The Liar's Paradox

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    I have only listened to 2 and a half lectures so far, but was just curious if this could be addressed (or is addressed in one of the lectures).

    I had at least one college professor who denied the Law of Excluded Middle, i.e. that every proposition is either true or false. One of the examples he would use to show the law is faulty is that which is typically called “the liar’s paradox.”

    The proposition is “this sentence is false.” If you say it is true or false you are in a bind. I have a feeling this proposition would not qualify under the definition, “Something that is affirmed or denied of something else” but I am not sure how to make the connection.

    Any help would be appreciated.


    This sentence is false is a meaningless string of words; it’s neither true nor false, since it doesn’t express any claim about reality. You can’t understand what it means without already understanding what it means.

    Jason Jewell

    In other words, what David is saying is that “This sentence is false” does not qualify as a proposition. It’s categorically the same as Lewis Carroll’s nonsense, “‘Twas brillig, and the slithy toves did gyre and gimble in the wave.” Thus it does not contradict the Law of the Excluded Middle.


    Yes that is the typical reply. But why does it fail Dr. Casey’s definition for proposition?

    If a proposition is “Something that is affirmed or denied of something else,” then why is “This sentence is false” not something that is affirmed or denied of something else?

    Couldn’t someone argue that the “something” is “this sentence” and the “something else” is “false.”

    I agree that it is meaningless, but I would love to be able to explain why it is NOT a proposition according to how we define propositions.


    One of the charms (is that the word I want?) of logic is that it leads to topics such as the one under discussion here. This topic, however, properly belongs either to metalogic or the philosophy of logic (it’s not always easy to distinguish one of these from the other). Metalogic, as the name suggests, takes formal logical systems and makes them the object of formal study. Philosophy of Logic (as in Susan Haack’s book, Philosophy of Logics – note the plural) also takes the various logics as objects of study but the questions it raises are of a philosophical rather than a formal nature. And then, arising from the perceived limitations of modern logic (such as the Paradoxes of Material Implication) we have the development of Modal Logic which then ramifies into a whole system of so-called deviant logics – epistemic logic, deontic logic, 3-valued logic, multi-valued logic and the seemingly paradoxical fuzzy logic!

    To return to the topic under discussion: a number of approaches have been taken to the so-called Liar Paradox. One approach is to outlaw self-referentiality so that propositions are not allowed to refer to themselves. This (probably) works but it has a disturbingly ad hoc air about it which may be intellectually unsatisfying. Another approach, to which I am drawn, is to hold that all propositions make an implicit claim to truth so that the Liar sentence simultaneously asserts and denies the same thing and so is a contradiction and so is false. But there are other attempts at a solution and no general agreement is to be found on how this is to be handled. These attempts at a solution and some others are discussed in this Wikipedia article

    When I was a student I found these kinds of questions practically irresistible, much to the detriment of my other studies and my social life. While I would encourage people to think hard about such topics, it probably isn’t a good idea to do so at the same time that one is attempting to learn the nuts and bolts of a primary formal system. Furthermore, to take part in current discussions you really need to familiarise yourself with modern Proposition and Predicate Logic, some Modal Logic, some metalogic and some philosophy of logic. If you want a short DIY introduction to elementary modern logic to get you started, send me an email ( and I’ll get my material to you.

    Finally, to see how logic can allow one to focus on aspects of philosophical discussions that might otherwise remain out of sight, consider the following snappy 6-line proof for the existence of God (I think I stole this from Charles Hartshorne but I’m not 100% certain):

    Let G stand for the proposition ‘God exists’; let § stand for the phrase (modal operator) ‘it is necessary that….;’ and let * stand for the phrase (modal operator) ‘it is possible that…’

    Line 1: *G (It is possible that God exists)
    Line 2: If G then §G (if God exists, then God necessarily exists)
    Line 3: If *G then *§G (If it is possible that God exists, then it is possible that God necessarily exists) We arrive at this by applying the * operator to each side of line 2
    Line 4: *§G (it is possible that God necessarily exists) We get this by putting lines 1 and 3 together, using the modal version of the rule called Modus Ponens
    Line 5: §G (God necessarily exists) We get this from line 4 by using a modal logic rule that allows us to dispense with all but the last modal operator in front of any given proposition.
    Line 6: G (God exists) This comes from the most basic rule in modal logic; whatever is necessarily true is true

    The premises of this argument are plausible; it doesn’t seem unreasonable to think that it is possible that God exists. Similarly, if God isn’t to be thought of as a rock or a tree, it seems reasonable to think that if He exists at all, He exists necessarily. Grant these two premises and the rules of modal logic (up to a certain level) and the conclusion (God exists) pops out!

    I like to tease my atheist friends with this bit of modal prestidigitation. The serious side of the argument is that it forces one to look again at whether the premises are, in fact, obviously true (perhaps it’s not even possible that God exists or perhaps God might exist but not necessarily!). On the more purely logical side, it leads to a consideration of the plausibility of the various systems of modal logic, in particular, the rule used in line 5 that allows you to reduce a proposition prefaced by a mixed string of modal operators to just the last operator. Here, as elsewhere on the borderlands of logic, our intuitions tend to run into the sand.

    There is much more that could by said on this and related topics but not by me tonight!


    I don’t think you have to hold that all self-referential sentences are meaningless in order to show that This sentence is false is meaningless. Sentences which refer to themselves qua linguistic entities (strings of words) seem to be fine; e.g., This sentence is in the present tense is probably meaningful (and true). But problems arise when a sentence refers to its own content. In order to understand what such a sentence even means, you need to already know what it means; therefore, it’s impossible to understand it. I don’t think this is “disturbingly ad hoc.”

    JohnD: To say that an English sentence is false is to say that its content is false. So This sentence is false really means The content of this sentence is false. But this sentence doesn’t even have an intelligible content, so there’s nothing of which anything is affirmed.


    David Konietzko: Thanks very much for your brief reply above. I agree that when someone says, “This sentence is false” it is in no way clear what the content of the sentence is, so there is nothing of which anything is affirmed. However, just because it is UNCLEAR that the sentence has an intelligible content, does not necessitate that it HAS NO INTELLIGIBLE content. So, it seems you are begging the question when you say “this sentence doesn’t even have an intelligible content.” Do you agree? I know this may seem like word games, but I am sincerely interested in your response to that.

    Dr. Casey: Thanks for a very well thought out and thorough response. I actually find your solution very satisfying (the one where you say the sentence implies contradiction either way and is therefore false). That has the ring of Russell’s paradox in Set Theory, and it certainly seems to be a viable argument.
    The thought of ALL THOSE DIFFERENT LOGICS is quite disturbing. I thought since logic is the science of necessary inference (proper thinking) that we could rely on it to yield certain truths. But with so many logics to choose from, where does that leave us???

    My attempt at Synthesis in Solving the Liar’s Paradox:
    (1) Let L = “This sentence is false”
    (2) It is not clear if L has intelligible content.
    (3) If L has no intelligible content, then it is not a proposition since it contains nothing that is affirmed or denied of something else.
    (4) If L has intelligible content, then employ Dr. Casey’s solution (i.e. If the sentence is false, then it is true [contradiction], and If it is true, then it is false [contradiction]) to show that the statement is meaningless. I.e. just stringing words together does not necessarily make a proposition, and in this case no proposition has been made. Just like in Set theory, merely defining a set with a property does not prove the set’s existence.
    (5) Since L is not a proposition, it does not fall within the realm of the law of excluded middle.


    Also, I credit both David and Dr. Casey for (3) and (4) in the solution above. Thanks!


    A sentence has a certain meaning precisely because people interpret it to have that meaning. Therefore, if it is in principle impossible to understand the meaning of a sentence, then it simply doesn’t have a meaning.


    David Konietzko: Apart from its perhaps being the case that outlawing self-referentiality is merely ad hoc, and thus unsatisfactory, it also has the effect, as you correctly note, of outlawing perfectly respectable sentences such as “This sentence is in the present tense”.

    JohnD: You write “The thought of ALL THOSE DIFFERENT LOGICS is quite disturbing. I thought since logic is the science of necessary inference (proper thinking) that we could rely on it to yield certain truths. But with so many logics to choose from, where does that leave us???”

    I can appreciate your concern. When you consider all the possibilities that I mentioned in my previous post, it can induce a kind of intellectual vertigo. Perhaps an analogy will help. The basic theory of modern physics seems to be in trouble. Just how many basic particles do we have? Is string theory the answer to our questions? However, whatever may be the case with theoretical physics, we have no difficult getting on with our daily lives, weighing things, measuring them, comparing them, and so on. The inferential core of logic is like our everyday practices of weighing and measuring; the fact that theoreticians may be at odds and special extreme cases may cause us headaches, doesn’t in any way affect, say, the invalidity of the fallacy of the undistributed middle or the validity of the conversion of E-type propositions. This analogy may (or may not!) help: all analogies limp. The disquiet expressed in your post was one reason why I was suggesting that while these topics which we are discussing are interesting and exciting, it can be confusing and distracting to focus on them at the same time as one is trying to come to grips with a basic formal logic system.

    David Konietzko: You write:”A sentence has a certain meaning precisely because people interpret it to have that meaning. Therefore, if it is in principle impossible to understand the meaning of a sentence, then it simply doesn’t have a meaning.”

    The syntactically-in-order sentence “The kronxite instrumentality befummels reflexiveness cromulently” doesn’t mean anything because some of its key words don’t mean anything in English. Failure to understand it is not a fault in anyone’s understanding; no one can possibly understand it. This is to agree with the second sentence in your post.

    The first sentence in your post is possibly ambiguous depending on whether ‘people’ means, assuming we’re using English, some particular special group of people who are competent in English or competent English speakers in general. A sentence could be well-formed in English and understandable to all competent English speakers; or it could be well-formed and understandable only to a proper subset of competent English speakers. In all cases, it would have meaning.


    Dr. Casey, thanks very much for your analogy. I do find it helpful. And for the record, I don’t think string theory is the answer.

    And I think you pointed out an important point above. Since if we were to judge the meaning of a sentence based on how “people” interpreted it, then it depends on which people we choose. Would it be better to define the meaning of a sentence as “the content that the sentence’s author intended to convey?” But then that gets all subjective too if you have an author that uses nonsense word order to convey meaning, like government code words and stuff. Oh boy this is confusing.

    It’s funny because I really do find all these specific cases and special circumstances fascinating, but I realize it would be more useful to learn the basics of the formal system first before getting into those things. Thanks again for the responses.

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