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.
