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What is Logic?

  • Nov 12, 2012 · 7:00 PM
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What is Logic?

'Contrariwise,' continued Tweedledee, 'if it was so, it might be;
and if it were so, it would be; but as it isn't, it ain't. That's logic.'
-Through the Looking Glass
Lewis Carroll

We most often think of logic as one element of sound reasoning. Other key elements might include: verifiable and relevant evidence (including appropriately used statistical information), repeatable demonstrations, applicable analogy or comparison, and inference through circumstantial evidence.

In one sense, logic appears to function without the other elements of sound reasoning (as in mathematics), but the other elements (such as evidence) generally require logic to make them applicable to an argument. This suggests the conclusion that logic is essential to argument.


- Formal logic requires our thoughts to be presented into structured sequences of propositions. However, we sometimes find it difficult to organize our thoughts and ideas into formal propositions. But when the formal propositions are structured and presented, there is an elegant inevitability to the results, although there is a legitimate question whether the conclusion of a formal logical argument was already contained within its initial proposition.
- Informal logic is used much more often in our reasoning and argumentation; however, even if significant fallacies are avoided, there are inevitable debates on assumptions, premises, definitions and context.
- Mathematical logic represents the foundations of mathematics as a scientific endeavour aimed at discovering new valid propositions.

Prior to logic as the basis for argumentation, we can also think of logic as the infrastructure for coherent communication, an apparatus without which argument would not be possible. If the infrastructure of logical reasoning is not sound, then the reasoning may itself be suspect. Consider certain basic assumptions we make in our conversation:
1. We talk about things which exist (even as ideas or dreams)
2. A thing is what it is
3. A thing is not at the same time another thing
4. A thing has at least one property or characteristic
5. A thing either has or does not have a property or characteristic (the middle is excluded)
6. A thing cannot both have and not have a property or characteristic (non-contradiction).

These assumptions concern the existence, identity, uniqueness and specificity of things, demanding that something be one thing or another and not both. These assumptions are, in effect, the axioms of rational discourse. But they are assumptions.

We might propose certain tests of these basic assumptions to assure ourselves of their validity (if we can set aside the difficulty that our tests themselves may rely on the assumptions we want to test). This is a good field for discussion but to start off consider, as examples, whether the assumptions:
• could be contradicted by experience (observation)
• form a consistent whole within themselves; i.e., they do not appear to contradict one another.
• provide a complete explanation of things and their properties.

The foundations of logic, for the most part, appear to meet these tests.

However, some argue that the principles of quantum physics suggest that our assumptions fail all three tests.

If, for example, there is a possibility that one thing can both exist and not exist at the same time, or can exist in two spaces at the same time, or can both have and not have certain properties, then one could challenge the assumptions of existence, non-contradiction and can be challenged.


For one other challenge to the distributive law in logic, consider the following citation from Wikipedia:
- "Quantum logic has some properties which clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:
- p and (q or r) = (p and q) or (p and r),
- where the symbols p, q and r are propositional variables. To illustrate why the distributive law fails, consider a particle moving on a line and let
- p = "the particle is moving to the right"
- q = "the particle is in the interval [-1,1]"
- r = "the particle is not in the interval [-1,1]"
- then the proposition "q or r" is true, so
- p and (q or r) = true
- On the other hand, the propositions "p and q" and "p and r" are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle. So,
- (p and q) or (p and r) = false
- Thus the distributive law fails."


If these challenges are valid, then our traditional infrastructure of logic may only describe how we experience our consensual and social reality. The rules and principles of logic are, in this view, neither immutable nor ineluctable; they are a human creation.


As well, in fuzzy logic systems propositions may be based on probability and likelihood rather than truth or falsehood. Where there are ‘degrees of truth’, the certainty of traditional logic is weakened.


There is an interesting further challenge which suggests that, even if the foundations of logic were unassailable, they remain a subjective cultural artifact: the principles were ‘discovered’ by humans, and are applied in argumentation by humans; there is nothing beyond human experience, so the argument goes, that suggests the foundations have any immutable or eternal value.  However, this challenge can be applied to any human experience or thought, and basically maintains that there is no objective reality beyond human experience. Such may or may not be the case, but it does leave the scent of solopsism in the air.


For our discussion, we could start with these questions:
- Do the principles of logic have objective value, or are they subjective human creations used to explain our experience?
- Is it important whether logic has objective reality?
- How do we test any logical principles or assumptions for validity?

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  • A former member
    A former member

    << Logic can be viewed in two ways: informally or formally. The latter means as a mathematical discipline >> Oh dear, I’m feeling definite. Formal logic is not exclusively the realm of mathematics. To argue such is to demean Aristotle, Aquinas, Spinoza, Schopenhauer, Wittgenstein, etc. O the humanity! O the humanity!

    November 13, 2012

    • A former member
      A former member

      maybe they were informal mathematicians ...

      November 13, 2012

  • Doug S.

    PART 6
    For good background on many of these topics I can recommend:
    http://plato.stanford.edu/contents.html

    http://mathworld.wolfram.com
    http://www.scholarpedia.org
    http://www.edge.org/
    I would view the TED talks as amusing food for thought. THATS ALL FOLKS

    November 13, 2012

    • A former member
      A former member

      thanks for the links...

      November 13, 2012

  • Doug S.

    Reflections After Our Discussion of Logic

    Logic can be viewed in two ways: informally or formally. The latter means as a mathematical discipline. Hence to discuss it formally, one needs to understand what this means; indeed, logic is the starting place for mathematical discussion. Math students take a course in logic at third year or higher levels. Without some such knowledge, you are in the same predicament as someone who can’t read music trying to understand Beethoven’s approach to orchestration.
    As I see it, the problem that kept arising in our discussion is that we unavoidably would get into concepts of considerable mathematical subtlety, eg Russell’s and Goedel’s results, not to mention quantum mechanics. I heard many comments that showed the speaker did not have a clear idea what these really were, and used terms very (far too) loosely. This is totally understandable, for I suspect very few of us have ever studied these complex subjects formally.
    MORE TO FOLLOW

    November 13, 2012

    • A former member
      A former member

      and sometimes we just make mistakes, Mike correctly pointed out to me after the discussion that I was saying inconsistent when what I meant was incomplete...

      November 13, 2012

    • A former member
      A former member

      the context was to do with the discussion on Peano arithmetic (natural number system defined by the Peano axioms) being incomplete, I said inconsistent incorrectly. In hind sight and in further more relaxed contemplation, Goedel's 2nd theorem which I believe states that the consistency of certain arithmetics, the Paeno arithmetic being one of those, can not be proven from within that (Paeno in this case) arithmetic. The interesting idea to me is if our perception of reality is such a "system" that satisfies Goedel's criteria, then does it not follow that we can not prove the consistency of our perceived reality from within in and that our perception of reality is as well incomplete, or our perception of reality is required to be non-deterministic.

      November 13, 2012

  • Doug S.

    PART 5
    The problem hinges around what the phrase “be in a place” or “see” means. Like other odd QM statements, like “P and not P” is true, it cannot be taken in the usual naïve sense. There are many caveats in QM that do not apply in the world we know. As for “seeing”, many people have “seen” UFOs, but no one can take a decent picture of one, so in what sense did they see them?. So I do not know exactly what he did – I am trying to track this down in the physics literature. It does not appear on physics sites that list all recent interesting results. I can give you pointers to other work that shows quantum phenomenae at the macroscopic level but within the proper bounds.
    I would welcome any discussion of these ideas. In fact, I would suggest that we could profitably spend at least one meeting discussing them!
    MORE

    November 13, 2012

  • Doug S.

    PART 4
    . (This is not the case in logic.) Probably the best we can do is to understand some of the experimental results, of which the key one is the double slit experiment. (Feynman said that this is ALL you need to understand, ie it comprises all that is strange about QM). At least this will give you an idea of what QM is trying to deal with. On Monday Kim told us about this TED talk by a guy who was saying he had somehow overcome a basic limitation of QM and “seen” something in two states at once, which QM says you can’t. http://blog.ted.com/2011/06/02/making-sense-of-a-visible-quantum-object-aaron-oconnell-on-ted/ Unfortunately he said he had “seen” something both “vibrating” and “not vibrating” at the same time, but did not say how. His little diagram of how atoms could be in two places at once would not make into a high school text.
    MORE

    November 13, 2012

  • Doug S.

    PART 3
    . So for example an analogy for Goedel’s incompleteness theorem, which says that in a certain system of reasoning you cannot do something, might be that in arithmetic you cannot represent the square root of 2 as a fraction (proof is sometimes given in high school). The usual upshot of such results is to create a more complex system in which this is possible, which for arithmetic lead to the invention of irrational numbers. Similarly, there are extensions of the assumptions of Goedel which partially solve some of these problems.
    So should we have a meeting on What is Quantum Mechanics? – probably the most abstruse subject imaginable. (Richard Feynman famously said “nobody understands quantum mechanics”). Quantum mechanics is a really hard nut. For years, one of my hobbies has be trying to understand it, fool that I am. The problem here is much greater: the experts themselves are still arguing about what it means, and how to understand it, since it is so weird.
    MORE

    November 13, 2012

  • Doug S.

    PART 2
    So what should we do? It is impossible for us to learn enough about these subjects to discuss them formally. But these are fascinating topics and I would encourage discussion of them. The trick then is to try to do it without using concepts beyond our knowledge. This is challenging, and I confess I don’t say very much on such topics because of this difficulty.
    Probably the best approach would be to have several people present particular subtopics informally and by analogy, and we could discuss these. For example, you might say logic is analogous to arithmetic, with which we are familiar. Arithmetic allows one to compute numbers; logic allows one to compute logical conclusions in an analogous way. Another thing that can be presented is famous results, ie to give some idea what they mean.
    MORE!

    November 13, 2012

  • A former member
    A former member

    fascinating & spirited discussion!

    November 12, 2012

  • Mike H

    Part 1

    That is an excellent question. I'd venture that a universal logic must at a minimum include the law of non-contradiction. In any event, perhaps the situation is not so dire. I had assumed the distributive law of the PC failed when it refers to certain quantum events, but perhaps it doesn't. The Wikipedia entry assumes that it is true that a quantum particle is either in interval [-1,1] or not. But, using its own assumptions, this is true only when the direction of the particle is unknown. If this suppressed premise were stated we'd have:

    q = "the particle is in the interval [-1,1]"
    r = "the particle is not in the interval [-1,1]"
    s = "it is not the case that the direction of movement is known"

    p= "the particle is moving to the right, which implies -s"

    (q or r) is thus properly unpacked to mean "(q or r) and s"
    P then means (p and -s)

    November 12, 2012

    • A former member
      A former member

      In quantum mechanics it isn't always the case that you either know or don't know, it is usually the case that your knowledge is "fuzzy", for example, yes if you know the position exactly, then yes you haven't the foggiest idea of its velocity and visa versa.. but it could also be that you know both values but neither accurately.. so your "s" is a special corner case... in effect you have chosen a "deterministic"­ case, that in which the position is known and the velocity is unknown or has infinite variance and is essentially a denial of any knowledge in a deterministic sense. A non deterministic "s" would be its position is at 0.9 plus or minus .2 units...

      November 12, 2012

  • Mike H

    I'd also suggest that if the quantum, or for that matter the macro world were completely non-deterministic there could be no laws (laws determine what is the case in a given context). However, quantum mechanics is full of laws. They may be probabilistic in nature but that does not make them any less law-like. I'm 100% not a physicist but I've heard it said that the quantum world is predictably unpredictable or, in other words, deterministically non-deterministic. For example, we do not know when a given atom of radon will decay but we know with certainty the rate of decay of an ounce of radon.

    November 12, 2012

    • A former member
      A former member

      The rate of decay of an ounce of radon gas is yes a well known value known to a certain degree of accuracy. But given say one ounce (a unit of force actually) of radon gas, you can not predict accurately which of the many atoms will decay and when. You know on average the rate will be some value... in any given specific experiment the measured value may be higher or lower.. but do it often enough and your average value will be the "rate'... it is non-deterministic if no matter how much data you have on all those atoms and on all the forces and factors on those atoms and no matter how much computational power you have you can not accurately say, these atoms will decay at this time, these at that time, etc.

      November 12, 2012

  • Mike H

    I question whether it is fair to call "classic" logic deterministic since (1) it is a formal system that itself implies nothing about whether the world is or is not deterministic, and (2) it can be used to express probabilistic claims. Even if the quantum world contains a certain irreducible unpredictability, I see no a priori reason to believe it undermines "classic" logic just because some systems of logic use more than two truth-values e.g. True, False, Unknown; or True=1, False= 0 and intermediate grades of certainty between 0 and 1. My point is that one could always code statements in the propositional calculus such that they state that it is true the x is unknown, or there is a probability of 0.6 that X is the case. The use of more then two truth values in other systems may be simply a matter of convenience rather than a blow struck at the heart of poor, old "classic" logic.

    November 12, 2012

    • A former member
      A former member

      by Deterministic I mean that given sufficient data and sufficient computational power one can state the outcome of an "experiment". Being non-deterministic to me does not imply no laws govern the interactions or experiment, only that we can't predict accurately what will happen. As to classical logic I meant those in which the distributive law and other such usually assumed laws hold true, the logic of "common sense", the logic of the macro world. we live in and relate bets towards. In regards to logic being deterministic I meant that each statement is made without any "fuzziness to it". For example, given a statement "a is a subset of b", that is a deterministic statement to me... "a is most likely a subset of b" is not. "If A then B, If also C = A then If C then B" is deterministic and is my understanding of classical logic. "If A then B, and if also C is 90% likely to be A, then if C then it is 90% likely that B occurs" is non-deterministic.

      November 12, 2012

  • Mike H

    OK, Radon is a gas so speaking of an ounce of it doesn't work. Like I said, I'm no physicist: just pick your favourite radioactive solid...

    November 12, 2012

  • A former member
    A former member

    My gut feel is that the many-worlds interpretation is just another of far too many examples of human reluctance to change aka we shouldn't be called homo-sapiens, but rather homo-stick-our-heads-in-the-sand ... if someone knows the Latin for that I'ld be really grateful...

    November 12, 2012

  • A former member
    A former member

    For me the key aspect of Quantum or Fuzzy or Probabilistic logic is that the validity of statements and/or even of the validity mechanism itself is explicitly acknowledged to be inexact. In other words they are non-deterministic logics whereas the classic logic the statements and/or validity mechanism are assumed to be exact, any inaccuracies are due to insufficient data and/or computational power aka a deterministic world view. The question then becomes is reality deterministic or non-deterministic, that is with sufficient data and sufficient computational power can everything be determined? The Copenhagen interpretation (my understanding of it) says reality is non-deterministic at least in the realm of sub-atomic particles, etc. The many-worlds interpretation (again my understanding) is that reality is deterministic, but to do so one needs this multi-universe reality and so far all we have evidence for is one universe.

    November 12, 2012

  • Mike H

    Part 2

    This gives: (p and -s) and ((q or r) and s) which is true only if s is both true and false. This is a contradiction that, by hypothesis, reflects the quantum mechanical principle that motivates the example. Hence when properly expressed the truths of quantum mechanics might well be consistent with the PC. That said, a quantum logic may still be desirable because it e.g. simplifies notation for the specific purposes of quantum theory. On this reading, the problem is not with the PC but with how the propositions are coded. Consider the claim: the present King of France is bald OR it is not the case that the present King of France if bald. Does this violate the law of excluded middle or merely imply that the ordinary language sentence compresses a more complex logical structure? Russell's view was the latter and I'd suggest that the same might be said of the Wikipedia example.

    November 12, 2012

  • A former member
    A former member

    If Quantum logic is valid (at least for some domain (in this case the microscopic)), then classic logic is not valid universally... is logic intrinsic to the universe or is it a human concept is I think one of the queries David presents. If logic is intrinsic and classical logic is not universal then what does that imply. One possible position would be that a very "basic" logic (whatever that may be) may be intrinsic and universal, with "local" areas which follow "higher" more restrictive logics (aka follow more laws like the distributive law).

    November 12, 2012

  • Mike H

    Part IV:


    While (on at least a certain interpretation of quantum mechanics) the PC may not be adequate to set out the logical relationships between propositions describing certain quantum states, it may well validly describe the logical relationship between propositions which refer to propositions that make statements about quantum states (e.g. "Heisenberg's indeterminacy principle is true") which a "quantum logic" which denies the distributive law could not. That one system cannot do everything should not be a surprise. For example, although Peano Arithmetic, if consistent, cannot prove its own consistency another system may well do so, and I understand has done so. The fact that we must rely on more than one formal system to further our scientific and logical ends does not imply that those systems are necessarily incompatible or in any sense subjective.

    November 12, 2012

  • Mike H

    Part III:

    At best, the failure of distributive law of the PC in respect of propositions which refers to quantum states implies that the PC is not adequate for the relationship which holds between propositions which describe such states. In one sense, this should not be a surprise since it is also not adequate to cover relationships as simple as those which hold between subject and predicate (the subject of the Predicate Calculus e.g. "All humans are mortal").

    November 12, 2012

  • Mike H

    Part II: Given this, (P+ Q) is true since, by hypothesis, both P and Q are true, and (P + R) is false since R is false. Therefore, (P + Q) v (P + R) is true since P + Q is true. Hence -[( P + Q) v (P + R)] would be true only if either P or Q is false. Therefore, the failure of the distributive law of the PC to hold when some propositions refer to certain quantum states implies that the distributive law must be dropped from the PC only if either it is not the case that quantum mechanics is a true theory or it is not the case that Heisenberg's determinacy principle is true.

    November 12, 2012

  • Mike H

    Part I: The quote from Wikipedia concerning the failure of the distributive law in the propositional calculus to take proper account of quantum states may, for all I know, raise questions of the proper interpretation of quantum theory. I will simply assume the Wikipedia quote is true and argue that it does not imply that we must discard "classic" logic. In what follows:

    + = and
    v = or (but not both).

    The supposed failure of the distributive law of the propositional calculus (PC) to hold when propositions refer to quantum states does not imply that the distributive law must be dropped from the PC. If it did, P + ( Q v R) would imply -[( P + Q) v (P + R)]. Consider therefore: P = Quantum mechanics is a true theory.

    Q = Heisenberg's indeterminacy principle is true.

    R = One equals two (in a base 10 system).

    November 12, 2012

  • A former member
    A former member

    In regard to Quantum Logic, Quantum mechanics (from what I understand and I do fall into that overly populated category of those who don't really understand Quantum Mechanics) has two main interpretations to this topic.
    From Wikipedia:
    The Copenhagen interpretation states that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta. The act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible value, This is called a wave function collapse.
    The many-worlds interpretation asserts that the objective reality of the universal wavefunction and that the collapse doesn't happen. It views reality as a many-branched tree (likely uncountably infinite), wherein every possible quantum outcome is realised.

    November 11, 2012

    • A former member
      A former member

      Yes, all I can do is base my judgements and beliefs on my own inaccurate model and on the feedback I get from other folks inaccurate models... and as I can not 100% prove anyone else exists...

      November 11, 2012

    • Dr S. Ranga S.

      right on!

      November 11, 2012

  • A former member
    A former member

    There is also the instrumentalist interpretation, it is summarized by the sentence "Shut up and calculate!" which maybe of relevance to physicists who don't want to talk philosophy, but to me that indicates they aren't really physicists as I see science as a branch of philosophy.

    November 11, 2012

  • A former member
    A former member

    Please delete me from this group. Although I'd love to join you all, it is not possible for me to attend Monday evenings as I have a standing family commitment.

    November 4, 2012

  • Dr S. Ranga S.

    I will attend. Thanks.
    -- ranga

    November 3, 2012

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