# What formal logical systems "resolve" the Liar Paradox?

## Short version of my question.

What formal logical systems can represent, and seem robust against, the Liar Paradox?

N.B. I would like to avoid reference to truth-values, except inasmuch as they provide semantics for the formal system. Specifically, within the logic, I would like to say "P is true" if and only if P is derivable from whichever premisses we decide to grant in the formal system; and I would like only to say "P is false" if and only if ¬ P is similarly derivable. For instance, I accept that this means that Peano Arithmetic has statements where P ∨ ¬ P is true, but where neither P nor ¬ P are true (nor false). What I am concerned with is derivability.

## Preamble

I would like to make clear precisely what I'm asking. Apologies for the length of the question: please suggest a way that I might make this question more concise.

This question is about formal logical systems, which I will take (somewhat fuzzily) to mean a formal system of manipulating symbols, which we suppose to have some sort of meaningful semantics in terms of "truth" and/or "falsehood", and where we have some plausible rules of inference involving &, ∨, ¬, etc. in their familiar roles as logical connectives. That is: fragments of this system at least are clearly intelligible as representing logical reasoning — there is a transformation which would allow us to obtain a string similar to P ∨ Q from either P or Q, either of which we could obtain from P & Q, and so forth.

For a typical paradox X of logic, we shall say that a logical system represents the paradox X if we judge that the formal system can capture the syntactical elements of the paradox; and we shall say that it suffers from the paradox X if the logical system is inconsistent (we may derive absolutely any well-formed formula) essentially as a result of the fact that it represents X. We will say that the formal system is (or seems) robust against X if we cannot demonstrate that it suffers from X.

## A simple formal model of self-reference

I want to consider formal logical systems in which we may represent the Liar Paradox, specifically in order to specify a way in which to treat the Liar Paradox as a feature (or a bug) of a formal logical system.

Consider a formal system in which propositions (i.e. strings of symbols) may be referred to by name. The names are labels which are allowed as propositions in well-formed formulae. The semantics of these labels being "names" arises from the fact that they are addressible in a straightforward way: the rules of inference allow for the name A of some proposition P to be substituted with the proposition P itself. (We might naively say that A ≡ P, though technically this would only be a tautology if we also allowed the substitution in reverse as well. We may consider systems in which this is allowed or not; I only assume that "expansion" is a valid transformation.)

We consider a proposition ¬ L (which is well-formed in this system) which we then give the label L. This is then a simple and formalized realization of the Liar Paradox.*

The classic question is what to make of L: is it true or false? In a formal system, the question is instead whether our formal system "suffers". What makes the Liar a "Paradox" is that classical logic does suffer from it. Consider a typical formalization of classical sentential logic. If we grant the Law of Excluded middle, we have

L ∨ ¬ L

from which (by various applications of arguing by dilemma and double negation elimination) we may infer

L & ¬ L

which represents the classical crisis of truth values for L — or more to the point, from which we can infer whatever we like via reductio ad absurdum. In this case, we may say (we must say) that L is both true and false; and furthermore that everything is both true and false.

* Obviously, this is a circular reference, but as something realizable by a formal system it is impeccable — it is left for us to struggle with providing semantics for logical systems in which such things are possible. If one is really concerned about it one can serialize the logical system to preclude circular references, but this is not what this exercise is about.

## Question

Using the above (or a similar) system of self-referential logic:

1. What systems of inference rules (i.e. what formal systems of logic) seem to be robust to the Liar Paradox?

2. What works consider such rules of inference?

I already know of some obvious candidates — my derivation above already hints heavily at two such systems, for example — but I hope for some help with specific references, to specific formal logical systems, with "semantics" being quite subordinate to syntax.

• @TomBoardman: paraconsistent logics are one obvious choice, as they are not "explosive". However, is there any particular article from the references in the Wikipedia section which you would recommend as a good presentation of a formal paraconsistent logic which highlights precisely what logical differences there are, to get a sense of the range of logical techniques which stop being valid as a result of its paraconsistency? Apr 4 '13 at 12:59
• If you give ¬ L the label L. Then you have that ¬ L = L which is a contradiction and not a paradox. May 23 '18 at 17:43
• @SigurdVojnov: The paradox is as most paradoxes are: that we have a situation that appears to produce a result which is not intelligible. In fact, many paradoxes are presented as though they are logical contradictions, though they mostly turn out to be logical contradictions only as a result of inferences from somewhat rigid premisses. TO deduce a logical contradiction is then usually considered an indication of a paradox: that some set of 'intuitive' premisses is in fact logically inconsistent. The Liar Paradox is such a contradiction, but perhaps a meta-logical one. May 23 '18 at 19:58
• I'd say it is resolvable in ordinary logic. But then I'm on of those who doesn't see it as a paradox but as a simple misuse of language,. . .
– user20253
Apr 10 '19 at 11:27

In Hartry Field's Saving Truth from Paradox (2009), he splits the resolution of the Liar paradox into two broadly distinct strategies. Either we can accept Classical logic, but need to restrict the class of propositions over which Truth can meaningfully operate, or we can weaken logical inference to block either the deduction of a contradiction from the Liar proposition or, accepting the existence of a contradiction, to deny the principle of Explosion - that whenever we have a contradiction, we can infer anything we like from it.

We might think that in order to rise to your challenge, a restriction strategy is of no use, since you want to be able to represent and robustly process the Liar proposition. If that's what we think, then a non-classical logic is essential.

## Non-Classical Ideas

Field's own preference is for De Morgan Logics featuring Explosion, but not Excluded Middle. The system he initially proposes for Classical Logic as an extension of Basic De Morgan (BDM) logic (also called First Degree Entailment) can be cashed out in a base natural deduction framework of conjunction and disjunction introductions, with specific clauses set aside for negations (Take `A`, `B`, `C` to be propositions and `Γ`, `Γ'` to be sequences of propositions). Importantly, Excluded middle and Explosion aren't merely logical principles, but primitive axioms that need to be added to the baseline BDM:

conjunctions `&-In` - `A, B Ⱶ A & B`

`&-Out` - `A & B Ⱶ A` , `A & B Ⱶ B`

`¬&-In` - `¬A Ⱶ ¬(A & B)` , `¬B Ⱶ ¬(A & B)`

`¬&-Out` - if `Γ, ¬A Ⱶ C` and `Γ, ¬B Ⱶ C`, then `Γ, ¬(A & B) Ⱶ C`

disjunctions `v-In` - `A Ⱶ A v B` , `B Ⱶ A v B`

`v-Out` - if `Γ, A Ⱶ C` and `Γ, B Ⱶ C`, then `Γ, (A v B) Ⱶ C`

`¬v-In` - `¬A, ¬B Ⱶ ¬(A v B)`

`¬v-Out` - `¬(A v B) Ⱶ ¬A` , `¬(A v B) Ⱶ ¬B`

double negations `¬¬-In` - `A Ⱶ ¬¬A`

`¬¬-Out` - `¬¬A Ⱶ A`

explosion `Exp` - `A & ¬A Ⱶ B`

excluded middle `Lem` - `Ⱶ A v ¬A`

We also assume some structural rules concerning the turnstile:

Order Agnosticism if `Γ Ⱶ C` then `Γ' Ⱶ C` (where `Γ'` is any permutation of the propositions of `Γ`)

Precondition Weakening - if `Γ Ⱶ C`, then `Γ, A Ⱶ C`

Transitivity - if `Γ Ⱶ B` and `Γ, B Ⱶ C` then `Γ Ⱶ C`

The idea with this logic is that to actually go wrong with the liar paradox is to prove that from the liar proposition, absurdity follows. Let's see how the proof goes using the above rules from the sentence satisfying `L == ¬L`.

1. `L Ⱶ ¬L` (definition of L)
2. `L, ¬L Ⱶ L & ¬L` (&-In with A:L and B:¬L)
3. `L Ⱶ L & ¬L` (Transitivity using 1. and 2.)
4. `¬L Ⱶ L` (definition of L)
5. `¬L, L Ⱶ L & ¬L` (&-In with A:¬L and B:L)
6. `¬L Ⱶ L & ¬L` (Transitivity using 4. and 5.)
7. `L v ¬L Ⱶ L & ¬L` (v-Out using 3. and 6.)
8. `Ⱶ L v ¬L` (Lem)
9. `Ⱶ L & ¬L` (Transitivity, 7., 8.)
10. `L & ¬L Ⱶ 0=1` (Exp)
11. `Ⱶ 0=1` (Transitivity, 9., 10.)

Either blocking Lem or Exp is sufficient to prevent the inference to 11 from 7. Blocking Lem gives you a theory of value 1 sentences in Kleene Strong 3-valued Logic (wikipedia); blocking Exp gives you Graham Priest's LP logic, as presented in his In Contradiction (1987), which preserves both the value 1 and value 1/2 sentences under the same semantics.

Some useful observations about each move:

• Blocking Lem removes the system's only means of asserting Tautologies, so on this account, there are no purely logical truths. If we add rules for a conditional operator, making it a reflexive (`Ⱶ A → A`) material implication with Modus Ponens would reintroduce the excluded middle as a theorem, and hence we either need to either abstain from the conditional, block mpp/reflexivity or specifically invalidate "`A → B` iff `¬A v B`"
• If we block Exp, we also need to refuse to add Disjunctive Syllogism to replace it, since each proves the other in the same background deduction system. However, we can still maintain a conditional (not a material one, since we do need to block Disjunctive Syllogism) with Modus Ponens in this structure - it's just much weaker than anything classical, since Reductio Ad Absurdum has a much more limited scope of application without adding more rules to explain where absurdities come from.
• Both strategies can be augmented by adding rules to capture classicality in the satisfaction of particular background assumptions. For instance, Field introduces a conditional that explicitly behaves classically whenever the Excluded Middle disjunction features as an additional premise, and there is nothing in Priest's system that prevents the explosive inference being true of some propositions. Arguments for the legitimacy of doing this are going to be semantic, and you're not wanting to get into that right now.

## The Classical Line

The reason why non-classical approaches seem necessary given your inquiries is that you've phrased the Liar in a strictly propositional way. This isn't the norm in current classical discussions on the Liar paradox. After Godel and Tarski, classical theories of paradox avoidance have generally invoked a first order theory of Truth, with the sentences or propositions of a language featuring as objects that the theory can quantify over. That is, rather than the liar sentence being `L: ¬L`, it generally takes the form:

`L: ¬Tr(<L>)`

(where `<L>` is the syntactic representation of the proposition or sentence L, and `Tr` is the first order property over codes of propositions or sentences and ideally obtains whenever the proposition or sentence that is coded is true and fails to obtain whenever the proposition or sentence is false)

Where's the difference? Well, it makes the notion of restriction much more tenable. We're no longer saying that there is some sentence asserting its own negation (we assume that the propositional liar is simply ill-formed), but rather that there is some sentence that attributes the property of failing to be true to the code that represents that same sentence. This syphons the paradoxicality of the liar off to the realm of syntactic representation of language while still retaining the idea that sentences with self-referential features can be expressed in a more indirect way. It remains open to a classical theorist hoping to construct a theory of truth that they might only have such a sentence being properly definable in the presence of very high large cardinal principles in set theory, for example.

Except, of course, Tarski's Theorem means that we're never going to have a complete internal account of truth in a language this way. If we're going classical, our theory is going to have to accept a cut-off point at some point; or, as Tarski puts it, our metalanguage is always going to have to be "essentially richer" than our object language. This isn't such a big deal for most mathematicians, but seems like a serious problem for any hope of giving a formal recovery of a classical theory of truth.

Many classical logicians following a similar line to Donald Davidson's reading of Tarski have tried to couch truth theories in an Axiomatic way - we capture the structure of a theory of truth satisfying particular desirable properties and remain agnostic as to whether any such theories correspond to any definable single property, or whether they capture everything that there is to say about a particular language or system of propositions, sentences or assertions.

If you're in principle happy to accept the arguments Field and Priest present, but would like to see more developed versions of the replacements for the principles they reject, these theories might be interesting in describing hypothetical systems of truth that some Paracomplete or Paraconsistent theory might ultimately instantiate. Meanwhile, classical logicians are evaluating them in abstraction to consider their proof-theoretic strengths and extracting potential applications. There's a slightly dated SEP article on some of the work in this field which should be of some interest if you're keen on exploring further.

The current answer doesn't seem to mention that your proposed formal system is already flawed by definition. You say that "¬L" is a valid proposition, which you then label as "L". There is nothing wrong with arbitrarily stipulating such proposition formation rules for a formal system, but it simply doesn't correspond to anything meaningful. Why? Because a proposition is intended to represent a factual assertion, so the first part of your argument fails, because before you have labelled anything as "L", you need "L" to already refer to some factual assertion. But it doesn't, so it's meaningless to follow your argument. An analogy would be to say "Let n = n + 1. (where n is an integer)". It's meaningless, because you can't use "n" before you've defined it.

In short, you cannot refer to something you have not defined. This is a principle that we hold to in any argument, not just in mathematics but also in philosophy. So the kind of formal system you propose doesn't make sense.

However, there's an interesting variant which the other answer mentions, namely, is it possible to find some proposition P such that "P" is equivalent to "¬Prov(P)"? "Prov" here denotes the meta-theoretic function such that "Prov(P)" is an arithmetical formula in the formal system in question that is satisfied by N if and only if the formal system proves P. Indeed this is true for any formal system that extends first-order PA (Peano Arithmetic), and is called the fixed point theorem in logic. But "¬Prov(⊥)" cannot be proven by PA even though it is satisfied by N, which was the intended structure we wanted PA to axiomatize. In general Godel's incompleteness theorem shows that what we believe N is cannot be captured by any formal system with decidable proof validity.

So one can say that analyzing the liar paradox properly gives rise to Godel's incompleteness theorem. Likewise analyzing Curry's paradox properly gives rise to Lob's theorem.

Finally, note that natural language offers Quine's paradox that, unlike the liar paradox, is completely without any circularity (no reference to undefined terms)!

" preceded by the quotation of itself is a false sentence." preceded by the quotation of itself is a false sentence.

This construction is essentially a quine. The difference is that if we assume that every well-defined sentence of the form "X is Y" has a truth value then this seems to be a robust counter-example. You can clearly see that the first half is a well-defined string, and that it preceded by the quotation of itself is another well-defined string, which the sentence asserts is a false sentence. The problem arises when one attempts to assign a truth value to this assertion.

• This system is not necessarily a system of propositions, though I'm obviously encouraging a conception of it pointing in that direction. There are no consequences of the Liar being true or false, due to its tight circle of reference; but then many words and concepts are only defined with respect to networks of other words, forming a strongly connected component. Wittgenstein would say that their definition is then given only by their role in some practical game: but perhaps we can conceive of axiomatic non-trivialising approaches to self-reference. This is a toy problem on those lines. Jun 1 '16 at 0:44
• @NieldeBeaudrap: Take what you want. You'll find that most logicians agree with me. Philosophers are another story. Jun 1 '16 at 5:01
• That's why I'm asking here --- not just to ask philosophers generally, but to try to unpack the thought process of that minority of logicians/philosophers who think that dialetheism (for example) is worth entertaining. Jun 19 '16 at 17:53
• @NieldeBeaudrap: Most philosophers do not understand logic and hence make useless remarks, Wittgenstein included. Even many top mathematicians at the time of Godel had difficulty understanding him, but philosophers today still have that problem. You have not provide any reason why one should accept validity of self-reference. It is not about finding a non-trivial system with self-reference of the sort you want, just like we're not just interested in a consistent formal system, since PA+¬Con(PA) is consistent but to us meaningless! Jun 20 '16 at 3:40
• @NieldeBeaudrap: You're also probably conflating expressions and their meaning (if any). Anyone can write "reidwocrpichxizizdruuela" down on a piece of paper, but whether or not it has meaning would depend on interpretation. If you use the 'standard' English language, this is just meaningless, and does not refer to any object or concept whatsoever. Similarly, the liar sentence is a grammatical English sentence but any logical interpretation that attempts to assign it a classical truth value will fail, meaning that it does not refer to any classical truth value. Jun 20 '16 at 3:46

In addition to what has been said, I propose a different slant: from a logical or mathematical viewpoint, a "flip-flop" between two or more answers is a problem only if you decide that it is so. You could enumerate all possible answers and that set might be considered a correct answer (or you could consider some form of "mean" between the answers). But you obviously want one answer.

But if you introduce time, it is not a problem at all: there can be, at any time t, only one answer. If it flip-flops, it becomes an "oscillation" (and note that if it seems a "paradox" to us, it is because we need to enumerate successive different answers as we go along with our calculations, something that requires time to do).

Hence the problem becomes "robustly" solvable if you introduce time into your model, as well as cause and effect (and feedback rules). And indeed, if we assign a time value of 1 to each computation of a proposition in the Liar's paradox (traditionally, modelled) we see a flip-flop as the clock is ticking.

The simplest application of "time" would be, of course, a system of time precedence between your propositions, stopping at the first one that gives a relevant answer. With automatic rules engines working on the basis of "first fit", a later contradicting proposition would be ignored. But that would remove the challenge in your question.

Another one (slightly more complex in the beginning, but perhaps less complicated in the end), is to define more accurately the propositions A, B and C: in other words they would be functions of other properties which could vary over time. So we would have A(t), B(t) and C(t). Then we would have to define the rules of cause and effect between these properties (principle of feedback).

The result could be a single, unequivocal answer, or it could flip-flop between two or more answers (oscillation), or it could become chaotic (which is generally unwanted) or it would diverge to infinity (which when applied to physics is often synonymous of "breaking apart"), etc.

That requires, of course, a slightly expanded type of mathematics than your formal model (though its basics would be the same): as a matter of fact it is simply the mathematical models of digital circuits. But perhaps the results might be more intuitive in the end.

Earlier answers explain the mechanisms of how this works, but I would like to address the motivation. You can accept logical fictionalism, in the sense that 'if there were truth, it would behave as our intuition dictates and satisfy Classical logic, but there is not truth'.

We know from the ways we interact that logic contains large areas of relative consistency that allow us to work with facts, but we also have the religious intuition we are "always safe an divinely protected" to some degree and that "miracles occur". We routinely ignore Hume's basic argument and we know almost everything has exceptions. Classical logic accommodates the former, but cannot meet paradox reasonably. Why not admit the latter?

Then the issue is not avoiding contradiction, in either of the two ways we attempt this. It is keeping your dependency in areas that avoid obvious pitfalls, and accommodate paradox ad hoc as it arises, presuming that our experience of the stability of natural language means that will not happen very often.

The corresponding logical system can simply be Classical logic restricted to observable deductions. You don't need to go all the way to Intuitionist or Constructivist extremes, nor do you even need to artificially sequence self-references. But when there is a conflict, you need an axiom that resolves it.

The simplest way out, which is left out of @PaulRoss's broad answer, is Quine's approach to the notion of wellfoundedness: invoking 'weak stratification'. You take the opposite of Zorn's Lemma as an axiom, which forces you to implicitly order all self-references and you choose a rule for addressing them in sequence.

• A nice extension, thanks. It does seem to fall somewhere between "weaken the logic" and "restrict the class of admissable truth sentences", because the well-ordering requirement does seem to imply a kind of hierarchy underlying everything, but at the same time the fact that we're not operating with standard ZFC makes us wonder whether work is being done in the model theory of our logic. I'll have to read more of it! Jul 1 '18 at 8:51

According to the Theory of Types, truth and falsehood have orders, and the truth and falsehood in layman's speech are ambiguous. When we say a proposition is true, we omitted to say the order of the truth. If the orders of truth and falsehood are spelt out, "I am lying" is no paradox.

People think with the Theory of Types all the time; that is why Russell said he discovered the Theory of Types - implying that is how people actually think. For example, when I say "the melting point of ice is t," one would expect t is a temperature; if I substitute a length, say 50 feet, for t the resulting sentence will be a nonsense.

Take "x is a man" for example: "the Moon is a man" is false, but nevertheless significant; however, "Socrates is a man is a man" makes no sense.

A propositional function f(x) corresponds to a totality {x | f(x) or ~f(x) }, each of whose members makes f(x) significant (either true or false). A totality cannot has itself as a member because a totality cannot be determined until each of its member is determined; if one of the totality's member the totality itself, a vicious circle ensue. It follows that a propositional function cannot have itself as an argument.

According to the Theory of Types, all the x's that make "x is a man" true or false constitute a type. All the p's that make "p has first order truth" true or false constitute another type; p's are such first order statements as "Socrates is a man." Both "Socrates has first order truth" and "Socrates is a man is a man" are nonsenses.

All the Q's that makes "Q has second order truth" constitute the third order type. Q's are such statements as "'Socrates is a man' has first order truth."

So on so forth.

Predicates about individuals are called first order propositions; predicates about other propositions are one order above the proposition they take in as arguments. Thus, the statement "p is false" is one order above p. In the liar's paradox "p is false" where p = "p is false" is a self-referential sentence, i.e. "p is false" is taking itself as argument - this is nonsense because an argument to a propositional function must be one of lower order propositions or individuals. According to the Theory of Types, the Liar's Paradox makes the same mistake as "Socrates is a man is a man." Taken literally "I am lying" is nonsense.

However, according to the Theory of Types, truth and falsehood has orders. Strictly speaking, "'Socrates is a man' is true" is not exact, it should be written as "'Socrates is a man' has first order truth." It follows that the statement "'Socrates is man' has first order truth" has second order truth.

If the order of truth is spelt out, there will be no contradiction, but "I am lying" becomes a simultaneous assertion of multiple statements:

`````` I'm asserting a false statement of the 1st order. --- p2
-False, because no first order statement is being asserted,
a statement about a statement is at least 2nd order.

I'm asserting a false statement of the 2nd order.  --- p3
-True because p2 is asserted and is false.

I'm asserting a false statement of the 3rd order.  --- p4
-False because p3 is asserted and is true.
``````

So on so forth.

We can see that, when the orders of truth and falsehood are spelt out, "I'm making a false statement of 2n+1 order" is false, while "I'm making a false statement of 2n order" is true, but there is no paradox.

• // Take "x is a man" for example: "the Moon is a man" is false, but nevertheless significant; however, "Socrates is a man is a man" makes no sense. // But ""Socrates is a man" is a man" makes sense, its actually false. May 23 '18 at 17:37
• @SigurdVojnov - Thanks for the comment. Suppose it is false, that means there are properties all men possess but "Socrates is a man" doe not. Take "weighing less then 20 ton" for example. If "Socrates is a man" weighs less then 20 ton is false, than the opposite should be true; the opposite is not true either, thus it is nonsense. In other words, "Socrates is a man" does not belong to the domain of "x is a man" because some properties which one can attribute to a man, being it true or false, cannot be attributed to a proposition. May 24 '18 at 13:50
• But stratifying all propositions is like shooting mosquitos with an elephant gun: You loose interesting sentences such the one you now have finished reading. May 24 '18 at 14:15
• @SigurdVojnov - Your sentences are interesting because they hit where it hurts. Russell always welcomed challenges. It is challenges like this that kept pushing the frontier of knowledge into provinces hitherto abandoned to philosophy. May 24 '18 at 16:08

Classical logic works fine!

We assume there is a sentence L such that L = "L is false"

1) L = "L is false"

Now we apply Leibniz law: https://en.wikipedia.org/wiki/Identity_of_indiscernibles

2) L is true IFF "L is false" is true

Using the definition of truth we get:

3) L is true IFF L is false

Our assumption was false! There is no L such that L = "L is false"! (QED)

But THAT is what is assumed to construct the Liar Paradox!

Example:

1 L is false

2 "L is false"= L

3 "L is false" is false

Hurkyl asks what the Liar Paradox IS and what can be considered its resolution. Also he search for a definition of propositions, what the rules are for constructing them and how these rules avoid paradoxes. A tall order :)

In the intro above I used the style of propositional logic where propositions are represented by single capital letters its not my ordinary style I was assuming it to be the norm in here.

Looking closer we have Paul Ross saying: "That is, rather than the liar sentence being L: ¬L, it generally takes the form:

L: ¬Tr() "

But still with propositional logic at bottom. Theres more formal systems in use, perhaps next interesting post will come from someone using Boolean Algebra?

I suggested that Classical Logic is enough for dealing with The Liar Paradox but we must at least allow for better resolution on propositions and allow first order predicate logic. Or some other system for representing subject/predicate propositions.

So let me use a subject variable x and a predicate variable Z and (to begin with) claim that any subject x (representing some object) can be joined together with any predicate Z (representing some property) to form the proposition Zx.

I assume that the ordinary rules of propositional logic are valid for the propositions so if we want molecular formulas involving distinct propositions we either introduce more variables or we index the variables we have. A moot question at present because the liar paradoxes usually arent expressed in molecular forms.

Liar Paradoxes are usually derived from an atomic sentence in the style:This sentence is false, Then we may represent it as: Zx, where x is "This sentence" and Z is "is false.

We should look at a more ordinary proposition and see how they work:

1) this sentence contains exactly six words
It becomes 1) Zx

WHAT SENTENCE?

2) this sentence is "this sentence contains exactly six words"

2) x = "Zx"

And now we can substitute!

3) "this sentence contains exactly six words" contains exactly six words

3) Z"Zx"

We have seen an Effective Method for finding the truth of elementary self referent sentences! We can extend it to ordinary typifiable propositions but were not interested in that right now!

Formally we have:

1) Zx (ASSUMPTION)

2) x = "Zx" (DEFINITION)

3) Z"Zx" (CONCLUSION)

We note that the conclusion is a so called ANALYTIC PROPOSITION! Just by inspecting it we can find its truth! (Given its semantics!)

We can now derive an interesting formula:

4) IF (x = "Zx") THEN (Zx IFF Z"Zx")

IF the right side of the implication is false then so is the left side: Which tells us that x is not identical with Zx, and that Zx is not a self referent proposition!

The expression: Zx IFF Z"Zx", can therefore be used (possibly even by computers) as a test for paradoxes.

We can now define a self referent proposition to be any object of the form "Zx" provided Zx IFF Z"Zx".

A nice thing about it, is that it is a Syntactic Definition!

Putting x ="this sentence" and Z = "is false the test becomes:

This sentence is false IFF "This sentence is false" is false

Which reduces to:

This sentence is false IFF This sentence is true.

The contradiction tells us that its not the case that: this sentence = "this sentence is false" ...

And "this sentence is false" is not a self referent proposition. (QED)

Hopefully this will be of some help to you Hurkyl :)

• I'm not sure this counts as a resolution. Rearranged, the Liar's paradox is "There can't be such an L" together with "There is such an L". The Big Problem is that reasonable-seeming rules for what constitutes the definition of a proposition turn out to allow explicit construction of such an L. A resolution of the Liar's paradox would, in my opinion, have to indicate how such a system of logic avoids having such constructions.
– user6559
May 20 '18 at 21:30
• I expanded my text a little trying to meet your program. May 20 '18 at 23:20
• First to mention that there's nothing specifically propositional about the Godel/Tarski approach. The diagonalization that is used to construct the Liar sentence is usually a consequence of a first order predicative syntax theory. Jun 30 '18 at 9:31
• Secondly, I would make the pedantic but important point that your biconditional expression is not a well-formed logical sentence. You've assumed x and Z as parameters, so properly understood, you need to introduce quantifiers. Something like `for all sentence-argument predicates Z and for all sentences x, Zx IFF Z"Zx"` would be closer to logically acceptable. Jun 30 '18 at 9:43
• So to my main objection. Can you prove that "Zx" is a sentence? Jun 30 '18 at 9:46

"I would like to say "P is true" if and only if P is derivable from whichever premisses we decide to grant in the formal system; and I would like only to say "P is false" if and only if ¬ P is similarly derivable." Niel de Beaudrap (original post).

That is the exact same basis that Wittgenstein used in his rebuttal of Gödel's 1931 Incompleteness Theorem. If true and false must be derived (which means that they must be proved) then unprovable only means untrue and nothing more.

Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics (Appendix III), 118-119. Cambridge, Massachusetts and London, England: The MIT Press

1. I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable."

Just as we ask: " 'provable' in what system?", so we must also ask: " 'true' in what system?" 'True in Russell's system' means, as was said: proved in Russell's system; and 'false in Russell's system' means: the opposite has been proved in Russell's system.-Now what does your "suppose it is false" mean? In the Russell sense it means 'suppose the opposite is proved in Russell's system'; if that is your assumption, you will now presumably give up the interpretation that it is unprovable.

And by 'this interpretation' I understand the translation into this English sentence.-If you assume that the proposition is provable in Russell's system, that means it' is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows...

"I would like to say "P is true" if and only if P is derivable from whichever premisses we decide to grant in the formal system; and I would like only to say "P is false" if and only if ¬ P is similarly derivable." Niel de Beaudrap (original post).

Here is an example of such a system:

When we define Analytical Knowledge as expressions of language verified as totally true entirely on the basis of their semantic meaning

and this semantic meaning is encoded as stipulated relations between the finite strings of this language (such as words being defined by other words)

then the process of verifying that a finite string has all of these stipulated relations is the membership algorithm of a recursive language.

This process generalizes the notion of a formal system such that a recursive language essentially is a formal system and its membership algorithm takes the place of axioms and rules-of-inference.

Membership indicates a finite string is a theorem of this formal system. When the formal system represents a body of knowledge then membership indicates that a finite string is true because all knowledge is true.

∀x ((x ∈ Analytical_Knowledge) ≡
(Analytical_Knowledge ⊢ x)) ≡
True(Analytical_Knowledge, x))

A language L on Σ is said to be recursive if there exists a Turing machine M that accepts L and halts on every w in Σ+. In other words, a language is recursive if and only if there exists a membership algorithm for it. (Linz 1990:288).

Linz, Peter 1990. An Introduction to Formal Languages and Automata. Lexington/Toronto: D. C. Heath and Company, 288.

• I like the general thought, but that suggests a contentment with a non-classical logic framework that most mathematicians will non accept. The theory that all proofs must be computable is very controversial. Apr 25 '20 at 12:47
• @SofieSelnes This is my life's work. There is more to come. The key aspect of this would be the halting problem. Apr 25 '20 at 15:25

I dont want to derail the conversation; perhaps my contribution here is ridiculous. Im coming at this from a much more naive direction, and instead of formal logic Im primarily looking at informal rhetoric, and the nature of language itself. Your discussion on formal logics is somewhat beyond me.

The only observation I can make, which you may or may not find insightful, is that not every statement requires a truth value. Does it? Maybe it does, but Im inclined to think not. Here is a short list of sentences that I think dont require a truth value:

1. Add 1 to 1. (a command)
2. What is the purpose of life? (a question, not an answer)
3. This statement is false. (a self-referential statement)

The first two might seem obvious, if you agree with my premise that not all statements require a truth value. You may dispute that they are statements at all. But why should the third have a truth value? What obligation is there? Because it has a period at the end? If you accept that not all statements carry logical value then surely we can find examples such as the liars paradox that might widen the scope of such logically meaningless statements.

Its self-referential, by its very nature it carries no meaning or impact on the world, and describes nothing but itself. Nothing substantive is communicated; there is no transference of useful or meaningful information to the external world.

I submit to you that such statements do not require a truth evaluation because they are meaningless. Just as other types of sentence structures - questions and commands, for example - which we all recognize require no truth value. I would argue that anything which references itself is immediately disregarded from any rational conversation, just as we refuse to carry logic on past a contradiction (because no insights can come from false reasoning), so too we can refuse to consider self-referential statements.

Self-reference is a lot like bootstrapping, or putting the cart before the horse. Or bureaucracy for that matter. If it exists solely to legitimize itself then there is no point to it and it cannot give itself meaning. Thats life in general, really; meaning is derived from interaction with the world beyond. Legitimizing a self-referential statement is as ridiculous in my opinion as trying to legitimize the existence of the universe itself. There is no meaning that transfers outside of itself. You can strike the paradox from existence and it would have no effect, nothing is lost or gained in the world if you did.

Its pretty clear that the "paradox" is no paradox at all. Its a statement with an inbuilt contradiction. Does that make it a paradox? Maybe, sure. But the point is, whenever you come across a statement P AND NOT P, youre supposed to stop and flush. Resolving the paradox is trying to fix a contradiction without rejecting the contradiction, and its logically absurd in my view.

This isnt an exercise in proof by contradiction where you make an assumption, find a contradiction, and ultimately reject the assumption. No, you dont have that leeway. You are given the contradiction up front as a premise. No argument stemming from it can be sound or valid.

Or put another way, the paradox is an artifact of semantics, a misuse or failure of language... not a logical puzzle.

Perhaps someone can prove me wrong, that there is something of value to be learned in analyzing a statement such as the liars paradox. Im not making a utilitarian argument, Im making a pragmatic one. Language is supposed to communicate a meaning and a sentence that fails to do so is not really worth considering (an argument for good grammar as well).

• Saying that the liar sentence is meaningless has consequences, though! To put it in common-sense terms, “What part of ‘This sentence is false’ don’t you understand?” Jul 16 '20 at 6:42
• The sentence on whole. IF youre being rhetorical. The sentence on whole carries no meaning. Its not about the words or the grammar... I didnt say the sentence wasnt comprehensible in its intent at least. The issue is that it carries no actual meaning; it has no impact on the world, it cannot be utilized nor is fruitbearing, it cannot be implemented and isnt actionable in any way. My point is that its no different than having never heard the sentence at all; nothing is added or taken from the world by piecing those words together. Jul 16 '20 at 14:31
• Im pretty sure I made this clear in my argument. Please do read it before responding again. You totally ignored half my argument at least just to put in your two cents. At least it seems that way. You havent actually confronted my points. Jul 16 '20 at 14:31
• Its less about the contradiction and more about the self-reference. This sentence ends in a period. Just as meaningless. If I removed the sentence it wouldnt impact the story one bit; the utility is in the information provided, conveyed by the sentence about the external world. Its inclusion contributes nothing. "The sky is blue." At least that communicates something. The paradox fails to do what language is meant to do. Jul 16 '20 at 14:40
• My pragmatic stance is precisely that not all declarative statements have a truth value, just as for instance not all infinite sums converge. However, just as an innovation was required to define the value of any infinite sums (and to distinguish between absolute and conditional convergence) --- thereby providing a useful response to Zeno's paradox --- I wondered what variations of formal logic might be fruitful head-on approaches to the Liar Paradox. Jul 16 '20 at 16:30