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Thread: The Laws of Non-Contradiction, Excluded Middle, and Bivalence: Laws of Logic (Laws of Thought)

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    Veteran Member Petros Agapetos's Avatar
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    Lightbulb The Laws of Non-Contradiction, Excluded Middle, and Bivalence: Laws of Logic (Laws of Thought)

    The Laws of Non-Contradiction, Excluded Middle, and Bivalence

    The Law of Non-Contradiction (LNC): ~ [X & ~X].
    • Nothing can both be and not be.
    • A proposition X and its logical negation ~X cannot both be true together.
    • A proposition X cannot be both true and false.
    • The joint affirmation of contradictories is denied!
    • Something cannot both be and not be.


    The Law of Excluded Middle (LEM): X V~X.
    • Either a proposition X is true or its negation ~X is true.
    • It cannot be the case that neither X is true nor ~X is true.
    • A proposition X cannot be neither true nor false (i.e., not true).
    • A proposition X and its negation ~X cannot both be false together!
    • Excluded middle logically excludes the "joint denialof contradictories (X, ~X)," also called "nor" operator, which stands forneither - nor:

    The Law of Bivalence (LOB):X xor ~X
    • A proposition can only bear/carry one truth value, that truth value being either true or false, not both, and not neither!
    • A proposition X and its negation ~X can neither be true together nor false together.
    • A proposition X is either true or false; where the "or" operator is to be understood as an exclusive-or [i.e., exclusive disjunction: = ‘xor’], which logically excludes both the “and” and the “nor” operations of contradictories X and ~X:
    • The conjunction (the “and” operation) of X and ~X is called the“joint affirmation” of contradictories (X,~X),which yields the both-and-option which states: both X and ~X are true. Therefore, the law of bivalence excludes this option: {i.e., ‘X is true’ and ‘~X is true’}. Therefore, the “joint affirmation” of X and ~X isdenied by the law of bivalence.
    • The “joint denial” of contradictories X and ~X is the neither-nor-optionthat says, “neither X is true nor ~X is true”. This joint denial is alsoexcluded by the law of bivalence. This neither-nor option is a result of the "nor"operation of contradictories (X, ~X):
    • [Xnor~X]= {‘X is false’, and ‘~X is false’};** i.e., “neitherXnor~X istrue”.
    • The law of bivalence excludes the options in which a proposition X and its negation ~X are both true together or both false together.
    • Thejoint affirmation(both-and-option) and thejoint denial(neither-nor-option) of contradictories are logically excluded by the law of bivalence.


    Comparing & Contrasting:
    Non-Contradiction (LNC) vs.
    Excluded Middle (LEM) vs.
    Bivalence (LOB)!

    Four a proposition X, the following options exist:
    [i].X
    [ii].~X
    [iii].Both X and ~X
    [iv].Neither X nor ~X

    Each option can be reformulated as follows:
    [i] = 1, [ii] = 2, [iii] = 3, [iv] = 4:

    1.X is true
    2.~X is true (i.e. X is false)
    3.X is both true and false
    4.X is neither true nor false

    • In classical logic, options (3/iii) and (4/iv) are forbidden, i.e., logically impermissible / excluded by logic.
    • Options3andiiiareexcludedby thelaw of non-contradiction.
    • Options4andivareexcludedby thelaw of excluded middle.


    Law of Non-Contradiction (LNC): ~ (X & ~X),
    (where “&” is logical conjunction: "and" operator).

    The law of non-contradiction (LNC) states the following logically equivalent statements:
    • It cannot be the case that a X and its negation ~X are true together (at the same time, in the same sense, simultaneously).
    • Non-contradiction excludes thejoint affirmationof X and its negation ~X: that is, it cannot be the case the both X and ~X are true.
    • If two propositions are direct logical negations of one another (X, ~X), then at least one of them is false, including the option that both are false and excluding both contradictories cannot be true.
    • A proposition X and its negation ~X cannot both be true.
    • Contradictions cannot be (i.e., are excluded or ruled out).
    • Contradictory propositions cannot both be true.
    • Nothing can both be and not be. That is, something cannot both be and not be.
    • The law of non-contradiction (LNC) can be reformulated as stating: A proposition X cannot be both true and false!
    • The law of non-contradiction does not exclude the case that both X is false and ~X is false!
    • The law of non-contradiction states at least one of X and ~X is false, including the option that both X and ~X are false together, but excluding the option that X and ~X are true together.
    • Out of two contradictories, at least one of them is false; they can both be false, but they cannot both be true.
    • Hence, the law of non-contradiction excludes only the joint affirmation of a pair of direct logical negations ("X is true" and "~X is true").



    Law of Excluded Middle (LEM): X V ~X,
    where V = inclusive disjunction ("or").

    LEMstates:

    • Either a proposition X is true or its negation ~X is true, where "or" is inclusive-or,i.e., LEM includes the conjunction (X & ~X).
    • LEMstates a proposition X is either true or not true (i.e., false), where "or" includes the option that: "X is both true and not true (i.e., false)". Since the inclusive-either-or (inclusive disjunction, "or") of X and ~X can be expressed as the negation (~) of the joint denial (neither-nor, "nor"): inclusive-either-or = not-neither-nor; therefore:
    • A proposition X and its negation ~X cannot be both false together.
    • LEMstates itcannot bethe case thatneitherX is truenor~X is true, which can be equivalently stated as follows:
    • A proposition X cannot be neither true nor false (i.e., not true).
    • LEM logically excludes the neither-nor option: the option generated from the “nor” operation of the two contradictories X and its negation ~X: [X nor ~X]. That is, the joint denial (i.e., “neither-nor”) of both X and ~X is excluded by the law of excluded middle.
    • The logical "nor" operation called "joint denial" of contradictories (X, ~X)! The joint denial of {'X is true' and '~X is true'} is the option that says neither X nor ~X is true; that is, (X is false, ~X is false). Denial of X means denying that X is true, and is not mere failing to accept that "X is true" (i.e. reject); quite to the contrary, to deny X is to accept that its logical negation ~X is true, which leads to therefore "X is false".
    • LEMdoesnotexclude the case thatbothX is trueand~X is true. LEM does not rule out contradictions!
    • LEMstates at most one of the contradictories X and ~X is false.
    • LEMstates at least one of the contradictories X and ~X is true.


    LEM states that at least one of X and ~X is true:
    I. {X is true and ~X is true} is excluded by non-contradiction (LNC) & bivalence (LOB)
    II. {X is true and ~X is false}
    III. {X is false and ~X is true}
    IV. {X is false and ~X is false} is excluded by excluded middle (LEM) & bivalence (LOB)

    The law of bivalence (henceforth, LOB) states that X is either true or false
    LOB includes exactly one of X and ~X is true, and the other false, and vice versa, and moreover excludes both the joint affirmation and the joint denial of contradictories (X, ~X).

    NotethatLOB does not have a negation operator (~)in its expression (whereasLEM does!)

    Further note that the law of bivalence can be expressed as: “X or ~X” where the "or" operator is to be understood as an exclusive-or (i.e., "xor", also denoted as "(+)"); therefore: LOB =X xor ~X.

    An exclusive disjunction [“xor”] of X and ~X is also called "The Exclusive Disjunction of Contradictories (X, ~X): [X xor ~X]”: = LOB
    LOB excludes both the 'joint affirmation' (i.e., X is true AND ~X is true) as well as excluding 'joint denial' (i.e., X is false AND ~X is false).

    A proposition X and its negation ~X form the following permutations
    (rows in the truth table)
    • {X is true and ~X is true} is excluded by non-contradiction (LNC) & bivalence (LOB)
    • {X is true and ~X is false}
    • {X is false and ~X is true}
    • {X is false and ~X is false} is excluded by excluded middle (LEM) & bivalence (LOB)


    LOB states, exactly one of (X, ~X) is true, and the other one false.
    • LOB states {either "X is true" or "~X is true"},
    • and it cannot be neither [X nor ~X],
    • and it cannot be both [X and ~X]!



    Therefore, the law of bivalence (LOB) can be reformulated as follows:

    "Something is not neither or both what it is (X) and what it is not (~X)".

    So, the law of bivalence excludes options (3/iii) and (4/iv)because
    LOB = LEM & LNC
    The law of bivalence is the conjunction of excluded middle and non-contradiction!
    LOB = LNC & LEM.

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    Veteran Member Petros Agapetos's Avatar
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    These are the rudimentary laws of classical logic, to which our thoughts obey. That is why they are also called "The Laws of Thought". They apply to anything you think about (any object of thought) and any thought itself. Moreover, they apply to propositions, declarative statements that can be either true or false.

    These Laws of Thought are:

    1. The Law of Identity
    2. The Law of Non-Contradiction
    3. The Law of Excluded Middle

    and a fourth one, which is the conjunction of Non-Contradiction and Excluded Middle called the Law of Bivalence, the definition of a proposition!

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    Veteran Member Petros Agapetos's Avatar
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    Is it possible for something to exist which violates any of these logical laws?

    Mind you, these are laws of logic, not of nature.

    Do you see how one might use these laws of logic to exclude something from the actual world (i.e., universe)? How about any possible world?
    Do you think logic alone can be sufficient to rule out the logical possibility of something?

    I am of the opinion that these laws of logic, the Laws of as Identity, Non-Contradiction, and Excluded Middle which can logically exclude the possibility of something in any possible world, not just our actual one. That is, I hold that demonstrating a conflict between something and any one of these rudimentary logical laws is sufficient to rule that thing out from existence - not just our actual universe - but any logically possible universe!

    Do you agree with me?
    Or do you think I am inferring or extrapolating too far?

    After all, doesn't it seem sort of ludicrous that laws of logic should dictate as to what can and cannot be possible in reality?

    What do you think?
    Last edited by Petros Agapetos; 10-26-2020 at 07:07 PM.

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    Bring me back good memories. I had a course on mathematical logic back in the University. We had to prove logical implications and equivalences using axiomatic systems and Hilbert deduction. The principles are quite fundamental and even obvious, but when you start combining lots of propositions things tend to get very messy. It was mostly a loss of time since i never use it today.

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    Quote Originally Posted by Petros Agapetos View Post
    These are the rudimentary laws of classical logic, to which our thoughts obey. That is why they are also called "The Laws of Thought". They apply to anything you think about (any object of thought) and any thought itself. Moreover, they apply to propositions, declarative statements that can be either true or false.

    These Laws of Thought are:

    1. The Law of Identity
    2. The Law of Non-Contradiction
    3. The Law of Excluded Middle

    and a fourth one, which is the conjunction of Non-Contradiction and Excluded Middle called the Law of Bivalence, the definition of a proposition!
    It is well known that quantum mechanics is fundamentally incompatible with the classical logic (i.e it follows a non-classical logic). For example, the wave-particle duality contradicts the law of non-contradiction. So if quantum theory follows, the classical logic is inconsistent by implication.

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    Veteran Member Petros Agapetos's Avatar
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    Quote Originally Posted by Token View Post
    Bring me back good memories. I had a course on mathematical logic back in the University. We had to prove logical implications and equivalences using axiomatic systems and Hilbert deduction. The principles are quite fundamental and even obvious, but when you start combining lots of propositions things tend to get very messy. It was mostly a loss of time since i never use it today.

    I am currently taking a course in the philosophy of symbolic formal logic; we have covered truth-functional logical connectives/operators, their truth tables (complete and partial), {implication (->), equivalence (<=>)} - both material and logical, logical operators: conjunction ("^") inclusive disjunction ("V") exclusive disjunction ("(+)", also XOR), NOR (joint denial), XNOR (iff.). I had studied logic gates in my 2nd year Chemical Engineering Program at the University of Alberta in Digital Logic Design (electronic circuit design). We called them "gates" AND-gate, NAND-gate, XOR-gate, XNOR-gate etc.

    Those were fun times!

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