Outer Limits of Reason Read Online Free PDF Page A

limitations of reason are Barrow 1999, Dewdney 2004, and Poundstone 1989. Sorensen 2003 is a wonderful history of paradoxes.

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    Language Paradoxes
    What we cannot speak about we must pass over in silence. 1
    â€”Ludwig Wittgenstein (1889–1951), Proposition 7 of Tractatus Logico-Philosophicus
    After all, Mr. Wittgenstein manages to say a good deal about what cannot be said.
    â€”Bertrand Russell (1872–1970), introduction to Wittgenstein’s Tractatus Logico-Philosophicus
    Half the lies they tell about me aren’t true.
    â€”Yogi Berra 2
    Rather than jumping headfirst into the limitations of reason, let us start by just getting our toes wet and examining the limitations of language. Language is a tool used to describe the world in which we live. However, don’t confuse the map with the territory! There is one major difference between the world we live in and language: whereas the real world is free of contradictions, the man-made linguistic descriptions of that world can have contradictions.
    In section 2.1 , we encounter the famous liar paradox and its many variants. These are relatively easy puzzles that will get us started. Section 2.2 contains a collection of self-referential paradoxes. I show that they all have the same form. In section 2.3 we meet several paradoxes involving descriptions of numbers.
    2.1  Liar! Liar!
    A linguistic paradox is a phrase or sentence that contradicts itself. A baby version of a linguistic paradox is an oxymoron (from the Greek oxys “sharp” and moros “stupid”—together they mean “pointedly foolish” or “pointedly dull”). These are phrases, usually consisting of two words, that contradict each other. Some examples are “original copies,” “open secret,” “clearly confused,” “militant pacifist,” “larger half,” “alone together,” and my favorite, “act naturally.” Even though these phrases do not really make sense, we human beings have no problem using them in common everyday speech.
    The classic example of a linguistic paradox is the famous Epimenides paradox . This dates back more than two and a half millennia to when Epimenides (600 BC), a philosopher and poet who lived in Crete, complained about his neighbors in a poem called Cretica . He wrote: “The Cretans, always liars, evil beasts, idle bellies!” This seems 3 paradoxical. If this statement is true, then since Epimenides is a Cretan, he is including himself as a liar and this line of the poem is false. In contrast, if it is false, then Epimenides is not a liar and the line is true.
    There are many linguistic paradoxes similar to Epimenides’ statement. The liar paradox is a simple sentence like
    I am lying.
    or
    This sentence is false.
    If these sentences are true, then they are false. Furthermore, if they are false, then they are true.
    The liar paradox is found in many different forms. For example, we can denote a sentence L 1 and then say that L 1 asserts its own falsehood:
    L 1 : L 1 is false.
    Again, if L 1 is true, then it is false. And if L 1 is false, then it is true. Other variations of the liar paradox have sentences that are not directly self-referential. Consider the following two sentences:
    L 2 : L 3 is false.
    L 3 : L 2 is true.
    If L 2 is true, then L 3 is false, which would mean that “ L 2 is true” is false and hence L 2 is false. In contrast, if L 2 is false, then L 3 is true and L 3 asserts that L 2 is true. Buzz! That’s a contradiction.
    It is important to note that just because sentences refer to themselves and their falsehoods does not mean there is a contradiction. Consider these two sentences:
    L 4 : L 5 is false.
    L 5 : L 4 is false.
    Let’s assume that L 4 is false. Then L 5 is true and L 4 is false. Similarly, if you start with the premise that L 4 is true, you get that L 5 is false, and hence L 4 is true. Neither assumption leads you to a