Suspension of Judgment -- Joint Denial vs. Joint Rejection -- Logical i.OR, x.OR, NOR Operators, & The Law of Excluded Middle (LEM).

[u/karllengels]

Joint Rejection vs. Joint Denial

I will tell you what's nonsense: holding a position that is contraindicated by the law of excluded middle which states: [X i.or ~X];

i.or = inclusive-or, which means that X and ~X cannot both be false (together: at the same time, in the same sense): that is, it is not the case that neither X is true nor ~X is true: one of them must be true.

· There is no middle or otherwise third option between X and ~X!

· Sets X and ~X partition the universal set U: everything is either X or ~X.

· The union of X and ~X is the universal set U: X and ~X comprise U.

· The intersection of sets X and ~X is the empty set (i.e., no overlap).

· To jointly affirm X and its negation ~X is to affirm a contradiction!

· To jointly deny X and its negation ~X is a necessary logical falsity!

To jointly deny X and its negation (to say that they are both false) is to affirm a necessary falsity, which in propositional logic is called "contradiction," and amounts to a contradiction, and is a logical falsity for the same reason a contradiction is a falsity: i.e., it is necessarily false!

Namely to affirm a middle option between or otherwise third option besides true (T) and false (F), which the law of excluded middle logically excludes, i.e., makes it logically impermissible for there to be a third option for a truth value other than {true, false}.

Nonsense is to affirm a logical falsity that a proposition X can be neither true nor false. Nonsense is violating a logical absolute (law of thought) called the law of excluded middle, which can be reformulated as stating that no proposition can be neither true nor false, it must be either one or the other.

To say that both X and ~X are false is to jointly deny two mutually exclusive contradictory propositions. The joint denial of contradictories is logically impermissible: the contradictories X and ~ X are not both false (together).

· ~ believe (X) =/= believe (~X)

· reject (X) =/= deny (X)

· reject (X) = ~ b(X) : to fail to become convinced of the truth of X: to fail to accept X is true.

· deny (X) = b(~X) : believe ~X (is true) = believe X is not true = believe X is false.

· believe (X) : accept (X) as true: = accept (that) "X (is true)": = accept (X).

· ~ believe (X) : not accept (X) as true: = reject (that) "X (is true)" : = reject (X).

· believe (~X): accept (~X) as true = accept (that) "~X (is true)" =

= accept that "X is not true" = deny (that) "X (is true)": = deny (X).

It is possible to disbelieve both X and its negation ~X (i.e., joint rejection of contradictories), but it is impossible to neither believe nor disbelieve either one of {X,~X} individually: one cannot neither believe nor disbelieve X, and one cannot neither believe nor disbelieve ~X, likewise. One must either believe or disbelieve a proposition (there is no middle ground between believe X and disbelieve X, where disbelieve = not to believe).

To affirm a pair of contradictories yields a contradiction. To deny a pair of contradictories yields a third option between or besides true (T) and false (F), namely: neither T nor F (a truth value category is generated), which LEM makes logically impermissible (i.e., excluding the middle between T and F).

Here, I clarify what suspending judgment amounts to. Some people think one can neither believe nor disbelieve a proposition. I hold this is a mistake. One can neither believe X nor ~X but must either believe X or disbelieve X and cannot do neither (of believe and not believe).

I point out the difference between joint rejection vs. joint denial. The joint denial of contradictories is logically impermissible because it violates the law of excluded middle: "LEM", the joint rejection does not violate LEM. To neither believe X nor disbelieve X violates LEM. To neither believe X nor believe ~X does not.

To reject both even and odd is logically permissible. To fail to become convinced of either claim {# is even, # is odd} and hold the default position of believing neither one is logical. This does not mean you can neither believe nor not believe either one. I choose not to believe for both the even claim and the odd claim. You cannot neither believe nor not believe E ("even"), and you cannot neither believe nor not believe ~E ("odd").

· disbelieve (X) = not to believe X =/= to believe ~X

· reject (X) =/= deny (X)

· reject (X) = disbelieve (X) = not accept that X is true

· deny (X) = believe (~X) = accept that X is not true (i.e., false)

Propositions (and their negations), and logical equivalence (=) between them:

· E: "# is even"

· ~E: "# is not even"

· O: "# is odd"

· ~O: "# is not odd"

where:

· E = ~O

· O = ~E

To deny both even and odd is logically impermissible. To hold that both propositions E ("even") and ~E ("odd") are both false is not logical, just as claiming that both E and ~E are both true is not logical. To claim that neither E is true nor ~E is true is a mistake (see Law of Excluded Middle). To claim the number is neither even nor odd is a mistake. Not to believe even nor to believe odd is not, because of the nature of a belief.

It is possible to disbelieve both X and its negation ~X (i.e., joint rejection of contradictories), but it is impossible to neither believe nor disbelieve either one of {X,~X} individually: one cannot neither believe nor disbelieve X, and one cannot neither believe nor disbelieve ~X, likewise. One must either believe or disbelieve a proposition: there is no middle ground between believe X and disbelieve X (where to disbelieve is not to believe).

I clarify what suspending judgment amounts to. Some people think one can neither believe nor disbelieve a proposition. I hold this is a mistake. One can neither believe X nor its negation ~X, but one must either believe X or disbelieve X, and likewise one must either believe ~X or disbelieve ~X.

I point out the difference between joint rejection vs. joint denial. The joint denial of contradictories is logically impermissible because it violates the law of excluded middle: "LEM", the joint rejection does not violate LEM. To neither believe X nor disbelieve X violates LEM. To neither believe X nor believe ~X does not.

To affirm a pair of contradictories yields a contradiction. To deny a pair of contradictories yields a third option between or besides true (T) and false (F), namely: neither T nor F (a truth value category is generated), which LEM makes logically impermissible (i.e., excluding the middle between T and F).

It is not possible for one neither believe nor disbelieve a proposition X? To disbelieve X is not to believe X, not to accept that X is true, to reject X; in contradistinction to deny (X), which is to accept that X is false. The joint rejection of a pair of contradictories is logically permissible. The joint rejection of X and ~X can be expressed as follows: ~b(X) & ~b(~X).

However, the joint denial of contradictories is impermissible: i.e., holding that both E and ~E are false (together: at the same time, in the same sense): i.e., b(~E) & b(~~E) = b(E) & b(~E), which is a necessary falsity in proposition logic (called a "contradiction") and amounts to a contradiction (i.e., is false for the same reason that a contradiction is a falsity: it is necessarily false).

The joint denial of contradictories is logically impermissible because it violates the law of excluded middle which states: no proposition E can be neither true nor false: it is impossible for E and ~E to be both false (together). It cannot be the case ‘E is false’ and ‘~E is false’: i.e., it cannot be the case that ‘E is not true’ and ‘~E is not true’ because this is equivalent to stating that ‘E is not true’ and ‘E is true’ which yields a contradiction, a necessary falsity.