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Thread: Something else than binary logic?

  1. #1 Something else than binary logic? 
    Vel
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    The basis for the whole mathematics is a set of axioms of logic. The mathematics that we usually study is based on binary logic, its axioms can be read in the first chapter of the first book of Bourbaki. For me only conception is important. Basically, everything reduces to deductive reasoning: If (A is true) and ( (from A follows B) is true) then (B is true).

    Then, axiomatic construction of any theory goes as follows: we declare what statements are true (axioms) and start to deduct about others statements. All the statements in the theory can be: true, false or such that it is impossible to conclude whether they are true or false (cf Godel theorem). The task of mathematician is to decompose all statements in these three classes. And only subclass of true/false statements is what we can learn from the theory.

    This was a binary logic, it is formalization of our every-day experience. My principal question is whether we can build other types of logics? I actually afraid that I cannot formulate the question properly because English also assumes binary logic. So my question assumes an answer: yes/no/we don't know. And probably in something else than binary logic my question is gibberish!

    I have some basic knowledge about other logics:
    1) -ary logic: each statement can be in one of states or in "don't know" state. Then we have to define more complicated operation than deduction. In fact, it is better to represent deduction as follows: if A, B than A&B. And deduction will be consequence of A, ( (not A) & B) -> A&(not A & B)=B. So it is (not A), basic unary operation, and (A&B), basic binary operation, which have to be generalized.
    2) "quantum logic": each statement is an element of Hilbert space (if you wish, each statement is a point in )

    Apart of idea that these constructions exist and certainly were studied, I know nothing more about them.

    Finally, here are my three more precise questions:

    A. Do you know nice explicit examples where 1) or 2) are really used? Short but meaningful answers will be very appreciated (link to 100-pages text is not a short answer, link to 1-page good text is a short answer).
    B. I think that 1) and 2) can be reduced to simple binary logic. Do you agree or even know a theorem?
    C. If what I think in B is true, then my question: can we invent something else, essentially different from binary logic?


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  3. #2  
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    Wouldn't the inclusion of "don't know" make it three-valued logic (we use this in electronics: 0, 1, X)?

    Isn't the difference that (mathematical) logic only deals with true/false and the undecidability is a "meta problem"; i.e. something that is worked out using the two-valued logic formalism? Which, of course, means that we have the problem that we can prove some things are undecidable/unprovable while there is another class where the unknown state cannot be proven to be true or false ...

    So it seems, intuitively, that formalizing a three-valued system would be useful. However, I am not aware of any formal work on three (or higher) valued logic (but then I am not a mathematician). There was some work on formalizing "fuzzy logic" when it was fashionable in the '80s (but I don't know much about that, either).


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    Quote Originally Posted by Vel View Post
    The basis for the whole mathematics is a set of axioms of logic....., everything reduces to deductive reasoning:
    Generally speaking, math is a system of equivalences which obey some rules, math is a language with a lexicon and a syntax.

    The sign equal expresses = the identity, the fundamental Law of the Universe which avoids Chaos and, strictly speaking, should be used only for tautologies : 3=3, 87 = 87
    The other relations are synonymical : in 2+4 = 6 we use the equal sign because the numerical value (the signified, the meaning, the real objects: six bananas, etc..) is the same, but in reality it is an equivalence of synonyms which have not all been listed in the dictionary, but can be obtained through general rules. The same applies to conditional equivalences: x = 8 (if x=2)
    In any other expressions: {a x (b+c)} = {ab + ac} ,
    {x - x/2} = {(2x-x):2}....etc ..., the sign of equivalence should be appropriate.

    I do not call this deductive reasoning.

    As you said in another thread, we can limit our vocabulary to fewer definitions, we need only 1, only one exists: (all is one and one is many; did you have Parmenides in mind when you said this?)

    1 ...........1
    ...
    111111111...9

    We do not need 0. Zero is the Nothingness (or at most the empty space): what does not exist, what is not. The linguistic sign zero/0 is literally meaning- less: the signified does not exist. And what does not exist, is not, cannot have properties, qualities, let alone be a number.
    0 was introduced as a marker for empty space [ ] 0, to avoid misunderstanding when writing on paper: 34 2 might not berightly read as 3402. It is useless if you use other markers, as ancient tribes did, or use for example colours.

    Generally speaking, moreover, even in logics you cannot deduce information which is not already known.
    The famous Aristotelian sillogism: "All men are mortal, Socrates is a man..." makes evident, underlines what is not explicitly written, it shows in the conclusion just the truism that a subset shares the quality of the set. The info is that 'Socrates is a man', for those who didn't know.

    I do not all of maths, you are a specialist, I'd like to learn what is there of real deductive reasoning, what are the exceptions you are aware of.
    At phisicsforum I read once of 'fancy logic' or similar peculiar, funny term for a logic with more than 2 values.

    EDIT: Binary logic is greatly reductive and is good only for limited cases of technical applications, but does someone think that if we add the excluded middle , we have solved the problem?
    If on the table there are 6 bananas is the proposition "On the table there are 5 bananas" false ?
    Last edited by logic; April 22nd, 2012 at 06:22 AM.
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    Something can be 30% correct
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    Hi Vel,
    If you're still interested, I remembered, it's called fuzzy logic
    bye
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    I devised new logic that isn't based on axioms, but only depends on new rules of inference. The rules are basic formulas saying how operators are to be used. The only issue it has is consistency. I can prove Modus Ponens (this is not at all obvious or basic), Modus Tollens and axioms of Constructive Logic with it.

    I have a problem with a basic axiom of mathematics: A -> (B -> A). This says "A follows from B given that ( B -> A) follows from A". It bothers me that the reasoning starts and ends with "A" and that "A" occurs recursively. Is this required for all of mathematics?
    It also matters what isn't there - Tao Te Ching interpreted.
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  8. #7  
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    Quote Originally Posted by Vel View Post
    The basis for the whole mathematics is a set of axioms of logic. The mathematics that we usually study is based on binary logic, its axioms can be read in the first chapter of the first book of Bourbaki. For me only conception is important. Basically, everything reduces to deductive reasoning: If (A is true) and ( (from A follows B) is true) then (B is true).

    Then, axiomatic construction of any theory goes as follows: we declare what statements are true (axioms) and start to deduct about others statements. All the statements in the theory can be: true, false or such that it is impossible to conclude whether they are true or false (cf Godel theorem). The task of mathematician is to decompose all statements in these three classes. And only subclass of true/false statements is what we can learn from the theory.

    This was a binary logic, it is formalization of our every-day experience. My principal question is whether we can build other types of logics?
    I dont want to prove it so I dont claim that every non binary consistent logic can be reduced to binary form.
    An idea behind such a proof could be a claim such that there actually is only one basic truth value: "TRUE",
    and that other values are limitations or extensions, in binary logic simplified in " NOT TRUE ".

    Since existence of undecidable statements seems to necessitate a second basic truth value
    I take the easy way out and refuse to believe in them.
    (Didnt Mr G say: Either there are undecidable sentences or...)
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