# Is this a resolution of the liar paradox? **URL:** \ **Category:** Factual Questions\ **Created:** [April 28, 2003, 11:10pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805 "2003-04-28T23:10:38Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![robertliguori](https://avatars.discourse-cdn.com/v4/letter/r/46a35a/32.png) [@robertliguori](https://boards.straightdope.com/u/robertliguori)\ **Post date:** [April 28, 2003, 11:10pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/1 "2003-04-28T23:10:38Z")
Possibly this is GD, although the fact that I’m looking for a yes/no is making me put this in GQ. Anyway, I am attempting to resolve the liar paradox. If this has already been done, than apologies for wasting time and hampster points. The liar paradox goes that the statement “This statement is false” cannot be true, because it states that it is false, and cannot be false, because if it were false, than it would be true, and it cannot be true. However, not every statement is either true or false. The statement “Ooga”, for instance, is also neither true nor false. If the statement “This statement is false” is neither true nor false, than it tells us nothing about anything, and can be therefore treated like “Ooga”. (I.e., ignored.) So I ask, am I right, and am I original? ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 28, 2003, 11:32pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/2 "2003-04-28T23:32:32Z")
The distinction between sentences and propositions is certainly not original. You will find a discussion in the early chapters of any textbook of logic. The basic ideas of the discussion are (i) Not all sentences express propositions. To take more orthodox examples than yours, neither “Hello!” nor “What time is it?” do so. (ii) Different sentences can express the same proposition. Thus the sentences “Tolstoy is dead.” “The author of _War and Peace_ is dead.” “Tolstoy est mort.” all make the same claim. Having made this distinction, we revise our definition and say that every proposition ( but not every sentence) is either true or false. The problem with this as a resolution of the Liar Paradox is that the sentence “This sentence is false.” clearly seems to be expressing a proposition. The usual approach is to establish a hierarchy of propositions. Firstly, we conceive of a realm of objects. Propositions about these objects we call ‘first-order propositions’. Propositions about first-order propositions are called second-order propositions, and so on. We place restrictions on the way propositions can be formed by ( for example) imagining the formation of propositions as a tempopral process. At each stage, we can only form propositions about objects and propositions created at an earlier stage. The sentence “This sentence is false” clearly breaks this rule, as it would be a proposition about itself. Thus the sentence is not a proposition and so the question of its truth or falsity does not arise. ---
**Author:** ![Captain\_Amazing](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@Captain\_Amazing](https://boards.straightdope.com/u/Captain_Amazing)\ **Post date:** [April 29, 2003, 12:07am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/3 "2003-04-29T00:07:03Z")
> [@](#): > > \*Originally posted by Jabba \* > \*\* At each stage, we can only form propositions about objects and propositions created at an earlier stage. The sentence “This sentence is false” clearly breaks this rule, as it would be a proposition about itself. Thus the sentence is not a proposition and so the question of its truth or falsity does not arise. \*\* Well, what about the sentence: “This sentence has five words.” That’s a sentence that’s a proposition about itself, and it’s true, right? ---
**Author:** ![Blake](https://avatars.discourse-cdn.com/v4/letter/b/a9a28c/32.png) [@Blake](https://boards.straightdope.com/u/Blake)\ **Post date:** [April 29, 2003, 12:43am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/4 "2003-04-29T00:43:16Z")
Two truisms: If any analysis provides insufficient information to answer the question asked, it is necessary to conduct further investigations in order to obtain the answer. Any investigation has the potential to interfere with its own analysis. For example I could investigate whether eating broken glass cured the sore throat caused by colds. I could do this by feeding people broken glass. The trouble is that the act of swallowing glass will almost certainly mask any changes in the cold symptoms. These sorts of situations are fairly common in the real world. To overcome such problems it’s necessary to eliminate the ‘interference’. Perhaps we could grind the glass up finely, perhaps we could place the glass directly into the stomach via an incision. The liars paradox is no different and no more paradoxical. The means of investigation we have adopted interferes with the analysis we are attempting. We need to find other ways of investigating the truth of the statement. The simplest solution would simply be to interrogate the speaker and ask “was the last statement you made correct?”. There is no paradox here that I can see, it’s just that the investigator is restricted to using invalid or insufficient investigative methods. It only remains paradoxical if all other means of investigation are prohibited. That being the case it’s not really any more paradoxical than asking someone to open a box with the key locked inside. That’s not a paradox, it’s just a really dumb place to put the key. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [April 29, 2003, 2:10am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/5 "2003-04-29T02:10:36Z")
I’m not a logician, but it seems wrong to me to assume that all propositions are either true or false. An analogy: If somebody asks what is the value of a real number x such that x^2 + 1 = 0, the answer is that there is no such real number. Similarly, if X is a boolean variable, and we are asked to find a value for X such that X \<–\> ~X, the answer is that there is no such value. Fundamentally, how is this different from evaluating the liar’s paradox? ---
**Author:** ![netscape\_6](https://avatars.discourse-cdn.com/v4/letter/n/8c91f0/32.png) [@netscape\_6](https://boards.straightdope.com/u/netscape_6)\ **Post date:** [April 29, 2003, 2:27am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/6 "2003-04-29T02:27:48Z")
> [@](#): > > \*Originally posted by Blake \* > \*\*not really any more paradoxical than asking someone to open a box with the key locked inside. That’s not a paradox, it’s just a really dumb place to put the key. \*\* might not be a paradox but it is a good dose of irony. ---
**Author:** ![Shalmanese](https://avatars.discourse-cdn.com/v4/letter/s/45deac/32.png) [@Shalmanese](https://boards.straightdope.com/u/Shalmanese)\ **Post date:** [April 29, 2003, 4:22am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/7 "2003-04-29T04:22:20Z")
Fuzzy logic states that the propostion “This statement is false” has a truth value of 0.5. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [April 29, 2003, 4:38am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/8 "2003-04-29T04:38:52Z")
**Jabba** , your distinction between first-order and second-order propositions, and the restrictions you place upon them, seems equivalent to the Zermelo-Fraenkel set theory proposed after Russel’s Paradox: It is constrained specifically to prevent sets from encompassing themselves, or to at least prevent paradoxes arising from sets encompassing themselves. Is that the case? (Russel’s Paradox was proposed by Bertrand Russel, and it runs like this: A set may be normal or self-encompassing. Normal sets do not list themselves as members, whereas self-encompassing sets do. Is the set of all normal sets itself normal? (If yes, it must be self-encompassing, but then it cannot contain itself, so it cannot be. If no, it must be normal, in which case see previous sentence.)) ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 29, 2003, 11:00am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/9 "2003-04-29T11:00:51Z")
**Captain Amazing** asked > [@](#): > > Well, what about the sentence: “This sentence has five words.” That’s a sentence that’s a proposition about itself, and it’s true, right? Discussing the truth or falsity of your sentence does not seem to lead us into any major problems, as it appears to be obviously true. I would mention two points, though. (i) We do not want the truth of our propositions to depend on the precise wording we use ( unless the change of wording results in a different propostion, of course). For example, the sentences Tolstoy is dead and Tolstoy is definitely dead express the same proposition. On the other hand, there is an obvious problem in saying that the sentences This sentence has five words and This sentence definitely has five words express the same proposition, and so have the same truth value. (ii) The aim of the step-by-step approach outlined above is to rule out propositions which lead to contradictory conclusions. Almost all the propositions we consider in maths, science, history and philosophy ( for example) are first- or second-order. Declaring improper a few sentences which appear to express true propositions is a small price to pay for ensuring that the resulting theory is consistent. **Derleth** wondered: > [@](#): > > **Jabba** , your distinction between first-order and second-order propositions, and the restrictions you place upon them, seems equivalent to the Zermelo-Fraenkel set theory proposed after Russel’s Paradox: It is constrained specifically to prevent sets from encompassing themselves, or to at least prevent paradoxes arising from sets encompassing themselves. Is that the case? It is a parallel process, yes. For example, Michael D. Potter, in his _Sets: An Introduction_ ( a book he says he was narrowly dissuaded from calling _The Joy of Sets_) constructs sets in a step-by step process: “The idea is that on the first day we start with the objects that already exist, i.e. the individuals; on each subsequent day we can collect together into a new collection any collections which were created on earlier days.” ( Potter, p. 16) This is obviously strongly analogous to the procedure I described in my first post. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 29, 2003, 11:05am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/10 "2003-04-29T11:05:14Z")
Sorry, forgot this one: **lucwarm** said > [@](#): > > I’m not a logician, but it seems wrong to me to assume that all propositions are either true or false. > > An analogy: If somebody asks what is the value of a real number x such that x^2 + 1 = 0, the answer is that there is no such real number. True enough, but What real number x satisfies x[sup]2[/sup] + 1 = 0? is a sentence but not a proposition. There is a real number x such that x[sup]2[/sup] + 1 = 0 is a proposition, and is false. ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [April 29, 2003, 11:40am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/11 "2003-04-29T11:40:29Z")
> [@](#): > > \*Originally posted by Derleth \* > **Jabba** , your distinction between first-order and second-order propositions, and the restrictions you place upon them, seems equivalent to the Zermelo-Fraenkel set theory proposed after Russel’s Paradox: It is constrained specifically to prevent sets from encompassing themselves, or to at least prevent paradoxes arising from sets encompassing themselves. Is that the case? That is one of the things Russell and Whitehead set out to do in their _Principia Mathematica_. ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [April 29, 2003, 11:41am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/12 "2003-04-29T11:41:45Z")
> [@](#): > > \*Originally posted by lucwarm \* > I’m not a logician, but it seems wrong to > me to assume that all propositions are either true or false. Yes they are, at least in classical or Boolean logic. ---
**Author:** ![robertliguori](https://avatars.discourse-cdn.com/v4/letter/r/46a35a/32.png) [@robertliguori](https://boards.straightdope.com/u/robertliguori)\ **Post date:** [April 29, 2003, 11:47am UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/13 "2003-04-29T11:47:31Z")
Urban Ranger: if “this statment is false” is neither true nor false, as I demonstrated, would it not follow that it is not a proposition? ---
**Author:** ![SCSimmons](https://avatars.discourse-cdn.com/v4/letter/s/e495f1/32.png) [@SCSimmons](https://boards.straightdope.com/u/SCSimmons)\ **Post date:** [April 29, 2003, 1:17pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/14 "2003-04-29T13:17:54Z")
> [@](#): > > \*Originally posted by Captain Amazing \* > \*\*Well, what about the sentence: “This sentence has five words.” That’s a sentence that’s a proposition about itself, and it’s true, right? \*\* No, and yes. The proposition expressed by that sentence would be ‘correctly’ expressed as “The sentence ‘This sentence has five words,’ has five words.” Using this ‘quotation’ method, one can arbitrarily define the truth-value of indeterminate propositions. Eg. “The sentence ‘This sentence is false,’ is false,” can be defined as expressing a true proposition. There’s no paradoxical self-reference there, because no sentence can _actually_ refer to itself. W.V.O. Quine killed that one, of course. True or false: “Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation. 😕 ---
**Author:** ![Dogface](https://avatars.discourse-cdn.com/v4/letter/d/9d8465/32.png) [@Dogface](https://boards.straightdope.com/u/Dogface)\ **Post date:** [April 29, 2003, 2:23pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/15 "2003-04-29T14:23:52Z")
> [@](#): > > \*Originally posted by robertliguori \* > \*\*The liar paradox goes that the statement “This statement is false” cannot be true, because it states that it is false, and cannot be false, because if it were false, than it would be true, and it cannot be true. However, not every statement is either true or false. The statement “Ooga”, for instance, is also neither true nor false. If the statement “This statement is false” is neither true nor false, than it tells us nothing about anything, and can be therefore treated like “Ooga”. (I.e., ignored.) > \*\* “Ooga” is not self-contradictory in the same way “This statement is false.” is self-contradictory. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 29, 2003, 2:29pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/16 "2003-04-29T14:29:50Z")
The OP has the same solution that I do. **Jabba** ’s objection is preceded by “The problem with this as a resolution of the Liar Paradox is that the sentence “This sentence is false.” clearly seems to be expressing a proposition.” My response is to note that many things seem to be what they aren’t, and IMO, this is one of them. By the way, if “This sentence has five words.” is OK, what about “This sentence does not have seven words.”? Which of it or its negation is true? ---
**Author:** ![robertliguori](https://avatars.discourse-cdn.com/v4/letter/r/46a35a/32.png) [@robertliguori](https://boards.straightdope.com/u/robertliguori)\ **Post date:** [April 29, 2003, 7:22pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/17 "2003-04-29T19:22:09Z")
Dogface: I am positing that rather than being both true and false, “This statement is false” is neither. Is there a way to show it true or false that does not depend on showing it non-false or non-true? ---
**Author:** ![Dogface](https://avatars.discourse-cdn.com/v4/letter/d/9d8465/32.png) [@Dogface](https://boards.straightdope.com/u/Dogface)\ **Post date:** [April 29, 2003, 8:29pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/18 "2003-04-29T20:29:05Z")
Truth or falsehood are irrelevant to evaluating “Ooga”. “This statement is false.” is a paradox because it is self-contradictory. “Ooga” is not self-contradictory. And in any case “This statement is false.” is not actually a paradox. It is a semantically true self-referential statement. By being nearly perfectly self-contradictory, it has rendered itself meaningless, thus being semantically false. Things that are semantically true have some sort of meaning. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [April 29, 2003, 10:22pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/19 "2003-04-29T22:22:52Z")
> [@](#): > > \*Originally posted by Jabba \* > \*\* > What real number x satisfies x[sup]2[/sup] + 1 = 0? > is a sentence but not a proposition. > There is a real number x such that x[sup]2[/sup] + 1 = 0 > is a proposition, and is false. \*\* So what? I certainly won’t disagree that some propositions are true and some are false. ---
**Author:** ![Homer](https://avatars.discourse-cdn.com/v4/letter/h/df788c/32.png) [@Homer](https://boards.straightdope.com/u/Homer)\ **Post date:** [April 29, 2003, 11:12pm UTC](https://boards.straightdope.com/t/is-this-a-resolution-of-the-liar-paradox/171805/20 "2003-04-29T23:12:51Z")
Read “Godel, Escher, Bach: An Eternal Golden Braid” by Douglass Hofstadter, for more insight into “Yields falsehood when preceeded by it’s quotation” yields falsehood when preceeded by it’s quotation, aka a “Quined” sentence. Hofstadter talks about the origin of self-reference and recursion in mathematics, music, and art. Another book which covers the subject is “A New Kind of Science” by Stephen Wolfram. Tim