Is it logically permissible to neither believe nor disbelieve a proposition X? Or does this violate the law of excluded middle?

Question

Given a proposition X, one can either believe it or disbelieve it.

• Is it logical however to neither believe X nor disbelieve X?
• Is it logical to neither believe proposition X nor its negation ~X?

I define 'believing X' as accepting that X is true. I define 'disbelieving X' as 'not believing X' (such as by failing to become convinced of the truth of proposition X). I define 'disbelieving X' as 'rejecting X,' or more precisely: "rejecting that X is true" (i.e., rejecting 'X is true' = 'not accepting' that 'X is true').

EXAMPLE: To be odd or not to be odd!

I have a jar of an unknown number of coins. The number of coins is either even or odd. Without sufficient information to determine the parity of the number, I disbelieve the number is even (where disbelieve = lack believe = do not believe). For the same reason, I also disbelieve that the number is odd, even though the number must in actuality have a single parity value, that value being either even or odd.

QUESTIONS:

Let: X: "the number... is even", then ~X: "the number...is not even" = "the number is...odd".

1. Is it possible to neither believe nor disbelieve a given proposition X: that is, is it possible to neither believe the number is even nor disbelieve the number is even?
2. Is it possible to neither believe X nor believe ~X: that is, is it possible to neither believe "the number is even" nor believe "the number is not even".
3. Does it not violate LEM to neither believe nor disbelieve either a proposition X or its negation ~X?
4. Does it not violate LEM to neither believe X nor ~X?

Please explain...

                 **Research that I have done so far**

Let (by definition):

• Let: LNC: = Law of Non-Contradiction
• Let: LEM: = Law of Excluded Middle
• Let: LOB: = Law of Bivalence

The law of excluded middle (henceforth LEM) states that either a proposition X is true or its negation ~X is true (where ~X = not X), which can be reformulated as "A proposition X is either true or not true, i.e., false, for a bivalent {two-valued (T,F)} proposition - a declarative statement capable of bearing only one truth-value at a time, that true value being either true or false.} LEM states the inclusive disjunction X V ~X.

Therefore LEM states X or ~X, where "or" is to be understood as an inclusive disjunction ("V"): LEM: = (X V ~X), where V = inclusive disjunction, as opposed to the law of bivalence (henceforth LOB) which states X or ~X, where the "or" operator is to be understood as an exclusive disjunction: LOB: = X (+) ~X, where (+) is 'XOR' (i.e. exclusive-or).

THE DIFFERENCE BETWEEN INCLUSIVE-OR ["V"] AND EXCLUSIVE-OR ["(+)"]:

Inclusive-or: includes the option that X is true and ~X is true. Exclusive-or: excludes the option that X is true and ~X is true.

Therefore, the law of excluded middle (LEM) states that either: LEM (i) X is true, or LEM (ii) ~X is true, or LEM (iii) Both X and ~X are true together, that is, the conjunction (X AND ~X) is true; where all "or" operators are inclusive.

A bivalent proposition is defined by the law of bivalence. The law of bivalence is the conjunction of the laws of non-contradiction and excluded middle.

A (bivalent) proposition cannot be both true and false (at the same time, in the same sense, simultaneously)---****Law of Non-Contradiction (LNC).

A (bivalent) proposition cannot be neither true nor false, but some other third or middle option. ---Law of Excluded Middle (LEM).

The law of bivalence states that a proposition X and its direct logical negation ~X cannot both be true together (LNC) or false together (LEM): that is, exactly one of the contradictory propositions (X,~X) is true and the other false:

The following conditions describe the law of bivalence:

• If X is true, then ~X is false.
• If X is false, then ~X is true.
• It cannot be the case that both 'X is true' and '~X is true': that is, X and ~X cannot be both true together.--- {the option "both X and ~X are true" is logically excluded by LNC!}.
• It cannot be the case that neither X is true nor ~X is true: that is, X and ~X cannot both be false together. --- {the option "neither X nor ~X is true" is logically excluded by LEM!}.

Whereas, the law of bivalence (LOB) states that EXACTLY ONE of X and ~X is true, and the other false. Therefore the law of bivalence satisfies the following conditions (in a truth table):

• LOB (i) X is true, then ~X is false
• LOB (ii) X is false, then ~X is true
• LOB (iii) It CANNOT be the case that both X and ~X are true together.
• LOB (iv) It CANNOT be the case that both X and ~X are false together. together.

A proposition is defined by the law of bivalence!

A proposition can be either

• (i) true, in which case its negation ~X is false, or
• (ii) false, in which case its negation ~X is true,

A proposition can be neither

• (iii) true and false,
• (iv) neither true nor false

In other words,

• (iii) A proposition cannot be both true and false
• (vi) A proposition cannot be neither true nor false.

Therefore LEM (inclusive-either-or) can be restated as the negation of the joint denial (not-neither-nor), i.e. LEM: = It is NOT the case that NEITHER X is true NOR ~X is true. That is, LEM can be reformulated as saying that X and ~X cannot both be FALSE together, in contradistinction to the Law of Non-Contradiction which states X and ~X cannot both be TRUE together!

NOTICE: It is demonstrable via a truth table that LOB = LEM AND LNC: where LOB excludes both the option that X and ~X are both true together (LNC), and the option that X and ~X are both false together (LEM).

Therefore, it would seem to violate LEM to say that it CAN be the case that neither X is true nor X is true.

I define denial as accepting that X is false, vs. rejection = not accepting that X is true (ex., such as by failing to become convinced of the truth of the proposition. A proposition is bivalent by definition: capable of carrying only one truth value, either true xor false.

Bi1. A proposition can have only one truth value.

Bi2. The truth value of a proposition can only be either true or false, where "or" is to be understood as an exclusive disjunction.

The above two theses of the law of bivalence taken together yield the "Law of Bivalence". A proposition is by definition bivalent: two-valued, those two truth values being true and false. A proposition can bear only one truth value, that single truth value being either true or false, where "or" is to be understood as exclusive. The law of excluded middle states X V ~X = ("X inclusive-or ~X"). The law of bivalence states X xor ~X.

Given: an exclusive disjunction (xor) outputs a truth value of true when exactly one of X and ~X is true and the other false. The options where X and ~X are both true together or both false together are logically excluded.

The logical complement of "xor" is xnor, where xnor = exclusive nor; where the nor operator is the joint denial of X and ~X; the option that X is false and ~X is false; which is the "neither-nor" option; the logical complement of inclusive-or. The inclusive-"either-or"-option is called an inclusive disjunction (simply, or), in contradistinction to the exclusive-"either-or"-option (xor), which excludes the option that both X and ~X are true (the contradiction: (X & ~X).

Exclusive-or (xor) means either X is true or ~X is true and it cannot be that case that both X and ~X are true, and it cannot be the case that neither X nor ~X is true; one of them has to be true, in which case the other is false: say X is true, then ~X is false; say X is false, then ~X is true, and it excludes the contradiction that "X is true" and "~X is true" (i.e., contradiction = joint affirmation: the option in which X is both true (X is true) and false (~X is true). An inclusive-or includes this contradiction (X and ~X), exclusive-or excludes it.

Answer 1

You seem to confuse belief (which is subjective) and the actual truth value of a proposition. The LEM only applies to the latter, not to the former. If you wish to stay inside a mathematical framework, one might view probabilities as being degrees of belief. This is the subjective probability interpretation, or the Bayesian view. In your example, we would simply give both options less than 100% probability each, to reflect that we do not know which one is true. Still, the number of coins is either even or odd, regardless of what we believe about it.

Answer 2

The logical opposite of "I believe X is true" is not "i believe X is false" but "I do not believe X is true".

While "X true" and "X false" are contradictory, and can't be both part of your beliefs without contradiction, not believe that X is either true or false is valid, and equivalent to "I don't know about X's truth".

It is in fact the most common and honest position one can have about most of the universe, since we don't know most of it.

NB: even "I believe X is true" and "i believe X is false" are not stricto sensu contradictory. The key word here is "believe". While "X true" and "X false" can't be true together, but could be both included in the beliefs of a person. Sure, that is not a very sound belief system, but people do believe contradictory stuff more often than not.

Answer 3

In your example with coins in a jar, the law of the excluded middle requires that you accept the proposition "the number of coins in the jar is even or the number of coins in the jar is odd". It does not require that you believe or disbelieve either half of that proposition in isolation nor does it speak to the validity of either half of that proposition in isolation.

Answer 4

A concrete case: There was a TV program about a woman in the USA, charged and convicted to a life sentence for murdering her two young children. She swears that an unknown person entered their home, attacked her, and then killed the kids.

I have not the slightest doubt that she is either guilty or innocent. However I don't believe she is guilty, and I don't believe she is innocent either. I believe that I don't have enough evidence to support either belief.

Answer 5

You only missed one thing: Believing ~p doesnt stand to believing p as how ~X stands to X. Basically, B~p and Bp arent related in the same way as ~X and X. That's because the proposition "i believe that p" is denied by denying the predicate altogether, rather than by denying a part of it. That is, "i believe that p" is contradicted not by "i believe that ~p" but by "i do NOT believe that p". That is, ~Bp. Bp v ~Bp is the proper instance of the LEM, and not Bp v B~p. The exact logical relations are:

Take "Bp" as "i believe that p is true", "B~p" as "i believe that p is not true", "~Bp" as "i dont believe that p is true" and "~B~p" as "i dont believe that p is not true

Bp and B~p are logical CONTRARIES, i.e Bp ^ B~p cannot hold, but ~Bp ^ ~B~p could hold ; Bp and ~Bp are logical CONTRADICTORIES, i.e neither Bp ^ ~Bp nor ~Bp ^ ~~Bp could hold ; ~B~p and ~Bp are logical SUBCONTRARIES, i.e ~B~p ^ ~Bp could hold but ~~B~p ^ ~~Bp cannot hold ; and {Bp, ~B~p} and {B~p, ~Bp} are logical SUBALTERNS, because Bp → ~B~p and B~p → ~Bp.

~Bp ^ ~B~p holds for all inanimate beings such as rocks and chairs, and also for humans in relation to certain subjects. Thus, if "p" means "God is real", then "~Bp ^ ~B~p" could hold for someone, such as for agnostics. What cannot hold in any circumstance is Bp ^ ~Bp (denies the LNC) and ~Bp ^ ~~Bp (denies the LEM). Basically, B~p and ~Bp differ. Bp and ~Bp are logical contradictories like X and ~X, whilst Bp and B~p are mere logical contraries. Bp and ~Bp (and X and ~X as well) are opposite in truth and falsity, whilst Bp and B~p are opposite merely in truth.