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The Impossibility of the Contrary
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9/24/2015 5:33:49 PM Posted: 1 year ago At 9/24/2015 12:43:38 AM, scmike2 wrote: I felt that while Mike's claim of divine revelation is one topic, this one should be separated as it is a different reasoning, one that is clearly circular, but interesting, nonetheless. So Mike, can you explain why this is not circular reasoning? Marrying a 6 year old and waiting until she reaches puberty and maturity before having consensual sex is better than walking up to a stranger in a bar and proceeding to have relations with no valid proof of the intent of the person. Muhammad wins. ~ Fatihah If they don't want to be killed then they have to subdue to the Islamic laws.  Uncung There would be peace if you obeyed us.~Uncung Without God, you are lower than sh!t. ~ SpiritandTruth 
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9/24/2015 9:33:22 PM Posted: 1 year ago At 9/24/2015 5:33:49 PM, DanneJeRusse wrote:At 9/24/2015 12:43:38 AM, scmike2 wrote: Are you kidding me? You feel the need to start a whole new thread when this is perfectly relevant to the one in progress? If this is a challenge to the above claims, then I respectfully request that you post it at the other thread and state it as such, as I'm having a hard enough time keeping up, as is, without bouncing around from OP to OP. There's still plenty of bandwidth left over there on the first thread you just started for us to hash this out. Besides, I'm still debating on whether or not to cut you loose anyway, as you haven't provided any rational argument thus far against the Christian claim. Maybe you can remedy that? Actually, Joe seems to be about the only one seeking a rational discourse over there and I want to focus the bulk of my energy on responding to him at this point so that his questions get the consideration they deserve. Anyway, if you have a rational objection, put it forth at the other thread. If not, thanks for your time. 
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9/24/2015 9:45:39 PM Posted: 1 year ago I guess you can take that as a no.

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9/24/2015 10:02:59 PM Posted: 1 year ago At 9/24/2015 5:33:49 PM, DanneJeRusse wrote:At 9/24/2015 12:43:38 AM, scmike2 wrote: How is it circular? It's A is contrary to not A. If not A is impossible then A is True. Do you know what circular means? 
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9/24/2015 10:32:24 PM Posted: 1 year ago At 9/24/2015 9:45:39 PM, dhardage wrote: I guess you're right. Most likely, he has no answer other than he has personally decided that circular reasoning is logical reasoning, which may not be valid here, but perhaps is in the Bizarro Universe. Marrying a 6 year old and waiting until she reaches puberty and maturity before having consensual sex is better than walking up to a stranger in a bar and proceeding to have relations with no valid proof of the intent of the person. Muhammad wins. ~ Fatihah If they don't want to be killed then they have to subdue to the Islamic laws.  Uncung There would be peace if you obeyed us.~Uncung Without God, you are lower than sh!t. ~ SpiritandTruth 
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9/24/2015 11:17:57 PM Posted: 1 year ago At 9/24/2015 5:33:49 PM, DanneJeRusse wrote:I felt that while Mike's claim of divine revelation is one topic, this one should be separated as it is a different reasoning, one that is clearly circular, but interesting, nonetheless.At 9/24/2015 12:43:38 AM, scmike2 wrote: The main problem with reductio ad absurdum arguments is that they rely on excluded middles to turn negative information into a positive conclusion. To ensure that excluding the middle is not a fallacy, you need to either argue by exhaustion, or state any assumptions  and assumptions may be implicit and overlooked. To see why this makes for dangerous reasoning, imagine that there are five suspects for a suspected homicide, and four have airtight alibis. Does that automatically make the fifth a murderer? Would you prosecute on that evidence? Could you convict? You usually couldn't convict on that argument alone. Instead you'd likely focus attention on the fifth suspect and try to find a constructive case that this person did commit the murder, rather than shifting the burden of evidence to the defendant. Philosophers and theologians generally like reductio ad absurdum arguments, and mathematicians tolerate them, but science, engineering and other contestable disciplines treat them with caution because they're not terribly diligent. Doctors generally won't perform irreversible surgery on excluded middle reasoning; courts resist convicting felonies on excluded middle alone; and air safety investigators often find excluded middle reasoning at the heart of air crashes. Finally, one of the problems with nonconstructive proofs is that you're never sure why they're true. There's a famous reductio ad absurdum proof that the square root of two (sqrt(2)) cannot be expressed as a fraction of integers, for example, that never shows you how to calculate sqrt(2). [http://www.mathsisfun.com...] So you may be no wiser for the argument, even if it's valid. In conclusion, unless you're really confident of having exhausted all reasonable possibilities in a wellknown domain, it's generally better to treat reductio ad absurdum arguments as conjectural, rather than conclusive. That doesn't mean they should be shunned, but they shouldn't be automatically trusted by themselves either. :) And even if such an argument is valid, that doesn't mean it's the last word in wisdom. :) 
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9/25/2015 1:39:26 AM Posted: 1 year ago At 9/24/2015 10:02:59 PM, Mhykiel wrote:At 9/24/2015 5:33:49 PM, DanneJeRusse wrote:At 9/24/2015 12:43:38 AM, scmike2 wrote: Yes, sorry, my mistake, it is indeed the reductio ad absurdum argument. Marrying a 6 year old and waiting until she reaches puberty and maturity before having consensual sex is better than walking up to a stranger in a bar and proceeding to have relations with no valid proof of the intent of the person. Muhammad wins. ~ Fatihah If they don't want to be killed then they have to subdue to the Islamic laws.  Uncung There would be peace if you obeyed us.~Uncung Without God, you are lower than sh!t. ~ SpiritandTruth 
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9/25/2015 5:40:11 AM Posted: 1 year ago At 9/24/2015 11:17:57 PM, RuvDraba wrote:At 9/24/2015 5:33:49 PM, DanneJeRusse wrote:I felt that while Mike's claim of divine revelation is one topic, this one should be separated as it is a different reasoning, one that is clearly circular, but interesting, nonetheless.At 9/24/2015 12:43:38 AM, scmike2 wrote: http://media.giphy.com... 
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9/25/2015 5:52:35 AM Posted: 1 year ago At 9/25/2015 5:40:11 AM, Envisage wrote:At 9/24/2015 11:17:57 PM, RuvDraba wrote:At 9/24/2015 5:33:49 PM, DanneJeRusse wrote:I felt that while Mike's claim of divine revelation is one topic, this one should be separated as it is a different reasoning, one that is clearly circular, but interesting, nonetheless.At 9/24/2015 12:43:38 AM, scmike2 wrote: It's always great to see you about, Envisage. You lift the tone. :D 
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9/25/2015 5:54:52 AM Posted: 1 year ago At 9/24/2015 11:17:57 PM, RuvDraba wrote:  What you're referring to in not really a reductio ad absurdum proof. The proof itself is undeniable. You're referring to some loose process of elimination that has more to do with intuition than formal logic & deductive reasoning. Reductio ad absurdum proofs in mathematics are not "tolerable", they are absolute. 
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9/25/2015 5:56:10 AM Posted: 1 year ago At 9/25/2015 5:40:11 AM, Envisage wrote:  Glad to see you live & kicking again. What are you so happy about?! 
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9/25/2015 6:18:16 AM Posted: 1 year ago At 9/25/2015 5:54:52 AM, Yassine wrote:At 9/24/2015 11:17:57 PM, RuvDraba wrote: Actually, their legitimacy is contested in mathematics, Yassine. See for example Constructivism [https://en.wikipedia.org...]. It's an advanced math topic, but there are also mathematical logics in which the statement 'A or not A' cannot be presumed true, and in those logics no reductio ad absurdum proof is valid. [https://en.wikipedia.org...] Arguably, those logics are a better fit for key problems in science and engineering than is classical logic  and they have interesting implications for epistemology too. But if the only proof you have is a reductio proof, that's generally tolerated as an advance  even if it's not the last word on why a theorem is true. :) Generally speaking, while reductio proofs are often simpler, if you can show how the defendant had means, motive and opportunity to commit the murder, that there are no other plausible suspects, and that the defendant cannot give a more reasonable account of his innocence, you get a much more insightful and diligent account of the crime than if you simply show that the victim had to have been murdered, and nobody else could have done it. :) Likewise, in maths, it's more useful to produce a decimal expansion of sqrt(2) and show that it doesn't ever repeat, than to argue that sqrt(2) is irrational without knowing how to produce the number in the first place. :D 
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9/25/2015 6:51:38 AM Posted: 1 year ago At 9/25/2015 6:18:16 AM, RuvDraba wrote:  You probably misunderstood what Constructivism refers to, & I assume it's the same term used in French. First of all, deductive proofs are still deductive. There is no difference between classical mathematics & constructivist mathematics when it comes to deductive proofs, including reductio ad absurdum proofs. Second of all, constructivist mathematics emerge in cases of infinite or arbitrary truth values, such as in probability & the likes.  There is a similar notion in Islamic logic as well. In constructivist mathematics, the existence of something is established by the possibly of its construction. In contrast, in Islamic logic, the existence of something relates to the degree of its necessity. In that sense, a cause would be more likely existent than its effect. :) Generally speaking, while reductio proofs are often simpler, if you can show how the defendant had means, motive and opportunity to commit the murder, that there are no other plausible suspects, and that the defendant cannot give a more reasonable account of his innocence, you get a much more insightful and diligent account of the crime than if you simply show that the victim had to have been murdered, and nobody else could have done it. :)  I don't disagree with this. It just has nothing to do with formal logic or mathematics. Likewise, in maths, it's more useful to produce a decimal expansion of sqrt(2) and show that it doesn't ever repeat, than to argue that sqrt(2) is irrational without knowing how to produce the number in the first place. :D  That's an example of constructivist mathematics. As I said, the truth values here are indeed infinite. However, in this very instance, the second method is actually better & conclusive. We can know that sqrt(2) is not rational because we designed it that way. & we don't really care how the decimal numbers behave. A more pertinent example would probably involve the different yet to be deductively proved prime numbers theorems. 
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9/25/2015 7:39:26 AM Posted: 1 year ago At 9/25/2015 6:51:38 AM, Yassine wrote:Unlikely, since I managed to convince my doctoral examiners that I knew what I was talking about. :)At 9/25/2015 6:18:16 AM, RuvDraba wrote: You probably misunderstood what Constructivism refers to, First of all, deductive proofs are still deductive. There is no difference between classical mathematics & constructivist mathematics when it comes to deductive proofs, including reductio ad absurdum proofs.I'm sorry, Yassine, but that's incorrect. Constructivist deduction still uses syllogism and so on, but limits what you can stipulate in ways that classical logic does not. In classical deduction you can always assume the Excluded Middle: that A or not A is true, for any wellformed formula A. As Mhykiel mentioned earlier, these are often injected into classical deduction proofs, especially reductio ad absurdum proofs, so you can wipe off one case by contradiction, and prove the other true. However, you can't assume that in the constructivist logic called Intuitionistic logic, for example. [https://en.wikipedia.org...] You have to stipulate which is true  A or not A, and demonstrate it to be so. Only then can you go on to make inferences with it. One of the reasons reductio ad absurdum proofs are invalid in intuitionistic logic is that they're hard to work unless you can write 'A or not A' somewhere  though that's not the only reason. :) Second of all, constructivist mathematics emerge in cases of infinite or arbitrary truth values, such as in probability & the likes.They emerge everywhere. So the Axiom of Choice, which operates over infinite sets, is a matter of constructivist concern, but there are intuitionistic versions of boolean propositional logic and other simple logics too  and they invalidate Excluded Middle and reductio ad absurdum (hint: read the earlier links.)  There is a similar notion in Islamic logic as well. In constructivist mathematics, the existence of something is established by the possibly of its construction. In contrast, in Islamic logic, the existence of something relates to the degree of its necessity. In that sense, a cause would be more likely existent than its effect. :)I'm delighted but unsurprised that Muslim philosophers tumbled to this distinction early, Yassine. Islam simply doesn't see enough credit for its many valuable contributions to empiricism and critical thought. :) Except for the history of logic and math, as shown in the links I supplied earlier.Generally speaking, while reductio proofs are often simpler, if you can show how the defendant had means, motive and opportunity to commit the murder, that there are no other plausible suspects, and that the defendant cannot give a more reasonable account of his innocence, you get a much more insightful and diligent account of the crime than if you simply show that the victim had to have been murdered, and nobody else could have done it. :) I don't disagree with this. It just has nothing to do with formal logic or mathematics. Yes indeed.Likewise, in maths, it's more useful to produce a decimal expansion of sqrt(2) and show that it doesn't ever repeat, than to argue that sqrt(2) is irrational without knowing how to produce the number in the first place. :D That's an example of constructivist mathematics. As I said, the truth values here are indeed infinite.No, the truth values aren't  they're just the usual binary 'true' and 'false'. Just the domain is. However, in this very instance, the second method is actually better & conclusive. We can know that sqrt(2) is not rational because we designed it that way.Agreed for this proof, though iirrational numbers were first discovered rather than devised. The earliest existence proof of irrational numbers is a reductio ad absurdum sometimes attributed to Pythagoras, sometimes the philosopher Hippasus. [https://en.wikipedia.org...] irrational numbers turn up a lot in triangles, and one of the ancient Greeks discovered that in certain isoceles triangles, either the hypotenuse or both of the legs had to be of irrational length. This discovery freaked out the Pythagoreans, by the way, who held that numbers were at the root of all metaphysics, and that all numbers could be represented as fractions. So by their definition, one of the sides of the triangle had to be 'not a number'! According to legend, the gods punished Hippasus for this impiety by drowning him at sea. :) [https://en.wikipedia.org...]) 
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9/25/2015 8:10:15 AM Posted: 1 year ago At 9/25/2015 7:39:26 AM, RuvDraba wrote:  What is it that you do? However, you can't assume that in the constructivist logic called Intuitionistic logic, for example. [https://en.wikipedia.org...] You have to stipulate which is true  A or not A, and demonstrate it to be so. Only then can you go on to make inferences with it. One of the reasons reductio ad absurdum proofs are invalid in intuitionistic logic is that they're hard to work unless you can write 'A or not A' somewhere  though that's not the only reason. :)  As I was saying, this applies to arbitrary or infinite truth value. Meaning, in cases other than where A is completely defined by notA. They emerge everywhere. So the Axiom of Choice, which operates over infinite sets, is a matter of constructivist concern, but there are intuitionistic versions of boolean propositional logic and other simple logics too  and they invalidate Excluded Middle and reductio ad absurdum (hint: read the earlier links.)  Boolean propositions that are not well formulated maybe. I'm delighted but unsurprised that Muslim philosophers tumbled to this distinction early, Yassine. Islam simply doesn't see enough credit for its many valuable contributions to empiricism and critical thought. :)  Well, thank you. Indeed, many of the modern ideas in logic & epistemology emerged earlier in the Islamic golden age. A good example here would be Hume's causality refutation, which has been done by alAsh'ari & alGhazali several centuries before. Except for the history of logic and math, as shown in the links I supplied earlier.  I am not sure what you mean by that. What does mathematics have to do with legal cases?! Btw, I tried to read it, but I just don't like wikipedia at all. No, the truth values aren't  they're just the usual binary 'true' and 'false'. Just the domain is.  Depends on how you formulate it. Agreed for this proof, though irrational numbers were first discovered rather than devised.  That's the difference I was referring to. The same can be observed in many of our theorems related to prime numbers for instance. The earliest existence proof of irrational numbers is a reductio ad absurdum sometimes attributed to Pythagoras, sometimes the philosopher Hippasus.[https://en.wikipedia.org...] irrational numbers turn up a lot in triangles, and one of the ancient Greeks discovered that in certain isoceles triangles, either the hypotenuse or both of the legs had to be of irrational length.  Interesting history. I always thought irrational numbers were first discovered by the Babylonians! 
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9/25/2015 9:06:51 AM Posted: 1 year ago Irrational numbers can always be found in places of worship and elsewhere.
You're welcome. No man ever believes that the Bible means what it says: He is always convinced that it says what he means. George Bernard Shaw 
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9/25/2015 7:33:14 PM Posted: 1 year ago At 9/25/2015 8:10:15 AM, Yassine wrote:I'm a bit of a dilettante. :) These days I run a consulting company parttime, and meanwhile write fiction and am learning to record/produce music. In former lives I've been a research scientist, academic, science manager, and consultant in a range of different informatic fields from symbolic logic and automated reasoning through data mining, data visualisation, information strategy, architecture, project management and assurance, contract management and procurement.At 9/25/2015 7:39:26 AM, RuvDraba wrote: However, you can't assume that in the constructivist logic called Intuitionistic logic, for example. [https://en.wikipedia.org...] You have to stipulate which is true  A or not A, and demonstrate it to be so. Only then can you go on to make inferences with it. One of the reasons reductio ad absurdum proofs are invalid in intuitionistic logic is that they're hard to work unless you can write 'A or not A' somewhere  though that's not the only reason. :) As I was saying, this applies to arbitrary or infinite truth value. Meaning, in cases other than where A is completely defined by notA.They emerge everywhere, my scholastic brother. :) Probably the most famous constructivist logic is intuitionistic logic, which like classical logic, has only two truthvalues: true and false. There's a famous result in intuitionistic logic that it's a twovalued logic, like classical logic. You don't need a third 'undefined' value to model it. (Please poke me for a link if this interests you.) But sure, there are multivalued logics (e.g. you hinted at fuzzy logic earlier, which acts much like probability does), but not all are constructivist; and not all constructivist logics are multivalued. But speaking of the infinite, another problem constructivism has is with the Axiom of Choice: the idea that in a bag with an infinite number of blue stones, and a finite number of red stones, you can reach in and somehow pull out a red one. The Axiom of Choice is used a lot in set theory, number theory, and some kinds of induction, and classical logic is usually fine with it. However if you presented that problem to an engineer, they'd ask you how you mean to find a red marble in finite time when the probability of finding a marble in any selection is effectively zero, and pulling blue marbles out doesn't simplify the search. Constructivist logicians may ask the same question: what's your procedure for finding a red marble in an infinite number of blue ones? So some constructivist logics reject the Axiom of Choice and thus put pressure on mathematical and philosophical idealism. :) Others weaken it, or grudgingly accept it. :)  Boolean propositions that are not well formulated maybe.Intuitionism takes the same grammar as classical logic, so everything is wellformed. Really what it's doing is insisting on a higher level of accountability for the logic than classical logic does. It's a kind of 'no BS' logic  to prove anything, you have to talk in detail about the thing you're proving  you can't talk around it, and that's why reductio proofs and excluded middle assumptions are rejected. We can also thank alHaytham (aka Alhazen) for his contributions to the scientific method, too [https://en.wikipedia.org...], and alKhwarizmi for his formal contributions to informatics and math. :) [https://en.wikipedia.org...]I'm delighted but unsurprised that Muslim philosophers tumbled to this distinction early, Yassine. Islam simply doesn't see enough credit for its many valuable contributions to empiricism and critical thought. :) Well, thank you. Indeed, many of the modern ideas in logic & epistemology emerged earlier in the Islamic golden age. A good example here would be Hume's causality refutation, which has been done by alAsh'ari & alGhazali several centuries before. My links showed that constructivism has played a major (and relatively recent) contribution to logic and math that shouldn't be ignored. What does mathematics have to do with legal cases?!Except for the history of logic and math, as shown in the links I supplied earlier.Generally speaking, while reductio proofs are often simpler, if you can show how the defendant had means, motive and opportunity to commit the murder, that there are no other plausible suspects, and that the defendant cannot give a more reasonable account of his innocence, you get a much more insightful and diligent account of the crime than if you simply show that the victim had to have been murdered, and nobody else could have done it. :) I don't disagree with this. It just has nothing to do with formal logic or mathematics. But more broadly, mathematics offers the same sort of contribution to law that it offers to business: business without math is blind, inefficient and ineffective; law without math is ignorant, needlessly contentious and unjust. It's offtopic, but I have some examples of this if you're interested. And there are mathematical insights about why constructivism provides better accountability and more reliable reasoning than reductio ad absurdumand other nonconstructive arguments. That doesn't mean we must always insist on constructivism, but it does suggest it's prudent to use constructive reasoning whenever we're ignorant of the domain. Constructivist thinking has now turned up in the sciences, it's all through engineering and it's making its way into law, business management  even less academic fields like cooking and winemaking. The earliest existence proof of irrational numbers is a reductio ad absurdum sometimes attributed to Pythagoras, sometimes the philosopher Hippasus.[https://en.wikipedia.org...] irrational numbers turn up a lot in triangles, and one of the ancient Greeks discovered that in certain isoceles triangles, either the hypotenuse or both of the legs had to be of irrational length. We owe the Babylonians (and their Egyptian neighbours) a debt for elementary arithmetic. They had some fractions; could extract square roots (useful for engineering); they could solve linear equations (handy for logistical problems); they could deal with triangles where the length of every side was a whole number (again, good for engineering); they had tables for cubic equations (useful for knowing what size ship or building you can build from planks of wood); had some understanding about the dimensions of a circle (useful for astronomy and navigation); and eventually developed astronomy comparable to that of the Greeks. But they had no zero, made elementary mistakes in geometry, and weren't comfortable enough with fractions to deal with weirdsized triangles, much less irrational numbers. [http://www.math.tamu.edu...] But still, they used base sixty in their arithmetic, which is very cool because it has so many convenient divisors  you can cut a plank of wood sixty fingers long into even planks without wastage in many different ways. :) 
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9/28/2015 12:22:20 AM Posted: 1 year ago At 9/25/2015 7:33:14 PM, RuvDraba wrote:  That is some resume. They emerge everywhere, my scholastic brother. :) Probably the most famous constructivist logic is intuitionistic logic, which like classical logic, has only two truthvalues: true and false. There's a famous result in intuitionistic logic that it's a twovalued logic, like classical logic. You don't need a third 'undefined' value to model it. (Please poke me for a link if this interests you.)  I am not sure what you're referring to here, but show me what you've got. In a branch of our Logic, called Q'yas (kinda like Analogical Reasoning), there are systems of 3, 5, 10, 15... truthvalues. But sure, there are multivalued logics (e.g. you hinted at fuzzy logic earlier, which acts much like probability does), but not all are constructivist; and not all constructivist logics are multivalued.  If it's well formulated, i.e. all the elements are identified, it's binary. Otherwise, it's either arbitrary, in which case you've one truth value, & its opposite(s) ; or it's infinite (or continuous). But speaking of the infinite, another problem constructivism has is with the Axiom of Choice: the idea that in a bag with an infinite number of blue stones, and a finite number of red stones, you can reach in and somehow pull out a red one.  Good thing reality is a lot simpler. There is a variant example where the cycles are infinite, & every time the picker either changes or loses his memory. It's quite mindbending. Constructivist logicians may ask the same question: what's your procedure for finding a red marble in an infinite number of blue ones? So some constructivist logics reject the Axiom of Choice and thus put pressure on mathematical and philosophical idealism. :) Others weaken it, or grudgingly accept it. :)  The problem with these problems is that they are not well formulated. There are gaps between the question you're posing & the answer you're looking for, i.e. there isn't a direct deductive link. Probably, in few hundred years, these problems will become ordinary in a much more sophisticated mathematical framework. Intuitionism takes the same grammar as classical logic, so everything is wellformed.  Not quite. Deductive reasoning deals only with structure, not with the degree of Truth. Nonbinary structures are a little messy at this point. The best way to deal with them is through Probabilistic reasoning, which is still messy. Really what it's doing is insisting on a higher level of accountability for the logic than classical logic does.  True. I would say it limits classical logic to what classical logic does well. It's a kind of 'no BS' logic  to prove anything, you have to talk in detail about the thing you're proving  you can't talk around it, and that's why reductio proofs and excluded middle assumptions are rejected.  They are not rejected. They are just not very useful in nonbinary structures. We can also thank alHaytham (aka Alhazen) for his contributions to the scientific method, too [https://en.wikipedia.org...], and alKhwarizmi for his formal contributions to informatics and math. :) [https://en.wikipedia.org...]  Of course. My links showed that constructivism has played a major (and relatively recent) contribution to logic and math that shouldn't be ignored.  Indeed. But more broadly, mathematics offers the same sort of contribution to law that it offers to business: business without math is blind, inefficient and ineffective; law without math is ignorant, needlessly contentious and unjust. It's offtopic, but I have some examples of this if you're interested.  You're probably referring to applied mathematics, not pure mathematics. And there are mathematical insights about why constructivism provides better accountability and more reliable reasoning than reductio ad absurdumand other nonconstructive arguments. That doesn't mean we must always insist on constructivism, but it does suggest it's prudent to use constructive reasoning whenever we're ignorant of the domain.  I think I get what you're trying to say here. Constructivist thinking has now turned up in the sciences, it's all through engineering and it's making its way into law, business management  even less academic fields like cooking and winemaking.  I don't see how this is relevant! We owe the Babylonians (and their Egyptian neighbours) a debt for elementary arithmetic. They had some fractions; could extract square roots (useful for engineering); they could solve linear equations (handy for logistical problems); they could deal with triangles where the length of every side was a whole number (again, good for engineering); they had tables for cubic equations (useful for knowing what size ship or building you can build from planks of wood); had some understanding about the dimensions of a circle (useful for astronomy and navigation); and eventually developed astronomy comparable to that of the Greeks.  In few hundred years, some people are probably gonna be talking about us, as we're now talking about those before us. 