**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 denial of
contradictories (X, ~X)," also called "nor" operator, which stands
for neither - 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 is denied by
the law of bivalence.
The “joint denial” of contradictories X and ~X is the neither-nor-option that says, “neither X is true nor ~X is true”. This joint denial is also excluded by the law of bivalence. This neither-nor option is a result of the "nor" operation of contradictories (X, ~X):
[X nor ~X] = {‘X is false’, and ‘~X is false’};** i.e., “neither X nor ~X is true”.
The law of bivalence excludes the options in which a proposition X and its negation ~X are both true together or both false together. The joint affirmation (both-and-option) and the joint 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.
Options 3 and iii are excluded by the law of non-contradiction.
Options 4 and iv are excluded by the law of excluded middle.
Law of Non-Contradiction (LNC): ~(X & ~X),
*where* & is logical conjunction ("and").
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 the joint affirmation of 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, but they both 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").
LEM states: 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).
LEM states 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.
- LEM states it cannot be the case that neither X is true nor ~X is true, which can equivalently stated as follows: A proposition X cannot be neither true nor not true (i.e., false).
- The neither-nor operation of the two following contradictories: [X
nor ~X]: that is, joint denial of both X and its negation ~X.
- 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".
- LEM does not exclude the case that both X is true and ~X is true. LEM does not rule out contradictions!
- LEM states at most one of the contradictories X and ~X is false.
- LEM states 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)
LEM states exactly one of X and ~X is true, and the other false, and vice versa, and moreover includes the option where both are true (contradiction), but excludes the option where both are false (joint denial).
The law of bivalence (henceforth, LOB) states that X is either true or false.
- Note that LOB does not have a negation operator (~) in its expression (whereas LEM 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 can be more clearly
expressed as: 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!
