express things through their creation.

hackits 10y ago
Correct me if I'm wrong, I've all ways seen Maths to be easy part. The translation of English Language (grammar, sentence structure) that can be ambiguous to a very precise meaning such as Math's to be the hardest part and where people with good translation skills excel.
- tel 10y ago
  I think much of maths is that translation. You learn a lot of mathematics and you recognize powerful, subtle, deep patterns. Then you spend your time looking for those patterns in the world. It's the same thing with even the most basic "translation problems" that kids in schools learn. They're armed with arithmetic and high-school algebra so they get arithmetic and high-school algebra "real world" scenarios to analyze. At higher levels its the same. The problems are just harder and the patterns aren't necessarily given to you. Sometimes, oftentimes, nobody even knows what they are.

cousin_it 10y ago

Gödel's arithmetization of logic is one hell of a beautiful mathematical idea, up there with linear algebra or probability theory. One of its big selling points for me was how easily it solves the unexpected hanging paradox. (The simplest arithmetization of the judge's self-referential statement is a statement about numbers that can be shown to be self-contradictory.) It's also cool how it turns Russell's paradox into Gödel's theorem, Curry's paradox into Löb's theorem, etc. The connections to algorithms and computability theory are also neat (Gödel's idea of "effectively axiomatized system" is any computer program that can print sentences, which has just the right amount of generality and connects to the halting problem in the obvious way). To me arithmetization is simply the right approach to logic, which easily subsumes everything that makes sense and rejects everything that doesn't.

westoncb 10y ago
Do you have a recommendation for a book that covers these topics that's technical but not necessarily a text book?
- empath75 10y ago
  The classic popular treatment is: Godel Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter
  - westoncb 10y ago
    I've read about a third of it. It's too long and too focused on other topics for me personally. I also read the book 'Gödel's Proof' a long time ago, before I had much familiarity with pure math, and remember it describing a very interesting sort of 'architecture' that Gödel had put together—but I left still wondering about any applications other than producing the incompleteness theorems. Maybe I'd get more out of it now though...
    - alasdair_ 10y ago
      GEB is well worth a second attempt. It's one of the very few books that completely changed my worldview.

catnaroek 10y ago

An article titled ”What is logic?” with no reference to computing? Does the author still live in the 19th century or what?

abtinf 10y ago

> “It’s logical, but logic has nothing to do with reality.” Logic is the art or skill of non-contradictory identification. Logic has a single law, the Law of Identity, and its various corollaries. If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. If so, then: a. things are not what they are; b. things can be and not be at the same time, in the same respect, i.e., reality is made up of contradictions. If so, by what means did anyone discover it? By illogical means. (This last is for sure.) The purpose of that notion is crudely obvious. Its actual meaning is not: “Logic has nothing to do with reality,” but: “I, the speaker, have nothing to do with logic (or with reality).” When people use that catch phrase, they mean either: “It’s logical, but I don’t choose to be logical” or: “It’s logical, but people are not logical, they don’t think—and I intend to pander to their irrationality.” http://aynrandlexicon.com/lexicon/logic.html http://aynrandlexicon.com/lexicon/logic.html

westoncb 10y ago
> If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. That may be the case. If our brain is only capable of forming self-consistent schemas (consider how it reacts when encountering new facts that contradict existing schemas: https://en.wikipedia.org/wiki/Cognitive_dissonance https://en.wikipedia.org/wiki/Cognitive_dissonance), then the world's apparent consistency may only be a consequence of our contemplating it through the human brain. (To be clear though, it's consistency within single schemas—not between, where contradictions abound.) > If so, then: a. things are not what they are; b. things can be and not be at the same time Can someone be a hero and not be a hero at the same time? I think there are some subtleties to this issue arising from the nature of language. Certainly at our scale (i.e. not quantum scales—although even there I don't really think 'real' inconsistency is happening, but it's less clear cut) a certain kind of consistency has very strong empirical support (it's probably our best supported hypothesis!), so it would be adaptive for us to build models of the world that would reflect that property—but this doesn't mean we've hit on something deep about the nature of reality.
- abtinf 10y ago
  To be honest with you, I just about fell out of my chair reading your comment. Seriously, this is the first time I've posted a relevant quote from Ayn Rand on the web and received a substantive and thoughtful response. I will have to think on the points you raise.

danielam 10y ago

Speaking of classical logic, I came across a book on Fred Sommers' work entitled "The Old New Logic" (David Oderberg is editor). I haven't read it yet, so I can't comment, but based on some excerpts and reviews, it sounds like a good read. I thought I'd mention it here for those interested. Link: https://mitpress.mit.edu/books/old-new-logic https://mitpress.mit.edu/books/old-new-logic Also, those interested in the fallout of Russell's paradox are encouraged to look into Lesniewski's mereology (this is a topic that occurs in my master's thesis where I attempted to formalize the structure of Aristotelian living things mereologically using many-sorted logic). In short, Lesniewski held that the axiomatic set theory of the day betrayed the intuitions of naive set theory and resulted in paradoxes. He attempted to dispense with the these paradoxes by proposing a formal system known as mereology that replaced the primitive element relation in set theory with the parthood relation (other relations can be defined in terms of parthood, e.g., overlap, disjoint, proper/improper parthood depending on which one you assume, and vice versa). Link: https://plato.stanford.edu/entries/lesniewski/ https://plato.stanford.edu/entries/lesniewski/

spacehacker 10y ago

I've been surprised to not find anything for ctrl-F "Bayes". It turns out that traditional deductive logic is simply a special case of applying Bayes rule in order to find out the state of some binary variables given the other variables in which the probabilities are only 0 or 1. See for example these examples from Barber's "Bayesian Reasoning and Machine Learning" (p. 39): https://i.imgur.com/W1b0Wem.png https://i.imgur.com/W1b0Wem.png https://i.imgur.com/JOFvd9C.png https://i.imgur.com/JOFvd9C.png

GregBuchholz 10y ago
I think you might enjoy E.T. Jaynes http://bayes.wustl.edu/ http://bayes.wustl.edu/ https://www.amazon.com/Probability-Theory-E-T-Jaynes/dp/0521592712 https://www.amazon.com/Probability-Theory-E-T-Jaynes/dp/0521...
cscurmudgeon 10y ago
Deductive logic is sort of like the machine code for reasoning. It encompasses everything, but it is hard to see in daily usage, just like machine code. Bayes Rule is just one part of it. If you look at the Mizar mathematical library you will find a formalization of Bayes Rule using plain FOL. In fact, one of my Logic II (an intermediate logic class) exercises was formalizing a version of Bayes' rule in classic first-order logic.

bluetwo 10y ago

Logic is a tool. Like all tools, they hold no value until given a purpose, then they take on the value of the task they solve. (Just my opinion. Agree or disagree. That's cool.)

samirillian 10y ago

Great summary/history lesson, but the title is a bit misleading. The author spends very little time discussing the nature of logic per se.

juskrey 10y ago

Just a tool to embrace the unknown.

woodandsteel 10y ago

Interesting article. I would add that Frege, Russell and Whitehead did not just want to logicize mathematics. They thought that reality itself followed this logic, and so were doing metaphysics. The later Wittgenstein is a decisive refutation of this idea, and this is a key reason logic has gone out of fashion.