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Sometimes, you can use the Law of the Excluded Middle (LEM) to validly prove things by contradiction (e.g. irrationality of sqrt(2)).

However, at other times, you can not, for example when you have self-referential statements (e.g. Liar Paradox).

My question is: Is there an algorithm/rule that allows one to decide when LEM can be validly used in a proof?

Zen
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    In classical logic, LEM is always valid; it is not so for e.g. Intuitionism. – Mauro ALLEGRANZA Jul 21 '16 at 12:49
  • For Liar etc, see Self-Reference and Liar Paradox: – Mauro ALLEGRANZA Jul 21 '16 at 12:52
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    Some proposed solutions to the Liar are based on non-classical logics that reject LEM. – Mauro ALLEGRANZA Jul 21 '16 at 12:55
  • Yes, but how can we know that we're using things that will not result in inconsistencies (like self-referencing statements) in classical logic? – Zen Jul 21 '16 at 13:36
  • Self-referential statements can cause "troubles"; thus (see Henning's answer) they are "banished". But everyday math does not need them at all... – Mauro ALLEGRANZA Jul 21 '16 at 13:51
  • Where do you get the idea that you can't use LEM when reasoning about self-referential statements? – Rob Arthan Jul 21 '16 at 21:57
  • Don't get bent out of shape over "This sentence is false." It is false. If $P\implies \neg P$ is true, then $P$ is false. – Dan Christensen Jul 22 '16 at 03:49
  • Construct the truth table of $P\implies \neg P \implies \neg P$ to show that it is a tautology. – Dan Christensen Jul 22 '16 at 05:05
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    @DanChristensen: That's nonsense. If it is false as you claim, then since it correctly says that itself is false, it is true. – user21820 Jul 22 '16 at 07:18
  • Try translating the consequential $\forall P,Q: (P \implies Q) \lor (Q \implies P)$ into English. – Henry Jul 22 '16 at 08:35
  • Well, it's valid to use LEM in any system that includes LEM, trivially. But the question "when can LEM validly be used?" might mean "when can we safely used LEM without worrying about proving a contradiction?". One answer is that LEM can be used safely in any proof that can be formalized in a consistent formal system that includes LEM. Unfortunately, natural language proof systems are inconsistent, so even though it is valid to use LEM in natural language deductions, you can derive contradictions. – Carl Mummert Jul 23 '16 at 14:53
  • I have not studied logic formally, so I don't understand all these P ⟹ Q V symbols. However from what I can gather, it is valid to use LEM, but you can get contradictions because language is inconsistent. But then when we use LEM 'validly', how do we know we aren't exploiting a contradiction to get a fallacious answer? For example with the Liar paradox, say you want to prove "This statement is false" True. Let's assume it is false. Then it is true. This is a contradiction. Therefore it is not false, and by LEM, it is true. Of course this is wrong because you can use the same logic to prove... – Zen Jul 24 '16 at 05:53
  • ...it false. So the solution we say is we can't have self-referential statements. Fine. But how can we prove a statement is not self-referential? And even if we can, how can we prove a given statement is not some other type of statement (like self-referential) that also doesn't work under LEM (a new type that would should ban too)? Sounds like all we can do is assume it works until shown that it doesn't, which doesn't sound very mathematical... – Zen Jul 24 '16 at 05:56
  • @Zen: that is an excellent question! Many people have thought and written about it. There is too much to say in an answer here, but I will write something. – Carl Mummert Jul 24 '16 at 13:40
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    You seem to think that the liar paradox depends on LEM. It does not; it can be derived in intuitionistic logic, which rejects LEM. Details: Suppose $P$ is equivalent to $\neg P$. Then from a temporary hypothesis $P$ we can deduce first $\neg P$ and thence "false". This proves (intuitionistically) $\neg P$, discharging the hypothesis $P$. But then we can deduce from this result that $P$ and thus a contradiction. – Andreas Blass Jul 24 '16 at 15:28
  • @AndreasBlass: Intuitionistic logic is just one possible logic that does not satisfy LEM. However, if you extend to a suitable 3-valued logic (such as Kleene's) where LEM no longer holds then it is possible to have a fixed-point of the propositional function $P \mapsto \neg P$, without the paradox that classical logic entails. This truth-value gap is the heart of a symmetric denial of LEM (unlike the intuitionistic one), so arguably LEM is important for the liar paradox. – user21820 Jul 24 '16 at 17:20

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In mainstream mathematics the law of excluded middle is always considered a valid proof step.

There are still some contexts where one is interested in the existence of proofs that avoid this law (as well as various other things deemed "non-constructive") but there's a firm expectation that one has to declare explicitly that one expects proofs in such a restricted style.

The mainstream response to the liar paradox is not to restrict the law of excluded middle, but to refuse completely to make self-referential statements as part of proofs.

Strictly speaking, what is forbidden is not self-reference, but any claims of the form "such-and-such is true" or "such-and-such is false" where "such-and-such" is not a concrete statement but instead an explanation of how to construct the statement we're talking about. This excludes both "this statement" and varieties such as "such-and-such preceded by a quotation of itself".

In practice we do allow writing things such as "Now, if equation (4) above is true, then bla bla bla", but only because the reader can easily unfold it to something where everything is explicit, as in "Now, if $x^2=e^{3x}-\sin^2\theta$, then bla bla bla".

Formulas that include "$\ldots$" are strictly speaking only recipes for making a statement, rather than statements themselves, and at such they are at risk of paradox unless we have a concrete definition that shows how to unfold them to something that does not refer to the truth of constructed statements -- often using an auxiliary recursive definition.

Mainstream mathematics can investigate self-referential statements by treating them as objects of study in an appropriately defined formal system -- running them "in a sandbox", as it were. They can be spoken about but not uttered as claims in their own right at the meta-level.

hmakholm left over Monica
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  • Yes. However what exactly do you mean by mainstream mathematics? We know that self-referential statements cause problems from the liar paradox, and so we don't allow them. That's fine, but how are we supposed to tell what "mathematics" will or will not cause problems like self-referential statements? There are other things such as Fuzzy Logic where LEM doesn't hold. How do we know when LEM holds in our mathematics without assuming it does until proven false? – Zen Jul 21 '16 at 13:25
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    @Zen: "Mainstream" mathematics is the kind of reasoning that will be accepted by the mathematics community in papers and areas that do not take any explicit stance about foundations. – hmakholm left over Monica Jul 21 '16 at 13:29
  • Sorry I pressed enter accidently without finishing my comment and have now edited it. – Zen Jul 21 '16 at 13:30
  • But isn't that like assuming the maths works until someone finds something wrong with it? – Zen Jul 21 '16 at 13:31
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    @Zen: Well, yes. You have to start by believing something. In practice it seems to be a pretty good touchstone for "mainstream mathematics" that things that can be formalized in ZFC with classical logic are accepted as mainstream, and conversely that everything that people accept as mainstream tends to turn out to be formalizable in ZFC with classical logic once you do the work. If you want to doubt the validity of that foundation, you're allowed to -- but then, by definition what you're doing is foundational mathematics. Then you can choose whichever restricted logic and theory ... – hmakholm left over Monica Jul 21 '16 at 13:41
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    ... you want for your investigations -- but you're still expected to say explicitly what your choices are if they differ from the mainstream ones. – hmakholm left over Monica Jul 21 '16 at 13:42
  • @HenningMakholm: I'm not sure your answer is sufficient. Self-reference is not necessary for paradoxes in natural language; see my answer. A much deeper issue has to be addressed, whether or not via the way I proposed. – user21820 Jul 22 '16 at 08:28
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    @user21820: I interpreted the question as being a practical one about whether one needs to be especially cautious about using the law of excluded middle in ordinary mathematics. The OP's reaction implies that was not what he intended to ask, which is fine. – hmakholm left over Monica Jul 23 '16 at 13:36
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Your underlying question is philosophical, but that's fine; logic is just as much philosophy as it is mathematics. Let me state your question:

How do we know that LEM is always valid?

The only reasonable answer to this is that there is only one reality, and when we consider totally precise and unambiguous statements about this reality, it is necessarily the case that every such statement is either true or false about this reality, meaning that either it correctly describes reality, or it does not correctly describe reality. We don't care about statements that are ambiguous or not well-defined, just like we don't care about nonsense like ß\EÂ{8ÄäÉ5¨5;-c1÷ÌOm¶ÑzYè:ÏÁôví2QêIxú·9Ñ5u¤­åÉ¡nçßów⧸}tì-Ì«ÞB8r%sHÛæW¯*".vD because it is meaningless.

Based on this, we get classical logic. In classical logic one is only allowed to refer to objects that exist. Because of that, it is impossible to construct the liar sentence, because it is equivalent to:

??? Let $P$ be a sentence such that $P$ is equivalent to $\neg P$.

Which is clearly invalid because we have not proven that such a sentence exists. Indeed, we can show that no such sentence exists. Same for other paradoxes using self-reference (such as Curry's paradox) and paradoxes that assume the existence of objects without proof (such as the Barber paradox).

However, there is another deeper paradox in natural language. Consider the following sentence $S$:

" preceded by the quotation of itself is not a true sentence." preceded by the quotation of itself is not a true sentence.

Flawed argument: Notice that $S$ is a grammatically well-formed sentence of the form X is not a Y., so it ought to be true or false. If $S$ is a true sentence, then by what it claims, $S$ is not a true sentence. Therefore $S$ is not a true sentence. But then by what it claims, $S$ is a true sentence. Thus we get a contradiction.

Where is the error? Think for a while before continuing!

The key is that we will never be able to justify that $S$ is a statement about reality, and so we cannot apply LEM to it.

Sentences can be considered as strings of symbols in reality, so we can rightly say that $S$ is a sentence, and that "$S$ is a sentence" is a sentence, but once we want to say something about truth, it may not be a statement about reality anymore. Let us see what we can and cannot say regarding $S$. (Note that we always interpret "true sentence" to mean "true statement about reality" unless otherwise specified.)

First note that $S$ is the exactly the same string as " preceded by the quotation of itself is not a true sentence." preceded by the quotation of itself.

Thus $S$ is equivalent to asserting that $S$ is not a true sentence.   [(*)]

If $S$ is a true sentence:

  $S$.   [We can state $S$ since it is a true sentence.]

  $S$ is not a true sentence.   [By (*).]

  Contradiction.

If $S$ is not a true sentence:

  $S$.   [By (*).]

  $S$ is a true sentence.   [What we can validly state must be a true sentence.]

  Contradiction.

But we do not have LEM for "$S$ is a true sentence", since we did not prove that it is a statement about reality!

Thus all we can say is that "$S$ is a true sentence" is not a statement about reality, and likewise $S$ is not a statement about reality.

In general this approach can be used to resolve all paradoxes, including Quine's paradox (the one above) and Berry's paradox and the Surprise test paradox.

Note that some things like the circularity of modus ponens are not paradoxes but are intrinsically circular and cannot be justified non-circularly.

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user21820
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  • Downvoter: Please explain what specifically is wrong with my answer. – user21820 Jul 23 '16 at 17:17
  • I think I might be lost, but how can one prove that any statement S is a statement that exists/a statement about reality? Maybe you could give an example? – Zen Jul 24 '16 at 05:41
  • @Zen: That would have to depend on your other assumptions. If you have no other assumptions then you of course cannot prove anything that pertains to reality. But there are many possible assumptions that you might want to include, such as "Every arithmetical sentence is a statement about reality." (Why? Because PA seems to have a physical interpretation in physical media with machines or circuits to implement arithmetic.) But if you only include justifiable assumptions, you will find that you cannot justify that Quine's sentence is a statement about reality. – user21820 Jul 24 '16 at 07:14
  • @Zen: Basically, there is no way to justify arbitrary sentences as being about reality. You could go beyond arithmetical sentences, for example you can justify that sentences about arithmetical sentences are about reality, and so on, but you'll never get to all possible sentences, and in particular never to Quine's sentence. We can safely assume that our world is a model of some consistent system (by its mere existence), and therefore if you ever manage to prove that Quine's sentence is about reality it must mean that you made a false assumption somewhere! – user21820 Jul 24 '16 at 07:20
  • @Zen: In fact, a proper analysis of Quine's paradox will actually yield either Tarski's undefinability theorem or Godel's first incompleteness theorem, while a suitable version of Curry's paradox will yield Lob's theorem. – user21820 Jul 24 '16 at 07:24
  • The claim "Every statement of PA is a statement about reality" is really not much different then "Every formula $\phi(n)$ in the language of arithmetic determines a set", which in turn is basically all we need to know PA is consistent. Put another way: if PA is inconsistent, but we accept set induction for the natural numbers, then there's a formula $\phi$ that doesn't actually determine a set of numbers (some kind of paradoxical formula). In that case, we can't say that $\phi$ is actually a statement about reality. This also is why second-order PA is easier to prove consistent than PA. – Carl Mummert Jul 24 '16 at 14:12
  • @CarlMummert: Your first sentence is incorrect because it presumes the meta-system to be a set theory. Whether or not PA is consistent is quite independent of any set theory, and is a meaningful question in many possible formal systems! Also, I didn't even say we must accept that arithmetical sentences are about reality; I merely gave it as an example that one might include. And if you downvoted my answer, please state precisely which part is invalid. – user21820 Jul 24 '16 at 17:07
  • @Zen: Frankly I think my answer is the only one so far that can really address your underlying question (how are we supposed to tell what "mathematics" will or will not cause problems like self-referential statements?), because any possible justification that mathematics is consistent has to be completely outside of mathematics (because if mathematics is inconsistent it will prove anything anyway including the 'theorem' that it is consistent). So you necessarily need to delve into philosophy. – user21820 Jul 24 '16 at 17:34
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This answer relates to the following clarified question from the OP in comments:

But how can we prove a statement is not self-referential? And even if we can, how can we prove a given statement is not some other type of statement (like self-referential) that also doesn't work under LEM (a new type that would should ban too)? Sounds like all we can do is assume it works until shown that it doesn't, which doesn't sound very mathematical.

This question has a long history. It occurred to many other mathematicians, especially in the 19th century, when within 100 years they (1) learned of the independence of the parallel postulate in geometry, (2) learned of the ambiguity of the previous foundations of calculus and real analysis, and (3) learned of set-theoretic paradoxes such as Russell's paradox.

The quoted question really comes down to the consistency problem: how do we know that the systems we work with are consistent? Unfortunately, the generally accepted answer is that, to some extent, we do have to "assume" consistency. That may sound unfortunate at first, but it does not turn out to prevent us from doing math. I will explain a little more.

The natural idea was that, as a first step, we should make sure that the systems we work in are fully specified, unlike natural language. This led to the development of formal logical systems such as first-order logic, and formal foundational theories such as Peano Arithmetic (PA) and Zermelo-Fraenkel set theory (ZFC). These systems are unambiguously specified, so that the question of their consistency is a precise mathematical statement. (This is not the case with natural language, which is not fully specified.)

Now, we can ask: is it possible to prove that a system such as PA or ZFC is consistent? This is related to "Hilbert's program" pursued by David Hilbert and colleagues in the early 20th century. There was a lot of progress on that question, including the famous "incompleteness theorems" of Kurt Gödel.

There are many theorems related to the problem. For example, there are two independent proofs (by completely different methods) of the consistency of PA. But both of these proofs do require nontrivial axioms or deduction rules to carry out. If we wanted to prove the consistency of those axioms or deductive rules, we would need to assume some other axioms or deductive rules. In the end, you have to take some axioms for granted, or you have an infinite regress.

Mathematicians differ on how to interpret this situation. Some mathematicians view the consistency problem as solved, in that they either view PA and ZFC and being sufficiently "proved" consistent, or they view it as impossible to prove and thus not a mathematical question. Others think that the consistency of PA and ZFC is still an open problem which might be resolved in the future.

This situation is not as bad as it may seem, however. Here is a common viewpoint among mathematical logicians today.

First, we have been working with PA and ZFC for a long time, and despite many efforts, nobody has found a contradiction in them. So, heuristically, they seem to have stood up to the test of time. We don't have any proofs of their consistency with "no" assumptions, but we do have several proofs of their consistency "with" assumptions, and these proofs at least help us see what else would have to go wrong in order for these systems to be inconsistent.

Second, if we did find a contradiction in a well known foundational system, it would be extremely interesting, because it would show us something we didn't know before. The most likely outcome is that we would modify the formal systems to avoid the new kind of paradox, and we would publish a lot of logic papers analyzing that paradox.

Most logicians believe that discovering an inconsistency in our foundational systems would not "break" mathematics, because most ordinary mathematics can be done in many different formal systems, so we would still be able keep all or almost all of the results of ordinary mathematics, although we might formalize them differently.

Going back to LEM in particular, there is another result of Gödel relating to Peano arithmetic and its cousin without excluded middle, Heyting arithmetic. Gödel showed that there is a completely effective procedure (an algorithm) that would take any formal proof of an inconsistency in Peano arithmetic and transform it into a formal proof an inconsistency in Heyting arithmetic. So, in the case of Peano arithmetic, if there is somehow an inconsistency, it isn't from LEM. This is just one of many, many theorems about the consistency problem.

Carl Mummert
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