Wednesday, January 25, 2006
Some further reflections on the last class.
A propos of the Liar's Paradox, it strikes me that there could be even a shorter version. Suppose I say to you: "I'm lying." Is this true or false? If the statement that I am lying is true, then I am not lying about my lying, so the statement about my lying is false. In short, it is true on the condition that it is false. If, on the other hand the statement that I am lying is false, then I am lying about my lying, which makes the statement about my lying true. In short, it is false on the condition that it is true. The truth value of such statements is apparently undecidable.
One thing we might note is that this and similar paradoxes seem to have self-referentiality in common. In other words, they refer to themselves. Suppose you came into the classroom and saw the following statement on the blackboard: "This statement is false." If it is true, then it is false in what it asserts. If it is false in what it asserts, the only alternative is that it is true. This follows from the so-called law of the excluded midde: Either A or not-A, but not A and not-A.
What exactly is it that is paradoxical about self-referentiality? (Note that there is implied self-reflexivity in the paradoxical statement that "everything is relative," because the categorical term everything includes the statement itself, and there is self-reflexivity in the statement "This statement is false." I suspect there is a connection, too, with subjectivity. The demonstrative "this" implies a subjective observer located in the immediate vicinity.
It is also noteworthy that the Liar's paradox, as well as my version of it ("I am lying") and the statement "This statement is false" all have the motif of falsehood. "This statement is true" is clearly not paradoxical, or is "I am telling the truth." Could it be that the paradox is related to the fact that linguistic utterances are, at a fundamental level, assumed to be true. What we say in language seems tacitly to be accompanied by a certain positivity. We say "what is," and under normal circumstances there is an assumption of truth to the speaker's assertions.
Two biblical references came up in the last class, the precise references to which I am now in a position to furnish.
1. "the still small voice " (I Kings, 19:12)
2. The judgment of Solomon (I Kings 3: 16-28)
One thing we might note is that this and similar paradoxes seem to have self-referentiality in common. In other words, they refer to themselves. Suppose you came into the classroom and saw the following statement on the blackboard: "This statement is false." If it is true, then it is false in what it asserts. If it is false in what it asserts, the only alternative is that it is true. This follows from the so-called law of the excluded midde: Either A or not-A, but not A and not-A.
What exactly is it that is paradoxical about self-referentiality? (Note that there is implied self-reflexivity in the paradoxical statement that "everything is relative," because the categorical term everything includes the statement itself, and there is self-reflexivity in the statement "This statement is false." I suspect there is a connection, too, with subjectivity. The demonstrative "this" implies a subjective observer located in the immediate vicinity.
It is also noteworthy that the Liar's paradox, as well as my version of it ("I am lying") and the statement "This statement is false" all have the motif of falsehood. "This statement is true" is clearly not paradoxical, or is "I am telling the truth." Could it be that the paradox is related to the fact that linguistic utterances are, at a fundamental level, assumed to be true. What we say in language seems tacitly to be accompanied by a certain positivity. We say "what is," and under normal circumstances there is an assumption of truth to the speaker's assertions.
Two biblical references came up in the last class, the precise references to which I am now in a position to furnish.
1. "the still small voice " (I Kings, 19:12)
2. The judgment of Solomon (I Kings 3: 16-28)
# posted by Michael @ 7:23 AM 
