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Thread: Justification of the mathematical definition of logical validity?

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    Contributor Speakpigeon's Avatar
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    Justification of the mathematical definition of logical validity?

    Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
    Here is the definition:
    Validity
    A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
    Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/
    Thanks for your answers.
    EB

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    I wasn't going to reply to this thread, but since you keep changing the definition in the other one, let us do this.


    First, I asked you in the other thread whether you believed all mathematical statements were either true or false. You replied:
    https://talkfreethought.org/showthre...l=1#post683824
    Quote Originally Posted by Speakpigeon
    I think all statements are either true or false.
    I will grant this for the sake of the argument.

    Second, by 'CML-valid' or 'valid according to classical mathematical logic' or similar expressions, I mean that a deduction (or argument, inference, or whatever one calls it) is valid according to the definition provided by you in the OP

    Namely, a deduction is CML-valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

    In particular, if A is a CML-valid mathematical argument with true premises P1,...,Pn and conclusion C, then C is true. Why? Because the premises are true, and the argument takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Therefore, C is not false. Given the previous hypothesis (i.e., every statement is either true or false), C is true.

    This gives us an important feature of CML-validity: it is truth-preserving. If one starts with truths, by CML-valid arguments one gets only more truths.

    Third, let us consider another proposed system of validity that is truth-preserving, say V. Since V is truth-preserving, it never is the case that an argument is V-valid but it takes a form such that it is possible for the premises to be true but the conclusion false. Hence, V-validity implies CML-validity. This gives us another important point: CML-validity is the strongest form of truth-preserving derivation of conclusions from premises.

    Fourth, while mathematicians sometimes make logical errors in applying CML, that is usually corrected before a paper is published, because the authors and other mathematicians check it repeatedly. Moreover, if some errors make it into a paper, readers - who are usually also mathematicians - will almost certainly sooner or later (very probably sooner, if the paper has readers) spot it. So, while the system is not perfert, it is generally very reliable in getting CML-validity right.

    Fifth, in order to prove things in mathematics, we need to start with something, right? Well, our starting points are true. If we were wrong about that, mathematics would be pretty much hopeless regardless of what method of deduction is used. We could discuss whether those starting points are true because they're self-evident, or because we just set up a hypothetical abstract scenario and stipulate that such-and-such things hold, so they do hold in the scenario we are considering, or it depends on the case (e.g., the natural numbers for the former; a Banach space for the latter), or something else, but at least we have true starting points (or first statements, or axioms, or whatever one calls them.)

    So, by CML, we find new mathematical truths, and we can find any truth that could be found from the same starting points by another truth-preserving method, but also more truths than any weaker method.

    So, that's a good reason to adopt CML-validity as the way of deriving statements from others: we find true statements from true statements, and it's the strongest method for doing that.


    Additionally, I would add that CML is intuitively right, for most mathematicians. This is not so because they were told so. Where I live, most mathematicians never take a course in mathematical logic (I think it would be a good thing if they did, but anyway), but those who do (or who decide to study it on their own), when they first encounter it, usually find it very intuitive, and in particular, they find the definition very intuitive.


    Now, do non-mathematicians find it intuitive?
    It depends on the person. But for that matter, there is a difference between trained and untrained intuitions, and the former are often better. For example, there are plenty of cases where the intuitions of people with no previous training, in nearly all cases, go wrong (purely for example, the Monty Hall Problem, where the folk probabilistic intuitions nearly always fail). On the other hand, after learning mathematics (more specifically in the example, probability theory), humans are considerably less likely to go wrong.

    Regardless, if CML-validity is not the same as the folk conception of validity, then for the previously given reasons, either CML-validity is stronger, or the folk conception fails to be truth-preserving. Either way, CML-validity is a superior tool for finding mathematical truths. On the other hand, if CML-validity is just the folk conception of validity, then no problem, either.

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    Quote Originally Posted by Speakpigeon View Post
    Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
    Here is the definition:
    Validity
    A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
    Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/
    Thanks for your answers.
    EB
    Math is not philosophy. Mathematicians develop rule based systems that are consistent, which means no ambiguities. By applying the rules in different ways and combinations you can never get two different answers. Given a problem in algebra no mater the order you apply addition, multi[plication, division, and subtraction you will always get the same valid result. It is consistent. Algebra is not a logical argument, it is a system based on rules and definitions.

    Arithmetic
    Geometry
    Linear Algebra
    And so on...

    The Incompleteness Theorem, which has a proof, says in any consistent rule based system there are truths improvable in the system. In plane geometry it is impossible to prove that the shortest distance between two points is a straight line. That requires calculus.

    As to deductive reasoning, it is generally not just either or deductive or inductive. Other than straightforward problems it is aleays a combination of the two.

    Inductive vs deductive is also called top down vs bottom up and backwards vs forward. It is all the same Aristotuilan logic, the difference being the starting point.

    The specific to the general and general to the specific is another desorption.

    When actually solving problems you pick a staring point and ether try inductive and deductive, If yiou start with inductive but di not get a solution, you then try deductive based aon the inductive results.

    When attempting a new math proof for which it is unknown if a proof exists, one goes back and forth, backwards forwards iterations looking for a solution..

    In practice for other than obvious solutions the direct application pf logic as in a syllogism does not work.

    Math like scientific theory on the end is demons red by usage over time. There is no guarantee that a proof in genral can have a lurking problem.

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    Quote Originally Posted by Speakpigeon View Post
    Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
    Here is the definition:
    Validity
    A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
    Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/
    Thanks for your answers.
    EB
    The definition you quoted is a definition which works for the sort of Tarskian, truth-functional classical notion of validity. It's one way, arguably shallow and uninteresting, to define validity for a particular logic.

    Logics which deviate from this, such as relevance logic, have their own definitions of validity.

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    Quote Originally Posted by A Toy Windmill View Post
    Quote Originally Posted by Speakpigeon View Post
    Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
    Here is the definition:
    Validity
    A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
    Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/
    Thanks for your answers.
    EB
    The definition you quoted is a definition which works for the sort of Tarskian, truth-functional classical notion of validity. It's one way, arguably shallow and uninteresting, to define validity for a particular logic.

    Logics which deviate from this, such as relevance logic, have their own definitions of validity.
    Yep, but I'm not sure you even understand the question.

    Anyway, you should perhaps know that this question came after an argument with Angra Mainyu about validity.

    I had started a thread on the validity of an argument (repeated below). Nearly all posters have voted either not valid or nonsensical, which amounts to not valid. So, I take this as a demonstration that the empirical evidence is that the argument is seen by most people as not valid. In fact, the only two who claimed the argument valid, Angra Mainuy and Bomb#20, both have a training in mathematical logic, which explains their claim and invalidates it as biased for the purpose of the poll.

    However, both Bomb#20 and Angra Mainuy at one point started to play teacher (repeated below) by authoritatively asserting validity to posters who had commented that the argument was not valid, and this on the basis that validity was a consequence of the definition of validity. This showed clearly that both thought the word "validity" means mathematical logic validity. Yet, my question was about validity without any reference to mathematical logic. Thus, the debate turned on to whether there was such a thing as a notion of logical validity outside mathematical logic. Given the answers by most posters, clearly there is, but our specialists of mathematical logic here disputed this. So, somewhere, I asked my question about the justification of the mathematical definition of logical validity/ Is there a justification that it is correct? Because if not, then people should abstain from pretending there's just the mathematical logic's notion of validity.
    EB


    Quote Originally Posted by Speakpigeon View Post
    This is a poll on the logical validity of the following argument:

    A squid is not a giraffe
    A giraffe is not an elephant
    An elephant is not a squid
    Joe is either a squid or a giraffe
    Joe is an elephant
    Therefore, Joe is a squid
    Is this argument logically valid?
    Either way, why?
    EB

    Quote Originally Posted by Angra Mainyu View Post
    Quote Originally Posted by jab View Post

    No because the stated premises say nothing about the relationship between squids or elephants as categories; therefore the conclusion that Joe is an elephant cannot be logically arrived at. It is like A human is not an animal; An animal is not a soul., Joe is either a human or an animal. Joe is a human. Therefore Joe is a soul. (the unstated premise is A Human is a soul--which most posters on this forum would debate. In the case of the elephant Joe; one would have to prove that, given the accepted meanings of the words, squids are all elephants, for the logic to hold.
    Bomb#20 already proved the conclusion from the premises, so it is valid.

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    Quote Originally Posted by Speakpigeon View Post
    I had started a thread on the validity of an argument (repeated below). Nearly all posters have voted either not valid or nonsensical, which amounts to not valid. So, I take this as a demonstration that the empirical evidence is that the argument is seen by most people as not valid. In fact, the only two who claimed the argument valid, Angra Mainuy and Bomb#20, both have a training in mathematical logic, which explains their claim and invalidates it as biased for the purpose of the poll.
    One might suggest that diagnosing validity requires training, and there are simple problems where people's untrained intuitions are wrong. I would say that understanding just the difference between validity and soundness requires some training, as does understanding the formalism developed by Aristotle.

    However, both Bomb#20 and Angra Mainuy at one point started to play teacher (repeated below) by authoritatively asserting validity to posters who had commented that the argument was not valid, and this on the basis that validity was a consequence of the definition of validity. This showed clearly that both thought the word "validity" means mathematical logic validity. Yet, my question was about validity without any reference to mathematical logic. Thus, the debate turned on to whether there was such a thing as a notion of logical validity outside mathematical logic. Given the answers by most posters, clearly there is, but our specialists of mathematical logic here disputed this. So, somewhere, I asked my question about the justification of the mathematical definition of logical validity/ Is there a justification that it is correct? Because if not, then people should abstain from pretending there's just the mathematical logic's notion of validity.
    I'm a specialist in mathematical logic, and like other specialists, I would tell you that there isn't one notion of validity in mathematical logic.

    You are right that the material implication has always been controversial. It trips up students all the time. Alternative logics have been developed to try to model implication in a way that better fits our pre-formal understanding. You have mentioned one before: relevant logic.

    Quote Originally Posted by Speakpigeon View Post
    This is a poll on the logical validity of the following argument:

    A squid is not a giraffe
    A giraffe is not an elephant
    An elephant is not a squid
    Joe is either a squid or a giraffe
    Joe is an elephant
    Therefore, Joe is a squid
    Is this argument logically valid?
    Either way, why?
    EB
    If you'll indulge me: here's the same argument with two of the premises deleted:

    A giraffe is not an elephant
    Joe is either a squid or a giraffe
    Joe is an elephant
    Therefore, Joe is a squid
    Is this invalid?

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    Quote Originally Posted by A Toy Windmill View Post
    One might suggest that diagnosing validity requires training, and there are simple problems where people's untrained intuitions are wrong. I would say that understanding just the difference between validity and soundness requires some training, as does understanding the formalism developed by Aristotle.
    Since all mathematicians abiding by the definition I provided of validity as used in mathematical logic will be systematically wrong about the validity of my squid argument, I think common sense validity wins hands down.

    And you're just wrong about the necessity of training.

    Quote Originally Posted by A Toy Windmill View Post
    I'm a specialist in mathematical logic, and like other specialists, I would tell you that there isn't one notion of validity in mathematical logic.
    I know that, but I can't possibly cater for all tastes. The situation in mathematical logic is just to complicated to follow in details. Do you even know exactly how many theories of logic there are? I've never seen anyone venturing a figure.

    Quote Originally Posted by A Toy Windmill View Post
    You are right that the material implication has always been controversial. It trips up students all the time. Alternative logics have been developed to try to model implication in a way that better fits our pre-formal understanding. You have mentioned one before: relevant logic.
    In other words, alternative logics try to better model our intuitive notion of validity, which straightforwardly contradict your initial claim about the necessity of training by implying that ordinary folks get it more often right than mathematicians abiding by the definition of validity consistent with material implication (i.e. most mathematicians, I think).

    As demonstrated here and elsewhere with my squid argument: Systematically, all untrained posters get it right saying the argument is not valid. Trained posters get it always wrong. QED, training makes you wrong. .

    Quote Originally Posted by A Toy Windmill View Post
    Quote Originally Posted by Speakpigeon View Post
    This is a poll on the logical validity of the following argument:

    A squid is not a giraffe
    A giraffe is not an elephant
    An elephant is not a squid
    Joe is either a squid or a giraffe
    Joe is an elephant
    Therefore, Joe is a squid
    Is this argument logically valid?
    Either way, why?
    EB
    If you'll indulge me: here's the same argument with two of the premises deleted:
    Why do you not answer my question first? Is my squid argument valid or not?

    Quote Originally Posted by A Toy Windmill View Post
    Quote Originally Posted by A Toy Windmill View Post
    A giraffe is not an elephant
    Joe is either a squid or a giraffe
    Joe is an elephant
    Therefore, Joe is a squid
    Is this invalid?
    There's no difficulty answering this one but I'll wait that you answer mine first.
    EB

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    Quote Originally Posted by Speakpigeon View Post
    There's no difficulty answering this one but I'll wait that you answer mine first.

    EB
    The argument is classically valid.

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    Joe is a squid
    Joe is not a squid
    Therefore, Joe is a walrus.

    Is that valid?

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    I ask the previous question to rule something out.

    If Joe is a person, Joe can walk.
    Joe is a person.
    If joe is a tree, then Joe cannot walk.
    Joe is a tree.
    Therefore, Joe can walk.

    That’s what I guess he would call classically valid.

    To me, it’s like mixing two puzzles together. So long as there’s enough ingredients in the premises to get to the conclusion, the argument is valid — contradictions and discrepancies be damned.

    But, my earlier question lacked the necessary ingredients but contained a contradiction.

    I’m trying to hone in on the thrust that gives rise to this “everything follows” mind set. It kind of reminds me of the notion that we have no rights if our rights can be altered—or something like that.

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