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What logics/philosophies deny LEM, the law of excluded middle (tertium non datur)?

This law is expressed as Philosophical Axiom 4.2:

Tertium non datur (Non est medium inter esse et non esse. ‑ foundation of all distinctions)

There is no third way, there can be no middle between being and non­being.


My question is inspired by the introduction to Semetsky, Edusemiotics — A Handbook (2006):

Eastern thought proclaims “the polar relationship of all opposites” (Capra [The Tao of Physics] 1975, p. 112). For Taosit philosopher Chuang Tzu, for example, ‘this’ is also ‘that’ and ‘that’ is also ‘this’. The apparent opposites are united, hence cease to be binaries but complement each other in the manner of yin and yang [☯], of body and mind, of material and spiritual, of intuitive wisdom and rational knowledge.

cf. "Are there languages with a third term describing the relationship between opposites?"

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    Intuitionistic logic is the standard-bearer, here, although three-valued logics (etc.) are also in the extended family of paracomplete systems. (The distinction, if my understanding does not mislead me, is that in e.g. 3VL one still can have a "law of excluded fourth," etc. so they are not inherently incomplete, or at least their completeness is rather different in proof-theoretic flavor compared to classical 2VL. But I have not been able to find clear definitions as such, so far.) Jun 10, 2023 at 4:38
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    MO thread Status of proof by contradiction and excluded middle throughout the history of mathematics? mentions intuitionism's precursors. More broadly, see LEM's rejection for future contingents (sometimes, even Aristotle is interpreted as restricting LEM for them) and vague predicates/sorites.
    – Conifold
    Jun 10, 2023 at 5:50
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    Minimal logic also does not have excluded middle. In effect it is fragment of intuitionistic logic. There are also several logics intermediate between intuitionistic and classical logic that lack LEM.
    – Bumble
    Jun 10, 2023 at 15:00

3 Answers 3

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Beyond True and False - Graham Priest

The Logic of Buddhist Philosophy

https://aeon.co/essays/the-logic-of-buddhist-philosophy-goes-beyond-simple-truth

Let’s start by turning back the clock. It is India in the fifth century BCE, the age of the historical Buddha, and a rather peculiar principle of reasoning appears to be in general use. This principle is called the catuskoti, meaning ‘four corners’. It insists that there are four possibilities regarding any statement: it might be true (and true only), false (and false only), both true and false, or neither true nor false.

At around the same time, 5,000km to the west in Ancient Athens, Aristotle was laying the foundations of Western logic along very different lines. Among his innovations were two singularly important rules. One of them was the Principle of Excluded Middle (PEM), which says that every claim must be either true or false with no other options (the Latin name for this rule, tertium non datur, means literally ‘a third is not given’). The other rule was the Principle of Non-Contradiction (PNC): nothing can be both true and false at the same time.

At the core of the explanation, one has to grasp a very basic mathematical distinction. I speak of the difference between a relation and a function. A relation is something that relates a certain kind of object to some number of others (zero, one, two, etc). A function, on the other hand, is a special kind of relation that links each such object to exactly one thing. Suppose we are talking about people. Mother of and father of are functions, because every person has exactly one (biological) mother and exactly one father. But son of and daughter of are relations, because parents might have any number of sons and daughters. Functions give a unique output; relations can give any number of outputs. Keep that distinction in mind; we’ll come back to it a lot.

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}
  ↗ ↖
 {T, F} { }
  ↖ ↗
  {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.

Fuzzy Logic - SEP

The Stanford Encyclopedia of Philosophy has articles on Fuzzy Logic, Vagueness, and the Sorties Paradox. Below is just a couple highlights from Fuzzy Logic article.

10. Fuzzy logic and vagueness

Modeling reasoning with vague predicates and propositions is often cited as the main motivation for introducing fuzzy logics. There are many alternative theories of vagueness, but there is a general agreement that the susceptibility to the sorites paradox is a main feature of vagueness.

1. Fuzzy Logic -

Fuzzy logic is intended to model logical reasoning with vague or imprecise statements like “Petr is young (rich, tall, hungry, etc.)”. It refers to a family of many-valued logics, where the truth-values are interpreted as degrees of truth. The truth-value of a logically compound proposition, like “Carles is tall and Chris is rich”, is determined by the truth-value of its components. In other words, like in classical logic, one imposes truth-functionality.

Fuzzy logic emerged in the context of the theory of fuzzy sets, introduced by Lotfi Zadeh (1965). A fuzzy set assigns a degree of membership, typically a real number from the interval [ 0 , 1 ] , to elements of a universe. Fuzzy logic arises by assigning degrees of truth to propositions. The standard set of truth-values (degrees) is the real unit interval [ 0 , 1 ] , where 0 represents “totally false”, 1 represents “totally true”, and the other values refer to partial truth, i.e., intermediate degrees of truth.1

“Fuzzy logic” is often understood in a very wide sense which includes all kinds of formalisms and techniques referring to the systematic handling of degrees of some kind (see, e.g., Nguyen & Walker 2000). In particular in engineering contexts (fuzzy control, fuzzy classification, soft computing) it is aimed at efficient computational methods tolerant to suboptimality and imprecision (see, e.g., Ross 2010). This entry focuses on fuzzy logic in a restricted sense, established as a discipline of mathematical logic following the seminal monograph by Petr Hájek (1998) and nowadays usually referred to as “mathematical fuzzy logic”. For details about the history of different variants of fuzzy logic we refer to Bělohlávek, Dauben, & Klir 2017.

Mathematical fuzzy logic focuses on logics based on a truth-functional account of partial truth and studies them in the spirit of classical mathematical logic, investigating syntax, model-theoretic semantics, proof systems, completeness, etc., both, at the propositional and the predicate level (see Cintula, Fermüller, Hájek, & Noguera 2011 and 2015).

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The logic which we actually use for evaluating claims is a 4-mode logic, and all four of these states violate LEM.

These four states are:

  1. we do not currently have sufficient justification to draw any conclusions about a claim.
  2. we have sufficient justification to consider the claim valid as a working hypothesis
  3. we have sufficient justification to consider the claim invalid
  4. we have sufficient justification to consider the claim to be non-evaluable

These fours states are the method by which we navigate the world, and also by which we do science. All four violate LEM.

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The three-valued logic of Jan Lukaseiwicz is one that denies the law of the excluded middle, as does his infinite valued version. His original intent was to account for the future contingent which Aristotle mentioned but did not directly address. His interpretation was challenged and not generally accepted, although the argument against it invoked the Law of the Excluded Middle. This does not demonstrate an inconsistency in his system as much as it does the fallacy of begging the question. Of course it is contradictory to apply the Law of the Excluded Middle to evaluate the consistency of a system in which it explicitly does not hold.

However, his original interpretation can be broadened to account for a more general form of equivocal uncertainty. Statements that are not definitely either true or false, and may possibly be either can be given the third truth value. This is distinct from statements which are either true or false, but it is not known which is the case.

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