What Is a Logical Equivalent of This Page Intentionally Left Blank?

[brian0918]

Hello,

I'm requesting a mathematical logic statement that equivalently represents the usage of the phrase "This page intentionally left blank" on blank pages. The phrase is a self-refuting meta-reference, in that it falsifies itself by its very existence on the page in question.

An example of this usage can be found parodied on this shirt:
http://www.just4yucks.com/catalog/5x/52013.phtml

Thanks!

 

[Orefa]

This reply was accidentally omitted.

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[Curious3141]

If I were designing this, I would alter the phrase to be "This page has been intentionally left blank except for this statement." Got to satisfy my OCD for semantic correctness.

 

[honestrosewater]

If you just want a sentence that is always false, a.k.a. a contradiction, then $(P \wedge \neg P)$ (read: "P and not P"), where P is any sentence, will work for any classical logic, the kind used for most things, including most math and the reasoning in this post.^*1 If you are working in a different logic and have trouble thinking of a contradiction in it on your own, an introduction to the logic should list several sentences that are contradictions in that logic. Note that whether a sentence is a contradiction depends on the rules by which it is evaluated, so the same sentence might be a contradiction in one logic but not in another.

I think your sentence is more interesting, and you could go several ways with it because there is more than one interpretation. First, notice that

1) This page intentionally left blank.

literally means that blank was intentionally left by the page (whatever that means), which isn't as interesting and almost certainly isn't the intended meaning. To convey the other meanings, you can choose an auxiliary verb (be, have, do) to make this page not the one doing the leaving. (You obviously did this mentally when you made sense of the sentence before -- cool, huh?) The tense (past, present, future, etc.) will make a big difference.

2) This page was intentionally left blank.
3) This page has been intentionally left blank.
4) This page is intentionally left blank.
5) This page will be intentionally left blank.

You also need to specify what other event the event of the sentence (the page being left blank) is being measured relative to. Is the page's being left blank being measured relative to the moment the sentence is uttered (spoken or typed or printed or ??), the moment the sentence is received (heard or read or ??), or some other event? Specifying this could give you, for instance

2a) This page was intentionally left blank before this sentence was added.

which, given the normal interpretation, isn't a contradiction.

Another interesting thing is to consider the distinction between an utterance and a sentence, the former being a physical manifestation of the latter (FYI: other terms are used to draw the same or a similar distinction; it's similar to the distinction between a numeral and a number). There is nothing strictly self-contradictory about a sentence claiming to have not been uttered. (And if you grant that a sentence must exist before it can be uttered (which might be a defensible position), that claim is indeed true of all sentences at some time. Of course, you could also just define things that way and call the rest philosophy. Anywho...) If you define things the right way -- and there is plenty of wiggle room in English -- this might be (edit: and I stress might be; I'm having doubts, but there's a better example below) stated as "the page that contains this sentence contains no utterance" or, more colloquially, "this page is blank".^*2
I think you might be interested in performative contradictions; try google -- or I'll try to find some good sites if you're interested.

Anywho, it might save you some disappointment to keep in mind that the flexibility and convenience (read: ambiguity) in English and other natural languages usually makes translations from them into the formal languages used in modern logic and math hairy and complicated -- if a translation is even possible. Just ask yourself how much information your brain filled in in going from reading (1) to thinking it was a contradiction.

*1. The negation of $(P \wedge \neg P)$ is an important sentence in logic and is commonly called both The Law (or Principle) of Contradiction and The Law (or Principle) of Non-contradiction -- though usually not at the same time. Teehee. (Well, I think that's funny.)

*2. I thought of a clearer example of how a sentence can contain a "latent" contradiction that is "activated" in some utterances of the sentence. Assume that if a sentence is a contradiction, then all of its utterances must also be contradictions (I can't imagine how this could be false, but just to be safe...).

6) This sentence cannot be spoken.

If (6) is spoken, that utterance is a contradiction. But if (6) is, for instance, written, it's not a contradiction. Indeed, any utterance of (6) that isn't spoken isn't a contradiction, so (6) is not a contradiction.

 

[fourier jr]

This reply was accidentally omitted.
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this whole thread cracks me up