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Thread: The deductive logic of arguments v. mathematical logic

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    The deductive logic of arguments v. mathematical logic

    Initially, mathematics was the systematic deduction of the logical consequences of axioms, axioms understood to constitute together, somehow, a model of some particular aspect of the real world. The most historically glorified example of that is probably Euclid's geometry:

    Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects.- Euclidean geometry - Wikipedia https://en.wikipedia.org/wiki/Euclidean_geometry
    The aspect of that which is still true today is that mathematics is a formal discipline: axioms are formal expressions, derivation of consequences is a formal process based on formal expressions, and the validity of the derivations is hopefully entirely justified on form.

    However, in the 20th century, mathematics has evolved its methodology in different directions. One aspect of that is that while before the 20th century mathematicians appear to have used the same logic to deduce consequences, now the picture is much more diverse and indeed complicated.

    The expression now favoured by mathematicians is not the "deduction of the consequences". It is the "derivation of consequences". Derivation is "the process of deducing a mathematical theorem, formula, etc, as a necessary consequence of a set of accepted statements". See https://www.collinsdictionary.com/us...ish/derivation

    A derivation is thus a process, or rather a procedure, and therefore essentially seen as a concrete operation, essentially like the derivation of the result 5 from the operation 2 + 3. It is also still an entirely formal operation, in that it depends only of the formal properties of these "accepted statements", as Collins calls them.

    As an entirely formal process, any derivation could be performed, at least in principle, by a computer, although the vast majority of mathematical proofs are still for now arrived at using, as Wikipedia puts it, "rigorous informal logic". See https://en.wikipedia.org/wiki/Mathematical_proof

    Deduction of course, at least in the context of a formal discipline, is itself a formal operation, and there is in effect little difference between the two notions. However, the term derivation allows mathematicians to emphasise the idea of a formal procedure, whereas deduction is essentially thought of, and indeed has always been thought of from Aristotle himself, as a particular kind of reasoning, that is, somehow, as an operation of the mind. See https://en.wikipedia.org/wiki/Deductive_reasoning

    Thus, while strictly speaking deduction is constrained by the kind of mind and, presumably, the kind of brain, humans have, derivation can be made to proceed according to any number of ad hoc rules that do not necessarily reflect human deduction. Thus, mathematicians themselves can decide which rules apply, and different mathematicians may decide to use different rules.

    In particular, mathematical logic, which is itself a discipline, not a method of logic, brings together mathematicians who use, and investigate the use of, different sets of rules. And there is indeed a large number of such systems, such as for example 1st order logics and second order logics, relevance logics and paraconsistent logics, constructive or intuitionistic logics.

    As a concrete example, we can mention intuitionistic logic, also called constructive logic, which differs from mainstream mathematical logic, itself usually and tellingly called "classical logic", by not including the law of excluded middle and the double negation elimination rule, which are, however, fundamental inference rules in classical logic. See https://en.wikipedia.org/wiki/Intuitionistic_logic

    Some systems of logic, for example multi-valued logics, will have different but analogous laws to stand in for the law of excluded middle. See https://en.wikipedia.org/wiki/Law_of_excluded_middle

    Thus, while there is only one logic of argument, as understood since Aristotle first formalised it in his syllogistic 2,500 years ago, there are now any number of ways that a mathematicians can choose to derive a theorem from a set of axioms.
    EB

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    Would every case of a deduction be a case of derivation yet not inversely?

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    Quote Originally Posted by fast View Post
    Would every case of a deduction be a case of derivation yet not inversely?
    Good question!

    Strictly speaking, deduction is something the human mind, or brain, does. Derivation is a procedure, performed by humans or by a machine. As a procedure performed by humans, a derivation requires deductions. However, all derivations also involve the application of formal rules, which are in effect the rules of your formal logic (to be understood as distinct from logic as a human capability).

    Each application of any one rule requires deduction, and so any derivation will usually involve many deductions. However, deductions in this case relate to the application of the rule, not directly to whatever premises the rule is applied to. The premise becomes an input to the rule. Deduction here applies to the input and the rule, not to the premise as such.

    To give an example, you can derive the result 5 from the input 2 + 3 by one rule of addition. To derive the "correct" result, you need to identify the appropriate rule and deduce the result not from the premise "2 + 3", but in effect from the rule and from "2 + 3" as an input to the rule rather than as a premise of your deduction. Thus, deduction is used to apply the rule rather than directly to deduce the correct result from "2 + 3", even if the result is the same, provided the rule is appropriate.

    Derivation may be seen as an intermediate machinerie you put between the premises and the conclusion, somewhat like using a computer to add 2 + 3 instead of doing the calculation in your mind. In this case, you no longer need to know what 2 + 3 is. Instead, you need to know how to use the computer. Doing mathematics now is exactly that. You need to know how to use one or any number of the machineries of derivation invented by mathematicians since the beginning of the 20th century. But, it is still true that apply the rules involves deduction.

    Deduction, as a process of the human mind or brain, may be seen itself as tantamount to a derivation. However, we only have one way to do it, whereas mathematics now effectively harbours different techniques of derivation. And more crucially, nobody seems to understand how deduction actually works. The various machineries of derivation used in mathematics rely on ad hoc rules invented by mathematicians and there is no good reason to believe that they do the same thing as human deduction. In fact, throughout the last few months, I have tested on this forum many examples where mathematical logic fails to produce the same results as human deduction.

    In this sense, no derivation is a deduction. Sometimes, a derivation will produce the same result as deduction, but you only know it's the same result because you recognise it as such. There's nothing in the derivation procedure itself that could possibly tell you that the result is correct. You still need the human mind to tell you that.

    We won't be able to say that a deduction really is equivalent to a derivation until we discover an algorithm--involving some sort of formal rules--that we could prove is the analogue to what the human mind does..For now, it seems clear nobody knows such an algorithm, notwithstanding claims to the contrary.
    EB

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    What is the mathematical logic you refer to? To me that is Boolean Algebra.

    Logic and reasoning is in general no different than anything else in math and science. One uses both deduction and induction. Formal logic may be used if appropriate. Same with syllogisms.

    Logic breaks down to AND, INCLUSIVE OR, EXCLUSIVE OR , and NOT or negation.

    When dong a proof when there is no clear path and it it s not known if the proof exists one uses deduction, induction, and trial and error.

    And there are modern forms of logic in use such as Fuzzy Logic and multilevel logic. There is statistical logic. Classical logic is of course part of it, but not all of it.

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    To me (and I say again, to me), to deduce is (oh, how can I say this) subtractive in nature. The phrase “process of elimination” comes to mind. Imagine taking a test and you have a multiple choice question. There are five possible answers. One option is obviously wrong and so I REDUCE the five options down to four viable options. Through reasoning (or noticing a conflict that’s inconsistent), I eliminate another option. I’m down to three. Through another bout of reasoning, I further narrow my original set of five down between two candidates.

    Now I’m stuck. I have to think through it again but from another third angle. I’m the Man Tracker on horseback and my human prey will not elude me by their trickery: walking in one direction (leaving tracks) and then carefully walking backwards making sure to step only where stepped before.

    I might deduce the prey has done just that—from the deeper than normal impressions, especially towards the front (and not heel) of the foot; between that and the occasional unexplained width expansion, but have I (have I really) deduced that?

    I can derive a conclusion through deduction, but then again, I can derive a conclusion through induction, and it seems we may sometimes incorporate various elements of reasoning amidst the complexity of it all.

    And through another line of reasoning, I’m not as stuck as I once was. Through deduction, I made much progress, and if I was mistake free through a purely deductive process, I should be overwhelmingly confident up to the point where I incorporate an inductive mindset—at which point my confidence will swing in accordance to the strength of the evidence.

    In any case, I have derived a conclusion. A purely deductive process will gain no insight except to shine a light on what was hard to see. Through an inductive process, we will gain no insight except to shine a light on what may be.

    And now, math.

    Ask me what’s 65 squared while a third grader types in 65*65. Me, well, I don’t know the answer right away, but it’s within me to figure it out fast-like. Because both numbers end in five, I know the final solution ends in 25. (5*5=25)

    Next, I know that 6 times one more than that is 42. (6*7=42). The child with the calculator will soon see 4225 display on his screen.

    Did either of us use logic? We both came to the same conclusion. We used an entirely different process, but we derived a solution that is undisputably correct. What’s the ole saying about seeing far when standing on the shoulders of giants? We were both able to arrive at a trusted conclusion so quickly because of those that came before us.

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    Quote Originally Posted by fast View Post
    Did either of us use logic? We both came to the same conclusion. We used an entirely different process, but we derived a solution that is undisputably correct. What’s the ole saying about seeing far when standing on the shoulders of giants? We were both able to arrive at a trusted conclusion so quickly because of those that came before us.
    We rely on logic whenever we infer.

    And p and not p implies q is not indisputably valid. It is indisputably not valid.

    Mathematicians use a mathematical method of logic like you will use a formal procedure to find out what 549980023 + 5677702311 is. We all trust this procedure and will trust it even for very large numbers, even if our lives depend on it. There is nothing else we can do in this matter outside opting to remain ignorant of the result.

    I see you think you are in the same situation vis à vis mathematical logic.

    However, that's not true at all. It's a fact that we don't need mathematical logic to decide on the validity of ordinary arguments, for example the Monkey argument, and people who get it wrong would also get it wrong using mathematical logic because it's even more difficult that the argument itself.

    Also, nearly everyone will correctly infer without even thinking about it that someone saying that all politicians are liars in the context of a conversation about Obama will mean that Obama is a liar.

    Also, there are arguments deemed valid in mathematical logic that most people will say are not valid, for example all argument with contradictory premises.

    There are also many other, more ordinary arguments, which is therefore also more of a problem. For example, the Therefore-there-is-a-God argument, if you remember that one. I was nearly lynched for merely suggesting that it was a logical truth! Yet, mathematical logic does say that this argument is valid. Think of the implication of that one! You can make up any number of arguments using the same form and infer anything you like! That Onassis killed John, that Trump is always right, whatever.

    So, why should you trust anyone who gives even once the wrong answer? But, you do as you please. Therefore, there is a God.
    EB

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    Do you have an argument that would be considered sound (“sound” as used by logicians or those that adhere to what you call mathematical logic) that would not be considered valid (the sense of “valid” you use) by you?

    The disconnect between our sense of “valid” that’s been with us for thousands of years and the newfound sense of ‘valid’, the corrupted version through mathematical molestation becomes inconsequential when we deflect away from the implication of the variance between the two when soundness used by logicians is compared to validity used by the layman.

    If I’m wrong, show me an argument I deem sound that you don’t deem valid. I still think language (and not any substantive issue) underlies the conflict. <imagining speakers discussing a topic through prankster interpreters>

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    Quote Originally Posted by steve_bank View Post
    What is the mathematical logic you refer to? To me that is Boolean Algebra.
    Speaking for him, mathematical logic is not a kind of math. Boolean algebra is type of algebra which is in turn a type of math. It is not a type of logic.

    Mathematical logic (if it were logical) would be a type of logic, but calling mathematical logic a type of logic is like calling counterfeit currency a type of legitimate currency. It is not; repeated, mathematical logic is not even a type of logic—let alone a type of math. If anything, it belongs under the umbrella of illogics.

    So, what is Mathematical Logic? It’s the result realized when mathematicians corrupt logic to an overwhelmingly egregious extent. For instance, consider what no untrained person would dare call valid. Then, let them absorb the teachings founded on the works of modern day mathematicians. Presto, magico, poof, it’s now considered valid. Under the guise of being shrouded in complexity, newbies buy into it, and to make matters worse, the teachers themselves have succumbed to their own trust in it.

    Did anyone miss the “speaking for him” part? Just sayin’.

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    Quote Originally Posted by fast View Post
    Quote Originally Posted by steve_bank View Post
    What is the mathematical logic you refer to? To me that is Boolean Algebra.
    Speaking for him, mathematical logic is not a kind of math. Boolean algebra is type of algebra which is in turn a type of math. It is not a type of logic.

    Mathematical logic (if it were logical) would be a type of logic, but calling mathematical logic a type of logic is like calling counterfeit currency a type of legitimate currency. It is not; repeated, mathematical logic is not even a type of logic—let alone a type of math. If anything, it belongs under the umbrella of illogics.

    So, what is Mathematical Logic? It’s the result realized when mathematicians corrupt logic to an overwhelmingly egregious extent. For instance, consider what no untrained person would dare call valid. Then, let them absorb the teachings founded on the works of modern day mathematicians. Presto, magico, poof, it’s now considered valid. Under the guise of being shrouded in complexity, newbies buy into it, and to make matters worse, the teachers themselves have succumbed to their own trust in it.

    Did anyone miss the “speaking for him” part? Just sayin’.
    LOL, that was ... fast.

    ____

    However, I do take mathematical logic to be mathematics, and then only that.

    Mathematicians got interested in a new toy, as proposed by Boole, quick-quick invented some simpleton axioms, and they won't ever look back because, essentially, they just don't get it. It's just not their job to care whether their theories fit the actual real world.

    There's one retired mathematician who posted a complete reworking of mathematical logic, a 600-page paper, apologising to his former colleagues for saying now that they are completely wrong! And it remains true that some mathematicians disagreed enough with so-miscalled "classical logic" that they developed different, contradictory, theories, unfortunately each more zany than the other. Three truth values anyone? True, False, and ... no even true or false? What does that even mean? Or, another one, true, false or ... true and false? Or, an infinity of truth values? And on and on and on. Sorry, that's not logic. It's mathematics. Everyday now there will be another junior mathematician willing to prove their worth by laying a new egg. Some university somewhere in the US. is proposing at the moment "classical logic" but without the principle of explosion! Right. Must be very underwhelming after all the fireworks!
    EB

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    Quote Originally Posted by Speakpigeon View Post
    However, I do take mathematical logic to be mathematics, and then only that.
    How dare a Frenchman question me on my authoritative skills to speak on their behalf!!

    Mathematical Logic is a subfield of math, but mathematical logic (my dear dictionary-loving friend) is not. Steer clear of those illogic inventing mathematicians that apply their newfound illogic in math.

    If a great logician conjures from the deep recesses of his inner brilliance a new method of reasoning but has his work stolen by a botonist, then it’s seems right a new highly misleading term will soon become apart of a new nonclemature when he applies it to every crevice of his field. Botanical logic my ass. But, Botanical Logic, no doubt.

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