Are there exceptions to the principle of the excluded middle?

Question

It is inherent in our truth tables that we configure a statement to be either true or false, but it seems that this might have certain exceptions, especially in conditionals.

For example, the commonly held tradition when faced with a conditional in which the antecedent never occurs or is 'false' is to hold that the conditional is automatically 'true'. So for example, if we have a conditional in which we say 'If it rains tomorrow (p), I will go for a jog (q).' In the event of the nonoccurrence of it raining tomorrow (p), then the entire conditional is considered true.

But it seems that from a certain view such a conditional would neither be true nor false. This appears to be the case in that the truth or falsity of a statement can only be determined by analysis of its whole meaning, but since the antecedent never occurs, it couldn't be said whether such a conditional, taken as a whole, is true or not.

This is because the consequent depends on the antecedent for its truth-value, in that the consequent of jogging is meaningful and relevant only insofar as a precondition is met in the antecedent that it rains tomorrow. Thus, if it doesn't rain tomorrow, the very truth-value and relevance of the consequent seems to be drastically diminished. As such, the whole of the statement is no longer determined and seems to rather be cast into a 'neutral' or undetermined relation to truth.

This of course means that the principle of the excluded middle has certain exceptions (namely those regarding conditionals). However, accepting this would apparently be a big sacrifice for logicians to make. As such, have any solutions been offered to appease the common sense notion that conditionals can sometimes be a supposed exemption from the principle of the excluded middle? Furthermore, if it is recognized that there are exceptions, how has the principle been interpreted in light of this?

Answer 1

There's two different things going in your question.

1. Should there be "exceptions" to the classical laws of logic (identity, excluded middle, and non-contradiction"?
2. How should we treat conditionals with false antecedents? (i.e. is the classic solution of considering them True problematic and if so what should be done instead).

To answer your question, I'm going to first give a brief overview of what the law of the excluded middle is (in part in order to say what it is not).

The principle of the excluded middle is an axiom of certain forms of logic. This has its origin in the West at least with Aristotle as one of his first principles (http://plato.stanford.edu/entries/contradiction/).

This joined with a principle of identity and a principle of non-contradiction give us a very simple system for logic that is largely effective. But these principles also by their very nature limit its applicability.

In other words, it's a tool that's part of a tool kit, and it's one we don't use constantly in our lives at every moment. Instead, it's useful when trying to resolve certain problems in a logical way.

---

How to treat conditionals with false antecedents is one of the places that shows a limitation with logic built around these three laws. I don't think it's best to think of it as requiring an "exception." But rather showing a potential limitation of logic built on three laws.

Logicians have suggested several solutions to the problems raised by the three law logic of Aristotle. One solution is to have a NULL value for things that do not compute. Another solution would be to refuse to translate English language conditionals for which if the antecedent is false, the value is undefined in this way. A third solution is to use different operators along with the three laws to avoid this problem.

In other words, you can either say that three laws are inadequate to conditional sentences or you can say that we need operators other than conditionals and must take great care in our translations of sentences into propositional forms.

---

Does this require there to be "exceptions" to the law of the excluded middle. On my view, the answer is no. It requires us instead to understand that the logic built around the three laws is a tool, which is very helpful when dealing with the right sorts of problems and very unhelpful when dealing with the wrong sorts of problems.

For instance, per the law of the excluded middle, the sentence "it is raining" (or more accurately the proposition behind the sentence) must be either true or false. But what if the reality is that it's "sleeting" or that there's some other form of wet precipitation happening or there's a mixture of rain and something else. Since the law of the excluded middle requires us to go two-valued, we have to make a choice between whether this is true or false.

Again, there's more than one way to fix it and make it more applicable. We can add many fine-grained distinctions to our definitions (narrowing or broadening the identity of "raining" vis-a-vis normal language) or we can refuse to try to fit that reality into propositional form.

Much of the reasoning we do is loosely bound by the three laws rather than strictly bound by the three laws. I take this to be the case, because it's a very clear way of making deductions and drawing conclusions.

But I think if we start making "exceptions", then we're breaking the tool where it works. Instead, we need to keep our heads on our shoulders and know when to apply this sort of logic strictly (for instance in a large amount of programming and math), when to apply it loosely, and when not to apply it all (arguments with my wife, perhaps?).

Answer 2

A better way to put it is not that the law of excluded middle (LEM) has exceptions, but rather the situations where it holds are an exception. This applies to all laws of classical logic, they require precise, ideal and unchangeable domain of discourse to hold. This is the case in mathematics and some mathematized scientific theories, and in some toy situations discussed in logic courses, but other than that the applicability of classical logic is greatly exaggerated, wherever there is vagueness or change it strictly speaking fails.

Some obvious situations where LEM/bivalence of truth fails are future uncertain events, which do not yet have a truth value (unless one believes in unchangeable Fate), statements with vague predicates like "is a heap" (are 10 grains a heap or not? how about 50?), and possibly statements that are in principle unknowable, such as "a city will never be built here", although that is controversial. But with LEM its universal applicability is disputed even in mathematics. According to intuitionists statement that can neither be proved nor disproved lack a truth value altogether, and the logic obtained from the classical logic by removing LEM (with related technical changes) is called intuitionistic. Intuitionistic logic and mathematics are well developed, and constructive mathematics is part of it. Philosophical extension of intuitionism beyond mathematics is called anti-realism.

But the issue you are raising seems to me closer related not to LEM but to another artifice of the classical logic, the material conditional. That the conditional of the natural language is not compositional, i.e. that its value is not based on truth values of its terms alone, is well known as well. For example, unless salt is actually placed in water "if salt is placed in water it does not dissolve" is true in classical logic, and so is "if the Sun is made of gas then 3 is a prime number", but not in natural reasoning. The conditional of natural if-then reasoning is called the indicative conditional, and as such it has no formal definition. Some approximation of it is given by relevance logic, which demands in particular that the premise be relevant to the conclusion for the conditional between them to hold, so in contrast to the classical logic "if the Sun is made of gas then 3 is a prime number" is false.

Another alternative known as the strict conditional was introduced by Lewis specifically to deal with counterfactuals like the one above. It combines the material conditional with the necessity operator in modal logic. Since "if salt is placed in water it does not dissolve" is not true necessarily, i.e. in every possible world, interpreted as the strict conditional it fails to be true. On the other hand, "if salt is placed in water it dissolves" obtains necessarily, trivially in the worlds where it is not placed in water, and due to the properties of salt and water where it is, so this strict conditional does hold, as intuitively expected.

Answer 3

The point is at what level one contemplates the systematic structure of any logic - at the level of defining its foundational elements, or at the level of its application. For example one says that the law of excluded middle is not valid in intuitionistic logic - but for defining the foundational elements of intuitionistic logic, the law of excluded middle is necessary, so there is no true fundamental non-dependence from this law.

Answer 4

Even in the strictest terms, Russel's paradox indicates the need for an exception to the Law of the Excluded Middle.

Take the set of all sets that do not contain themselves as elements

R = {x: x ∉ x}

and try to decide whether it is in itself,

R ∈ R ?

You can't. If it is in itself, then by definition it is not in itself. So it is not true that either it is, or it is not an element of itself:

¬ ( R ∈ R ⋎ R ∉ R )

You can diagnose the issue by claiming the problem here caused by self-references in universal quantification, and Quine has laid out a logic which carefully controls self-references and evades this example through weak orderings of potential references.

But maybe those are only the cases we see. We did not see the original problem for centuries, so why would we assume that diagnosis is correct?

---

On the other hand, from another point of view, your example is not really about the LEM itself. It is really about 'modal' logic, where the whole universe is not yet determined as an object of reasoning. The fact is that it might rain tomorrow, or it might not, which throws you into a modal frame, and not a Classical one.

'Might' is traditionally called a 'mood' and creates 'modality' in the logic. It creates a slippery space between those things which I might or might not do, and those things I will or will not actually do. There is a general structure for modal reasoning, which keeps the other expectations of ordinary logic, including the LEM, while expanding the syntax to cover the undetermined cases.

In a modal sense, we can know X ⋎ ¬X. But, where the diamond stands for 'might (if something)', these three:

♢(X ⋎ ¬X)

♢X ⋎ ♢¬X

♢X ⋎ ¬♢X

all mean slightly different things and are not equivalent.

---

Formal logics (like Constructive Mathematics or Intuitionism) that admit exceptions to the LEM generally create the equivalent of a mood meaning 'can be safely assumed, but is not certain'. If E is certain (necessary, proven) ¬¬E can be assumed not to result in a contradiction, but it is not certain, since it has no independent verification. After all, we cannot necessarily derive it from E: the proper expression of E might contain something deep inside like "R ∈ R" that will blow up when we combine it with the right other propositions and then try to negate it.

Keeping these notions separate, we should not build upon facts that are just 'safely assumed', but upon those that are proven. Meanwhile, we assume the assumed things are safe, and guide our investigations accordingly, with the ultimate goal of either proving them or finding approaches that do not depend upon them.

---

In the same way the law of contradiction is not iron-clad. Legal and moral systems with a lot of formality often have the complementary problem: you are put under mutually contradictory constraints. Local laws may contradict national ones (e.g. marijuana), laws may contradict moral codes (e.g. Quakers, Good Order and Pacifism), etc.

So there is also a variant of modality for the mood of 'should' called paraconsistency. It allows apparent contradictions when all the modal dressing is removed, but protects the logic of negotiation by bracketing clauses and pulling in disambiguating requirements before an actual decision can be reached. It is much more complex and not as well understood.

Answer 5

Short version: no. If you give up the law of the excluded middle, you're not allowed to have nice things anymore (figuratively speaking of course). Classical logic assumes this principle and it all falls apart when you question it - and it's damn common to question it, especially when you first learn formal logic. While there are workarounds (see comments), which I've edited this post to be more accepting of, your question seems to deal more with how the material conditional works, not the law of the excluded middle.

Long version: what you're expressing is a common concern, but it's actually unrelated to the law of the excluded middle. DeMorgan's law says that: p->q ~ not p ^ q. That is, the two are logically equivalent! And coincidentally, it's saying the exact same thing you're saying:

Thus, if it doesn't rain tomorrow, the very truth-value and relevance of the consequent seems to be drastically diminished

If you know not p then by the above equivalence you don't care about q. However, you draw the wrong conclusion:

As such, the whole of the statement is no longer determined and seems to rather be cast into a 'neutral' or undetermined relation to truth.

Since not p ^ q ~ not p and you know not p then by p->q ~ (not p) ^ q you know that p->q is still true! This of course relies on some logical maneuvering that you may not have seen before this post, but be assured that you have not come up with a novel or convincing critique of the law of the excluded middle.