The Liar Paradox Deconstructed

Edit: The way plato.stanford.edu presents the paradox is also addressed as an addendum to this blog post. It rests on the same errors as the Wiki. http://plato.stanford.edu/entries/liar-paradox/#SimFalLia

 

Personally, I am convinced that the Liar Paradox is a myth. This blog post demonstrates why the Liar Paradox poses no actual problem for the realm of logic. I will show step by step why the argument ultimately rests on a hypothetical syllogism where the conclusion does not follow from its premises. To be thorough, I will deconstruct all the relevant passages in the Wiki.

http://en.wikipedia.org/wiki/Liar_paradox

This so called paradox is shown:

“If “This sentence is false.” is true, then the sentence is false, but then if “This sentence is false.” is false, then the sentence is true, and so on.”

and

“The simplest version of the paradox is the sentence:

This statement is false. (A)

If (A) is true, then “This statement is false” is true. Therefore (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then “This statement is false” is false. Therefore (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.”

Each sentence is numbered according to the original order at which they appeared in the article:

1. This statement is false. (A)

2. If (A) is true then “This statement is false” is true

3. Therefore (A) must be false

4. The hypothesis that (A) is true leads

5. to the conclusion that (A) is false, a contradiction.

I will now reduce the sentences into their simple logical form as subject and predicate.

1. This statement is false. (A)

Clearly, this signifies the intention of (A).

*1. (A) is “This statement is false”

2. If (A) is true then “This statement is false” is true

Erasing the unneeded “then”

*2. If (A) is true, “This statement is false” is true.

3. Therefore (A) must be false

Rearranging into simple logical form

*3. (A) is false

Switching (1) and (2) so the order of the premisses in the more popular form [Note 1] of the hypothetical syllogism.

*2. If (A) is true, “This statement is false” is true.

*1. (A) is “This statement is false”

*3. (A) is false

Well here we have it folks. The conclusion simply does not follow from the premisses. We may bypass the rest (4) and (5).

4. The hypothesis that (A) is true leads

5. to the conclusion that (A) is false, a contradiction.

The antecedent in the hypothetical proposition of the major premiss (*2) is “If (A) is true.” In this form of proposition, the truth of the consequent is dependent on the truth of the of this antecedent. The minor premiss (*1) did not express the antecedent of the major premiss. Therefore, the conclusion (*3) does not follow. That was easy. Why do people take this so seriously?

Even though I copied and pasted the exact words from the Wikipeida article, some may think that the Liar’s Paradox is ill representative. Therefore, I will form the argument in another way, which I think is more representative of the standard.

1. A is false.

2. ‘A is false’ is true.

3. A is true.

Now we have asserted the contradictory and alas, a coup de grace! But look again, (3) can not possibly follow from either (1), (2), or a syllogistic combination of them. From the law of excluded middle, if a proposition (A) is false, then its contradictory is true vice versa. Also, there is no syllogistic combination of (1) and (2) since they have no term in common. Therefore, the proper argument can only look like what’s below because we can only apply the law of excluded middle on (1) or (2):

1. A is false.

2. ‘A is false’ is true.

*3a. The contradictory of A is true.  (excluded middle from 1)

*3b. ‘A is true’ is false. (excluded middle from 2)

(3) is contradictory to both (3a) and (3b). Therefore to posit (3) along with (3a) or (3b) violates the law of contradiction. Furthermore, (1) and (2) are completely redundant to each other, along with (3a) and (3b). They say the exact same thing. This redundancy provides no additional information. The proposition was asserted because its true. There’s no need to qualify this again by saying the proposition is true. Lets fully illustrate with an example starting with the incorrect argument:

1. “All lions can fly” is false.

2. “‘All lions can fly’ is false” is true.

3. “All lions can fly” is true.

And the corrected argument:

1. “All lions can fly” is false.

2. “‘All lions can fly’ is false” is true.

3a. “Some lions can not fly” is true.

3b. “‘All lions can fly’ is true” is false.

Removing the useless redundancies:

1. “All lions can fly” is false.

3a. “Some lions can not fly” is true.

This corrected “inference” stands on its own. Strictly speaking its not really even an inference. It tells us nothing we did not already know, we merely reformed the original proposition (1) to attain (3a).

Ultimately, however, we know that in fact no lions can fly. The truth of the universal contains the truth of the particular so that we may also say some lions can not fly. However, how did I know this? Because I am cognizant of the concept of the term “lion” from my past experience. So I may merely analyze this intention and discover that no lions can fly. The point is that I don’t even need to analyze the formal argument at all. The truth of all the terms depend on their corresponding real objects. Truth does not depend on the formal structure of syllogism, although its possible to discover new truths through such syllogisms.

Notice how the original argument gave us “This statement is false.” Practically speaking, the only way a person may assert the truth of this determination is if they know what the statement contains. They can only know the statement is false if they know what the terms signify in reality thereby enabling them to determine its falsity. Therefore, the whole proposition “This statement is false” MUST rest on us having knowledge of “The statement.” There is no instance where we would refer to some generality, “the statement” and say “its false” randomly. Therefore, we may in fact bypass everything said thus far in this blog post and say that if a person knows that “the statement” is false, then they obviously must know what the statement is. Therefore, we may merely analyze the statement to see how we may categorically determine the subject and eliminate any need for such useless analyzing of propositions.

Ironically, this greatly simplifies our problems much more than any attempt to simplify a logical system has done. Notice how this sort of method (the direct account of the real objects) is not derived from any system at all. Its derived from the universal notions (e.g. All lions cannot fly) formed throughout life. Clearly, no simplified logical system endows us with these abilities. “The statement” can only be known in reference to reality. This reality can only exists separately by the action of our conceptual faculties.

[Note 1]

Form of the hypothetical syllogism:

1. If A is B, C is D.

2. A is B.

3. C is D.

Very simple stuff. Same as these sources:

https://faculty.unlv.edu/beisecker/Courses/Phi-102/HypotheticalSyllogisms.htm

http://www.wwnorton.com/college/phil/logic3/ch10/hyposyll.htm

http://en.wikipedia.org/wiki/Hypothetical_syllogism

 

Presentation of the Paradox by plato.stanford.edu

Again, here’s the link: http://plato.stanford.edu/entries/liar-paradox/#SimFalLia

For the sake of accuracy, I won’t initially omit any words by the authors. I will show every step. Let me know if I skipped a step.

“Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false.

FLair: FLiar is false.

This seems to lead to contradiction as follows. If the sentence ‘FLiar is false’ is true, then FLiar is false. But if FLiar is false, then the sentence ‘FLiar is false’ is true. Since FLiar just is the sentence ‘FLiar is false’, we have it that FLiar is false if and only if FLiar is true. But, now, if every sentence is true or false, FLiar itself is either true or false, in which case—given our reasoning above—it is both true and false. This is a contradiction. Contradictions, according to many logical theories (e.g., classical logic, intuitionistic logic, and much more) imply absurdity—triviality, that is, that every sentence is true.”

I will consider every individual sentence to extract all of the propositions (at least the relevant ones) contained in them.

“Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false.”

No proposition is contained here.

“FLair: FLiar is false.”

Here we have our first proposition:

1. FLair is “FLair is false.”

You can see no omissions here are made. However the copula is added because that is indubitably implied in the clause.

“This seems to lead to contradiction as follows.”

No proposition here pertaining to the argument.

“If the sentence ‘FLiar is false’ is true, then FLiar is false.”

Do I see a hypothetical proposition? This constitutes our second proposition:

2. If the sentence ‘FLiar is false’ is true, then FLiar is false.”

Next sentence: “But if FLiar is false, then the sentence ‘FLiar is false’ is true.”

We have now have our next proposition:

3. But if FLiar is false, then the sentence ‘FLiar is false’ is true.

Next clause in the next sentence constitutes a fourth proposition, “Since FLiar just is the sentence ‘FLiar is false’,” as:

4. Since FLiar just is the sentence ‘FLiar is false’.

Next clause: “we have it that FLiar is false if and only if FLiar is true.”

5. we have it that FLiar is false if and only if FLiar is true.

Next clause: “if every sentence is true or false, FLiar itself is either true or false.”

6. if every sentence is true or false, FLiar itself is either true or false.

Next: “in which case—given our reasoning above—it is both true and false”

7. in which case—given our reasoning above—it is both true and false.

Next: “This is a contradiction.”

8. This is a contradiction.

This is sufficient for the argument. I now present the unaltered clauses in the order as presented:

1. FLair is “FLair is false.”

2. If the sentence ‘FLiar is false’ is true, then FLiar is false.

3. But if FLiar is false, then the sentence ‘FLiar is false’ is true.

4. Since FLiar just is the sentence ‘FLiar is false’.

5. we have it that FLiar is false if and only if FLiar is true.

6. if every sentence is true or false, FLiar itself is either true or false.

7. in which case—given our reasoning above—it is both true and false.

8. This is a contradiction.

Now I will simplify the sentences to get rid of superfluous words reducing them to their logical form. I will specifically enumerate all the alterations made.

(1) This was already put in subject predicate form.

(2) I will erase “then” since its unneeded. I will do this for all the propositions alike in this manner. I will also remove the qualifier “the sentence” because it contributes nothing to the intention of “FLiar is false.” Its already known to be a sentence due to it having quotes.

(3) “But” adds no additional intention. It will be removed from all.

(4) “Since” and “just” similarly add nothing and are removed from all.

(5) “we have it that” is extraneous.

(6) “Itself” is unneeded.

(7) “in which case—given our reasoning above” is superfluous. “It” is FLair.

(8) “This” is in reference to the proposition (7). It is replaced by (7).

Simplified propositions:

1. FLair is “FLair is false.”

2. If “FLiar is false” is true, FLiar is false.

3. If FLiar is false, “FLiar is false” is true.

4. FLiar is “FLiar is false.”

5. FLiar is false if and only if FLiar is true.

6. If every sentence is true or false, FLiar is either true or false.

7. FLiar is both true and false.

8. “FLiar is both true and false” is a contradiction.

We now have our the simplified propositions in the order of presentation. All these contain the full intention of the words used in the original paragraph. If there are disputable alterations, you can let me know. All changes were transparently demonstrated without exception. We may make the following observations:

(1) is the exact same as (4), this redundancy serves no purpose and one may be neglected.

(1), (2), and (3) may all be potential premisses. However, (1) has no terms in common with (2) and (3), and therefore cannot be syllogistically combined.

(5) must be derived from at least two of these premises and the only premisses left to combine are (2) and (3).

The only syllogistic form that admits a conclusion from two hypothetical propositions is the pure hypothetical proposition. (See https://faculty.unlv.edu/beisecker/Courses/Phi-102/HypotheticalSyllogisms.htm) The sole and only form for this syllogism is:

1. If C is D, E is F.

2. If A is B, C is D.

3. If A is B, E is F.

Now to employ this form, we use propositions (2) and (3):

2. If “FLiar is false” is true, FLiar is false.

3. If FLiar is false, “FLiar is false” is true.

Giving us:

If “FLiar is false” is true, “FLiar is false” is true.

We may also switch the order of the premisses and get:

If FLiar is false, FLiar is false.

These are clearly not the same as either (5):

5. FLiar is false if and only if FLiar is true.

Therefore, plato.stanford.edu is guilty of processing conclusions which do not follow from their premisses. It may be possible that the rest of their examples may commit these logical errors. This oversight is astonishing to me.

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9 thoughts on “The Liar Paradox Deconstructed”

  1. The core of your rearrangement is this:

    > Rearranging into simple logical form and correcting the order of the premisses to the more popular form of the hypothetical syllogism.

    > *1. If (A) is true, “This statement is false” is true.

    > *2. (A) is “This statement is false”

    > *3. (A) is false

    This isn’t a hypothetical syllogism, not in form or anything else. It’s a series of disconnected statements. No one was ever arguing that (A) would have to be false because (A) is some statement and statements are true if they’re true.

    Both of your premises focus on the relationship between (A) and the given statement when that’s totally extraneous to the argument at hand. (A) is just a rewriting of the statement at hand, and the mere fact of writing it as (A) certainly doesn’t imply anything of interest.

    To understand the paradox I’d suggest you completely eliminate any reference to ‘(A)’ and just focus on the proposition itself.

  2. I didn’t arrange it. I rearranged it so it looked clear. I knew that if I skipped a step, it might be less clear. I will edit the post. Here is the original three statements not made by me:

    1. This statement is false. (A)

    2. If (A) is true then “This statement is false” is true

    3. Therefore (A) must be false

    I switched (1) and (2), because in a syllogism the order of the premises are unimportant, it just puts it into its popular form that people usually see.

    So switching (1) and (2)…

    1. If (A) is true then “This statement is false” is true

    2. This statement is false. (A)

    3. Therefore (A) must be false

    Now I can hope you can see that I can simplify these sentences without omitting any meaning whatsoever.

    1. If (A) is true, “This statement is false” is true

    2. (A) is “The statement is false.”

    3. (A) is false.

    This is most definitely in the form of a hypothetical syllogism. Same form as:

    1. If A is B, C is D.

    2. A is B.

    3. C is D.

    Very simple stuff. Same as these sources:

    https://faculty.unlv.edu/beisecker/Courses/Phi-102/HypotheticalSyllogisms.htm

    http://www.wwnorton.com/college/phil/logic3/ch10/hyposyll.htm

    http://en.wikipedia.org/wiki/Hypothetical_syllogism

    1. Look at the form you’ve provided of a hypothetical syllogism, specifically the element ‘B’. What you’ve produced is evidently not the same form, since “This statement is false” is not the same thing as “true”.

      1. So replace all the D’s with B’s. This does not pose a problem. In fact, replace all the letters with “B” and it would still follow. Use any letter you want, it does not change the form of the syllogism.

      2. I see what you are saying now. I am not the one who posed (3) as the conclusion. They posed (3) as the conclusion when they said:

        “Therefore (A) must be false”

        They presented this as the conclusion of the two previous premisses. However we can even switch (2) and (3) if you like, even though this not how the argument was presented.

        1. If (A) is true, “This statement is false” is true

        2. (A) is false.

        3. (A) is “The statement is false.”

        Even though this is not the argument presented, the conclusion still does not follow.

  3. Well, again, you’re focusing on the connection between (A) and “This statement is false” to the exclusion of everything else when that connection is irrelevant to the argument.

    Leave what Wikipedia says aside for the moment. You’re misinterpreting it, but that’s not relevant for now. The point of the liar’s paradox is this:

    If “This statement is false” is true, then it follows tautologically that “This statement is false” is false. (This is the half of the argument that you’re rearranging.)

    However, if “This statement is false” is false, then it follows by the principle of the excluded middle that “This statement is false” is true.

    I can rephrase these as syllogisms. Let’s take a more general case first.

    1. If any proposition is true, then it is true. (Tautology)
    2. “P is false” is true.
    3. P is false.

    In the second:

    1. Any proposition is either true or false. (Principle of the excluded middle)
    2. “P is false” is false.
    3. P is true.

    The liar’s paradox arises when P is “This statement” (or “This proposition” etc.) — in other words when the proposition is referring to itself. Then, by simple substitution, we get:

    1. If any proposition is true, then it is true. (Tautology)
    2. “This statement is false” is true.
    3. “This statement is false” is false.

    In the second:

    1. Any proposition is either true or false. (Principle of the excluded middle)
    2. “This statement is false” is false.
    3. “This statement is false” is true.

    Clearly both of these are contradictions, but they are both formally valid, hence the paradox.

    Hope that clears things up a bit.

    1. “If “This statement is false” is true, then it follows tautologically that “This statement is false” is false. (This is the half of the argument that you’re rearranging.)”

      This is absolutely incorrect. You should try reading my article, I know I am a terrible writer.

      “1. A is false.

      2. ‘A is false’ is true.

      3. A is true.

      Now we have asserted the contradictory and alas, a coup de grace! But again, (3) does not follow from (2). From the law of excluded middle, if a proposition (A) is false, then its contradictory is true. Therefore, the proper argument looks like:

      1. A is false.

      2. ‘A is false’ is true.

      *3. The contradictory of A is true.

      (3) and (*3) are not the same. Its very easy to see that to assert both (3) and (*3) as true violates the law of contradiction. Furthermore, (1) and (2) are completely redundant. They say the exact same thing. This redundancy provides no additional information. The proposition was asserted because its true. There’s no need to qualify this again by saying the proposition is true.”

    2. How you formulate it is a simple contradiction. Easily shown:

      ““If “This statement is false” is true, then it follows tautologically that “This statement is false” is false. (This is the half of the argument that you’re rearranging.)””

      Two statements:

      “This statement is false” is true

      “This statement is false” is false

      These are direct contradictions as easily shown. I am sorry but I don’t make the rules of logic. There is no way in which a tautology can be contradictory unless you have a very distorted understanding of “tautology.”

  4. To clarify, you commit two inexplicable errors:

    1. If “This statement is false” is true, then it follows tautologically that “This statement is false” is false.

    These are obviously not tautologous because they are contradictory. ““This statement is false” is true” is the contradiction of ““This statement is false” is false.”

    2. However, if “This statement is false” is false, then it follows by the principle of the excluded middle that “This statement is false” is true.

    This is incorrect. The law of excluded middle says of two contradictory propositions, one must be true, the other false. You attribute “true” and “false” to the same proposition, again committing a contradiction. The law of excluded middle does not allow for contradiction.

    “This statement is false” is false

    What DOES NOT follow (from the law of excluded middle):

    “This statement is false” is true

    What DOES follow (from the law of excluded middle):

    “This statement is true” is true

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