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

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?

``````                 **Research that I have done so far**
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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.

• You misunderstood what propositions are. When we say a proposition is true or false there is a specific CONTEXT. Not just any ole context. There are different types of truths. An objective truth is what we use to describe propositions when we mean the truth value dies not change. There are also contingent TRUTHS. There are propositions that do change truth value from true on Monday and false on Thursday. BELIEF has nothing to do with an objective truth. A proposition is objectively true or false even if you are not aware of it. So if you confuse the definition you can get wrong conclusions. Jun 26, 2020 at 17:15
• If you let Mathematical logic people define proposition or the LEM you will get what you have. You have to be a pretty literal reader to think this way. If you understood CONCEPTS instead which are propositions then you would interpret LEM correctly. The proposition x can not be simultaneously true and false in the same respect (such as time, location, position, size, depth, classification, behavior, substance, etc) and in the same context used the same language. That if you tell me proposition x is true on one side and proposition x is false over there, THEN something is not detailed enough. Jun 26, 2020 at 17:22
• A proposition cannot both be true and false AND a proposition cannot be neither true nor false, but some other third or middle option.: X and ~X cannot both be true together or false together. Jun 26, 2020 at 21:42
• Denying X V ~X does violate LEM, but denying B(X) V B(~X) does not, the negation does not commute with the belief operator. It is not only possible but quite common. When done deliberately it is called suspension of judgment. Trial juries are even required to neither believe nor disbelieve the person's guilt until the end of trial. Moreover, if one simply does not care to consider a proposition they can neither believe nor disbelieve it. People unfamiliar with set theory have no beliefs concerning the continuum hypothesis, for example. Jun 26, 2020 at 23:53
• Regardless of whether we accept LEM, it's being misapplied here. The negation of "I believe X" is "I do not believe X," not "I believe not-X." And these are clearly different things. Jan 18, 2021 at 19:19

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.

• LEM states that either a proposition X is true or its negation ~X is true (i.e. true or not true), where: not true = 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.} Jun 26, 2020 at 21:19
• A bivalent proposition is defined by the Law of Bivalence (conjunction of the laws of non-contradiction and excluded middle.). A proposition cannot both be true and false (Law of Non-Contradiction) AND a proposition cannot be neither true nor false, but some other third or middle option. (The Law of Excluded Middle): X and ~X cannot both be true together or false together. Jun 26, 2020 at 21:20
• @Karen karapetlyan, all statements do not have a truth value. So some statements do not have an either true or false value. Some statements are neither true or false but they do NOT Express a proposition. So because x is not true does not necessarily make x false. You ought to state if you are using only a Mathematical logic context or deductive reasoning. There is a difference. In a Rhetoric context can be a whole lot different results than Mathematical logic. Context makes a huge difference. Jun 26, 2020 at 22:30

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.

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.

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.

• Non-contradiction is a tautology ~(X^~X), Contradiction is a necessary falsity. X and ~X cannot both be true means X cannot both be true and false. X i.or ~X means that X and ~X cannot be both false together; where i.or = inclusive disjunction (incl.-or). If ~[X ^ ~X] means X cannot be both true and false, then X V~X must mean that X cannot be neither true nor false, but another middle or otherwise third option other than true and/or false. To say I neither believe nor disbelieve(i.e.,do not believe) X implies one believes neither X nor ~X, the converse is not necessarily true. Jan 20, 2021 at 4:26
• A -> B : implication "forward". B ->A: converse implication. I can grant that to neither believe nor not believe X implies neither to believe X nor ~X, but this doesn't mean that the converse necessarily follows, namely that to believe X and ~X are not true necessitates neither believing nor disbelieving either one: believe(X) NOR not believe(X) =/= believe(X) NOR believe(not_X). Jan 20, 2021 at 4:31