Where can I find a comprehensive discussion of the idea that some propositions are neither true nor false?

I know of Łukasiewicz but his discussion seems rather limited.

Which philosopher if any tried to prove that propositions cannot possibly be neither true nor false?

Thanks for scholarly references.


Contributors may not be native speakers of English and as a result possibly confused by the neither-nor structure. Before we endeavour to understand logic, may be we should try to speak English, so allow me to explain.

Many logic pundits think of propositions as either true or false. That is to say, they would say that a proposition is possibly true and that it possibly false, but that it cannot be neither true nor false. The idea that a proposition is neither true nor false is the idea that it is not true and not false. Thus, it is the idea that the proposition is something other than true or false, namely, neither true nor false.

EDIT 2 I will take a proposition here to be what is meant by a meaningful declarative sentence.

This acception does not imply that a proposition is necessarily either true or false.


3 Answers 3


Firstly, as the SEP article on propositions notes, philosophers use the term 'proposition' to mean a number of related but different things, including:

  • A statement, or declarative sentence that has a truth value.
  • A statement or declarative sentence with the indexicals and names unambiguously resolved.
  • The semantic content or language-independent meaning of a meaningful declarative or descriptive sentence.
  • The object of propositional attitudes, i.e. the things that stand in place of P in expressions like "believes that P", "hopes that P", "fears that P", etc.
  • The object of that-clauses, i.e. the things that stand in place of P in expressions like "the fact that P", "the possibility that P", etc.

Suppose we take it that minimally a proposition is something that has a truth value, without worrying about whether this something is a sentence, an utterance or an abstract object with a meaning. The principle that all propositions have one of two values is called the principle of bivalence. Usually these two values are taken to be true and false. So your question amounts to asking, what arguments have there been for and against the principle of bivalence? I say argument, rather than proof, since general principles like this are seldom amenable to proof.

Often it is simply assumed, or taken by convention, that propositions are true or false with no other possibility. Classical logic, which is the most commonly used logic, is bivalent, at least in its standard form. But there are some areas where bivalence seems implausible. This in turn has led to the development of logics that are not bivalent.

  1. There are vague statements, such as "John is tall". This is not simply either true or false but a matter of degree. Treating vague statements as bivalent leads to the sorites paradox. There is considerable discussion of this paradox, and one popular response is to reject bivalence and use fuzzy logic.

  2. Future contingents are often treated as not being bivalent. Aristotle in his passage about the sea battle that might happen tomorrow is usually understood to be rejecting the claim that it is either true or false today that the battle will take place tomorrow. Indeed there are arguments for fatalism that proceed from this claim and a considerable literature exists on how to circumvent them. Some writers have argued that we can maintain bivalence with future contingents by the use of modalities and modal logic. The SEP article on Gregory of Rimini notes that he was one of the first to argue in this way.

  3. Bivalence within mathematics is rejected by intuitionists. According to this approach to the philosophy of mathematics, mathematics is concerned with what rational agents can prove, and so mathematical truth cannot extend beyond what is provable. But there are mathematical propositions that are neither provable nor disprovable, by the standards of intuitionists at least. So intuitionism introduces its own distinctive logic that lacks bivalence.

  4. Michael Dummett, who was an advocate of intuitionism, applied the same principle outside mathematics. There are many traditional debates within philosophy that take the form of a realist position in opposition to an antirealist one. E.g. realism vs. nominalism about univerals, moral cognitivism vs. non-cognitivism, scientic realism vs. constructivism, realism about the external world vs. idealism, etc. Dummett advanced the view that bivalence is the logical criterion that distinguishes realist positions from antirealist ones. If we are realist about some subject, this implies we believe that propositions about it are true or false independently of ourselves and our knowledge, and hence bivalence applies. If we are antirealist, this implies that there is no independent fact of the matter, and so bivalence does not apply. This means that potentially there are lots of ways for bivalence to fail, depending on your philosophical views.


EDIT - Wikipedia - Tautology, Contradiction, and Contingency


The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue.

Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent.


In philosophy and logic, contingency is the status of propositions that are neither true under every possible valuation (i.e. tautologies) nor false under every possible valuation (i.e. contradictions). A contingent proposition is neither necessarily true nor necessarily false.

SEP - Aristotle's Logic

  1. Premises: The Structures of Assertions


The pair consisting of an affirmation and its corresponding denial is a contradiction (antiphasis). In general, Aristotle holds, exactly one member of any contradiction is true and one false: they cannot both be true, and they cannot both be false.

Graham Priest - Beyond True and False


Now, in logic, one is generally interested in whether a given claim is true or false. Logicians call true and false truth values. Normally, and following Aristotle, it is assumed that ‘value of’ is a function: the value of any given assertion is exactly one of true (or T), and false (or F). In this way, the principles of excluded middle (PEM) and non-contradiction (PNC) are built into the mathematics from the start. But they needn’t be.

To get back to something that the Buddha might recognise, all we need to do is make value of into a relation instead of a function. Thus T might be a value of a sentence, as can F, both, or neither. We now have four possibilities: {T}, {F}, {T,F} and { }. The curly brackets, by the way, indicate that we are dealing with sets of truth values rather than individual ones, as befits a relation rather than a function. The last pair of brackets denotes what mathematicians call the empty set: it is a collection with no members, like the set of humans with 17 legs. It would be conventional in mathematics to represent our four values using something called a Hasse diagram, like so:

↗ ↖
{T, F} { }
↖ ↗

Thus the four kotis (corners) of the catuskoti appear before us.

In case this all sounds rather convenient for the purposes of Buddhist apologism, I should mention that the logic I have just described is called First Degree Entailment (FDE). It was originally constructed in the 1960s in an area called relevant logic. Exactly what this is need not concern us, but the US logician Nuel Belnap argued that FDE was a sensible system for databases that might have been fed inconsistent or incomplete information. All of which is to say, it had nothing to do with Buddhism whatsoever.

Lotfi Zadeh and the Birth of Fuzzy Logic


Fuzzy technology, Zadeh explained, is a means of computing with words—bigger, smaller, taller, shorter. For example, small can be multiplied by a few and added to large, or colder can be added to warmer to get something in between.

The Japanese pioneered the use of fuzzy computing. Fuzzy systems are now in common use. The words hot, warm, and cold, for example, are distinct. But the concepts overlap as vague or fuzzy categories. I call a vague or fuzzy distinction an indistinct distinction. By the way I have much experience with swimming pools. People do not feel the temperature of the water consistently. They feel the difference in temperature between the air, water, and condition of their body. So the same temperature water can feel warm or cold at different times to the same person.

  • "exactly one member of any contradiction is true and one false: they cannot both be true, and they cannot both be false." This is irrelevant to my question. Please remove this part of your answer. Feb 19 at 9:41
  • Your answer doesn't address my question. Please delete it or edit it to make it relevant to my question. Feb 19 at 9:44

Which philosopher if any tried to prove that propositions cannot possibly be neither true nor false?

I'd say no one, that is just not a philosophical problem per se. (I think) a philosopher might indeed retort: isn't it the very definition of "proposition" that it is a statement of fact and, as such, can only be one of true or false depending on whether the statement is faithful to the actual fact or not? (A definition that, modulo subtleties, holds both in general Logic, i.e. the logic of/in natural language, as well as in formal logic.)

(Separate issue is that propositions may be not decidable, which is about our lack of ability to decide some propositions, not about the definition of proposition per se. Indeed, we could rather doubt/question that undecidable propositions are propositions at all.)

  • Yes, and, the largest room in the world is the room for nonsense, which, really, most everything is. "Colorless green ideas sleep furiously."
    – Scott Rowe
    Feb 18 at 22:27
  • @JulioDiEgidio Thank you, you're at least one who seems to have understood the question! - 2. "I'd say no one" Maybe no philosopher successfully proved it, but I distinctly remember one example of one who tried. Whether he was successful, I cannot say, and I don't remember who it was, and possibly it was not someone well-known. Feb 19 at 9:34
  • 1
    @ScottRowe "Colorless green ideas sleep furiously." If the meaning of the single words is their meaning in English, that utterance is just meaningless, since "colorless" and "green" are incompatible predicates. More problematic would be "Green ideas sleep furiously", yet I'd say that is a proposition (by the form of it) as long as it is given some meaning (in some figurative sense!). IOW, that we can give it a meaning is preliminary to an utterance being a proposition or not: so, that is no problem at all. (Most humbly, I do disagree with Chomsky as to the analysis of that example.) Feb 19 at 17:41

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