• TheMadFool
    3.9k
    Liar Paradox concerns the following statement:

    1) This sentence is false.

    Everybody knows the paradox well enough but just to refresh everyone's mind: If the sentence is true, then it's false. If the sentence is false, then it's true. We can't seem to ascertain the truth value of the statement. Hence the paradox.

    One way to solve the paradox is to consider it a non-statement. Non-statements don't have truth values and so we don't have to worry about finding the truth value of the statement.

    I have a different view on the matter though and I'l like you all to comment on it.

    First...

    Consider the statement below:

    2. This sentence is both true and false.

    Statement 2 is our classic contradiction. It can NEVER be true. It is ALWAYS false. It's claim, which is important for my ''solution'' to the Liar paradox, is that the sentence 2 is true and false in the same respect (includes time and space). I'll focus on time here. The statement ''I'm sleeping'' may be true at 12 midnight and the statement ''I'm not sleeping' may be true at 10 AM. However, both can't be true at the same time - that would be a contradiction. In short, contradiction requires that the opposing claims be made at the same time.

    Now, look at sentence 1. We know it's not true, because if it is, it's also false. We also know it's not false, because if it is, it's true.

    Since sentence 1 can't be false it is NOT a contradiction. Contradictions have to be false. Now, here's the tricky part. If sentence 1 is NOT a contradiction, it's not making the claim, truth and falsity, at the same time. In other words, there's a time gap between true and false states of sentence 1. If it is true at t0, it becomes false at t1, and then true at t2, and then false at t3, and so on...like a switch alternating between on and off states.

    So, sentence 1 is a statement AND has a truth value, only it alternates like a pendulum between true and false, as time passes. At any one instant it is either true or false but the next instant it switches states.

    Another way to look at it is in a causal context. Sentence 1 being true causes it to become false and vice versa, and we all know causation, most, takes a nonzero amount of time.
  • Banno
    5.7k
    It's the Liar.

    Time doesn't enter into it.

    The solution involves avoiding self reference in truth statements, either by using a meta language (Tarski), or by specifying before hand the sorts of sentences to which 'is true' can be predicated (Kripke).
  • Michael
    8k
    Perhaps we could approach the liar paradox (which isn't Russell's paradox, as fishfry has mentioned), by first providing an account of what it means to be true, and then rephrasing the sentence appropriately.

    For example: this sentence does not correspond to a state of affairs.

    This gives us two options:

    1. "this sentence does not correspond to a state of affairs" corresponds to a state of affairs.
    2. "this sentence does not correspond to a state of affairs" does not correspond to a state of affairs.

    They can't both be contradictions, can they? At first glance, the latter actually seems redundant.

    I should note that this formulation is actually closer to "this sentence is not true" rather than "this sentence is false". Which I think is a good thing, as the former is considered a stronger variation of the liar paradox as it attempts to counter the solution that "this sentence is false" is neither true nor false. Although if the second option is indeed redundant as I suggested then the solution is the same regardless.
  • TheMadFool
    3.9k
    Sorry, I made the correction but it does apply to Russell's paradox too (I think).
  • TheMadFool
    3.9k
    Time doesn't enter into it.Banno

    It does. The statements ''I'm hungry'' and ''I'm not hungry'' contradict each other if made at the same time. However, as we all know, everyday, at different times they're true and false.
  • TheMadFool
    3.9k
    Can you make it clearer. Thanks.
  • Banno
    5.7k
    That this involves time is a function of hunger, not logic. Hunger changes over time, but "This sentence is both true and false" doesn't.
  • Michael
    8k


    Let's say that to be true is to refer to a fact and to be false is to refer to a fiction. The liar sentence is then "this sentence refers to a fiction".

    We have four (not necessarily exclusive) options:

    1. "this sentence refers to a fiction" refers to a fact.
    2. "this sentence refers to a fiction" doesn't refer to a fact.
    3. "this sentence refers to a fiction" refers to a fiction.
    4. "this sentence refers to a fiction" doesn't refer to a fiction.

    I think option 3. is redundant, rather than a contradiction, and comparable to:

    5. "this sentence contains 36 characters" contains 36 characters.
  • TheMadFool
    3.9k
    The same reasoning applies to Russell's paradox.

    A) The set of all sets that doesn't contain itself.

    B) If it contains itself, then it doesn't contain itself.

    C) If it doesn't contain itself, it contains itself

    Argument B:
    1. P......premise
    2. P > ~P.....assumptiom
    3. ~P......1, 2 MP
    4. P & ~P....1, 3 conjunction
    5. ~(P > ~P)....2 to 4 reductio


    Argument C:
    1. ~P.....premise
    2. ~P > P........assumption
    3. P.......1, 2 MP
    4. P & ~P....1, 3 conjunction
    5. ~(~P > P)...2 to 4 reductio

    So, the connection between ''it contains itself'' and ''it doesn't contain itself'' is NOT logical implication (simultaneous) because it leads to a contradiction. Therefore, as for the liar paradox, there's a time gap between ''it contains itself'' and ''it doesn't contain itself''.
  • TheMadFool
    3.9k
    That this involves time is a function of hunger, not logic. Hunger changes over time, but "This sentence is both true and false" doesn't.Banno

    Exactly. But the liar statement can't be false, because if it is, then it's true. So, it can't be a contradiction, because contradictions are false.
  • Banno
    5.7k
    So - it is malformed.
  • fishfry
    789
    Therefore, as for the liar paradox, there's a time gap between ''it contains itself'' and ''it doesn't contain itself''.TheMadFool

    No. The assumption that we can form such a set leads to a contradiction, showing that unrestricted comprehension can not be allowed. Please read the Wiki article on Russell's paradox that I linked earlier.
  • TheMadFool
    3.9k
    o - it is malformedBanno

    Another way to look at it. Yes.
  • Banno
    5.7k
    Hence the two solutions I listed, both of which show how to avoid the malformation.
  • TheMadFool
    3.9k
    The assumption that we can form such a set leads to a contradiction,fishfry

    IF you assume that ''set contains itself'' and ''the set doesn't contain itself'' imply each other THEN we have a contradiction.

    But contradictions are impossible

    Therefore, ''the set contains itself'' and ''the set doesn't contain itself'' don't imply each other.

    We may say, they cause each other, with a time gap between ''set contains itself'' and ''the set doesn't contain itself''. No contradiction.
  • fishfry
    789
    IF you assume that ''set contains itself'' and ''the set doesn't contain itself'' imply each other THEN we have a contradiction.TheMadFool

    A contradiction in sentential (aka propositional) logic, a contradiction is the statement "P ^ not-P" for some proposition P.

    There is no requirement that they "imply each other." This is something you are making up.

    I don't mean to say that you are wrong. You are entitled to make up your own rules of logic, or to use the word "contradiction" in a different way than logicians do.

    But when you do so, you are abandoning the generally agreed-on meaning of contradiction.

    Do you understand this? Again, I'm very open-minded. Tell me the rules of your system of logic and we'll play. But we're not doing standard logic then. Because in standard logic, a contradiction is when you can prove P and also not-P.

    But contradictions are impossibleTheMadFool

    On the contrary we use contradictions all the time. You could hardly do math without the famous technique of proof by contradiction, or reductio ad absurdum. The idea is to assume something and show that the assumption leads to a contradiction. We then conclude that the thing we assumed could NOT have been true.

    Everyone's seen the classic proof (which I won't reproduce here) that the square root of 2 is irrational. We start by assuming sqrt(2) is rational, and we derive a contradiction. We conclude that sqrt(2) can not be rational.

    In the case of Russell, we assume that we may form a set by taking all the objects that satisfy some predicate. For example if G(x) stands for the statement "x is a giraffe" then we may form the set {x : G(x)}. This is read as "The set of all x such that x is a giraffe." In other words this is the set of all giraffes.

    It's intuitively tempting to want to say that we can always form a set from a predicate. But in fact we can NOT do this; because if we let P(x) mean "x is not an element of x" then we get a contradiction.

    This shows that we may NOT in fact be so permissive with predicates.

    The solution in formal set theory is the axiom schema of specification, which says that if X is a set and P is a predicate, we may form the set {x in X : P(x)}. In other words if we start with a set we can always cut it down by a predicate. But we can not just have a predicate by itself. That way leads to contradiction.

    Russell himself proposed type theory, which remained somewhat of a backwater for most of the twentieth century, but is now getting renewed interest from the field of computer science.

    When you say "contradictions are impossible" and then conclude that they "don't imply each other," that is not standard logic, that's Mad Fool logic, or MF logic if you will.

    In standard logic, the contradiction shows that we can not use unrestricted comprehension, as it's called, to form sets.
  • TheMadFool
    3.9k
    that's Mad Fool logic, or MF logic if you will.fishfry
    :D I knew I was sailing into unknown territory. Anyway, be lenient.

    A: The set of all sets that don't contain itself.

    Does A contain itself? Yes/No?

    YES:
    1) A contains itself. Then A doesn't contain itself.

    In sentential logic:

    1a) IF A contains itself THEN A doesn't contain itself

    NO:
    2) A doesn't contain itself. Then A contains itself

    In sentential logic:

    2a) IF A contains itself THEN A doesn't contain itself

    3) The two implications 1a and 2a are necessary to derive the contradiction (A contains itself AND A doesn't contain itself)

    In sentential logic:

    3a) IF 1a and 2a are true THEN a contradiction follows

    Reductio ad absurdum:

    1a and 2a are false

    The connection between ''A contains itself'' and ''A doesn't contain itself'' is not logical implication. It's something else. I like to think of it as ''causation''. A contains itself'' causes ''A doesn't contain itself'' and the converse. At one moment A contains itself. At the next moment A doesn't contain itself...and so on. There's no contradiction because the two opposing states happen at different times.

    One could think of it as a rule:
    1. If you see X in the box, then take it out
    2. If you see X outside the box, then put it in

    X being inside AND outside is a contradiction. But the rule doesn't say that. It simply alternates the two states on a timeline, at different points of time.
  • fishfry
    789
    X being inside AND outside is a contradiction. But the rule doesn't say that. It simply alternates the two states on a timeline, at different points of time.TheMadFool

    Logical implication is not causation. They're two completely different things.

    Have you ever seen the truth table for implication?

    I am wondering, do you think your ideas are how logic works? Or do you understand that you are making up your own logic that's different than standard logic?

    There is no time or causation in sentential logic. If P and Q are propositions, we say P => Q is true in case either P is false or Q is true. That's all it means. There need be no causality or connection between them. If 2 + 2 = 5 then I am the Pope. That's a true implication, because the antecedent is false.

    This is not a difference of opinion between us. The way I described it is how sentential logic is done.

    If you're using some other system of logic like MF logic, please make that clear.

    If you are under the impression that standard sentential logic is as you say it is, you need to understand that you are wrong about that. There is no time and no causality in sentential logic or set theory.
  • TheMadFool
    3.9k
    There is no time and no causality in sentential logic or set theoryfishfry

    You're wrong.

    Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "One cannot say of something that it is and that it is not in the same respect and at the same time. — Wikipedia

    Sentential logic is ''stuck'' at one instant in time or else the law of noncontradiction doesn't make sense.

    Russell's paradox is totally dependent on deriving a contradiction, which implies, it's also predicated on time not changing.

    I'm saying if we simply switch from ''pause mode'' (sentential logic) to ''play mode'', as we do when watching movies, the contradiction disappears because time changes.
  • fishfry
    789
    I'm saying if we simply switch from ''pause mode'' (sentential logic) to ''play mode'', as we do when watching movies, the contradiction disappears because time changes.TheMadFool

    Yes I think I see your point. If we think of it as a process, as in the execution of a computer program, then we are just flipping states back and forth. Is that what you mean?

    The problem is that we're no longer doing standard logic and set theory. We're doing something else.

    The Wikipedia quote you gave seems misleading. "At the same time" doesn't really have to do with time. Perhaps the Wiki article misspoke itself. You can't take an inaccurate sentence in Wikipedia and extrapolate an entirely different meaning of sentential logic than the rest of the world uses.

    I still want to clarify. Are you confused about standard logic or just objecting to it? Do you understand that you say one thing and the entire rest of the world says another?

    That's ok if that's what you're doing. But if you think your understanding is the standard one, you should study some logic so at least you can understand what you're objecting to.

    Also when you quote Wiki, can you please link the article? I'd like to see the context of that quote since it is profoundly misleading. There is no time involved in sentential logic. It's really important for you to understand that.

    ps -- I should mention for clarity that when it comes to the Liar paradox, I have no opinion and virtually no interest in the topic. I take no position with your reformulation of the problem in terms of time and causality.

    But on Russell's paradox, you are factually wrong on both the history and the math. Russell's paradox and set theory take place in the context of logic in which time and causality are not relevant. Russell's paradox is a historical event in the early history of set theory and it means what it means and not something else.
  • TheMadFool
    3.9k
    Well, I too think my grasp of Russell's paradox isn't up to mark to continue the discussion into anything fruitful.

    As for time being involved in logic, I think I'm correct. Contradiction doesn't make sense otherwise. Take for example the statements ''It's 2:00 PM'' and ''It's not 2:00 PM''. As a day passes by both these statements acquire values true and false. It is only at exactly 2:00 PM that, together, they become a contradiction. Also, note that when it's not 2:00 PM, the contradiction occurs. Anyway...time is absolutely essential for contradiction to make sense. In fact, Russell's paradox is reliant on time being paused to generate the contradiction.

    Here are two links re time and logic, specifically contradiction.

    1. Stanford

    2. Wikipedia
  • fishfry
    789
    Well, I too think my grasp of Russell's paradox isn't up to mark to continue the discussion into anything fruitful.TheMadFool

    Ok. But now that I've explained it to you, your understanding should be excellent :-)

    As for time being involved in logic, I think I'm correct.TheMadFool

    It's true that there are logics where time is modeled, for example temporal logic. But ordinary sentential logic doesn't involve time. Can't add any more, it's all out there on Wiki.
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