madnak
10-22-2006, 09:57 AM
Because everyone, including David, seems to be making this error, I'm starting a thread to point it out.
"True" and "false" are, according to some highly-structured standards of notation, such as formal logic and programming languages, very well-defined. They basically amount to evaluations of terms based on some operators. So long as the evaluation of these statements doesn't become recursive they're relatively easy to evaluate.
In these contexts, any given expression may be evaluated as either true or false, always one or the other. As a result of this, many people like to make statements such as "either A is true or A is false." But this is unjustifiable for a few reasons:
First, even if A is a statement of formal logic that's not always the case - for example a syntax may be infinitely recursive or may otherwise fail to evaluate according to the rules. These statements can be thought of as resembling division by 0 in some cases, and as resembling taking the root of a negative number in other cases. A simple example is the definition of statement A as "not A." Or, "this statement is false."
Second, A is rarely a logical statement. Or anything even resembling it. Organic language is vague by design and necessity. It's open to interpretation, it's complex, and it has elements of color and context that are critical to its interpretation. Even the most seemingly simple statements are usually open to interpretation. And at any rate no inherent rules for the evaluation of semantics exist in natural language, so such rules would have to be agreed upon before something like "either A is true or A is false" could be considered. A good example is "the sky is blue." Is it? "That depends on what your definition of 'is' is." No, really. In order to make clear statements we have to enter some recognized structural context - to really nail this statement down we have to define everything from the sun's radiation through the Earth's atmosphere all the way to the observer's optic nerve (either that or define the qualia, which is much harder). And even then there would be "wiggle room."
Finally, A is almost always a compound statement that represents part of a network of other statements and assumptions and values. It typically includes a number of component statements, the premises upon which they're based, and even a few conclusions that may be derived from them, along with implied connections to other "common sense" assumptions. Statement A can rarely be taken effectively in isolation. It's not even meaningful to say that "A is true if and only if every component of A is true." And if it were meaningful, it would be wrong. Often even the person positing A isn't suggesting that every component of A is true - only that some component of A is true, or even that A is true in some limited sense.
For all these reasons (and more that aren't relevant enough to mention), it's an outrageous error to treat a normal English statement such that it must be either true or false!
"True" and "false" are, according to some highly-structured standards of notation, such as formal logic and programming languages, very well-defined. They basically amount to evaluations of terms based on some operators. So long as the evaluation of these statements doesn't become recursive they're relatively easy to evaluate.
In these contexts, any given expression may be evaluated as either true or false, always one or the other. As a result of this, many people like to make statements such as "either A is true or A is false." But this is unjustifiable for a few reasons:
First, even if A is a statement of formal logic that's not always the case - for example a syntax may be infinitely recursive or may otherwise fail to evaluate according to the rules. These statements can be thought of as resembling division by 0 in some cases, and as resembling taking the root of a negative number in other cases. A simple example is the definition of statement A as "not A." Or, "this statement is false."
Second, A is rarely a logical statement. Or anything even resembling it. Organic language is vague by design and necessity. It's open to interpretation, it's complex, and it has elements of color and context that are critical to its interpretation. Even the most seemingly simple statements are usually open to interpretation. And at any rate no inherent rules for the evaluation of semantics exist in natural language, so such rules would have to be agreed upon before something like "either A is true or A is false" could be considered. A good example is "the sky is blue." Is it? "That depends on what your definition of 'is' is." No, really. In order to make clear statements we have to enter some recognized structural context - to really nail this statement down we have to define everything from the sun's radiation through the Earth's atmosphere all the way to the observer's optic nerve (either that or define the qualia, which is much harder). And even then there would be "wiggle room."
Finally, A is almost always a compound statement that represents part of a network of other statements and assumptions and values. It typically includes a number of component statements, the premises upon which they're based, and even a few conclusions that may be derived from them, along with implied connections to other "common sense" assumptions. Statement A can rarely be taken effectively in isolation. It's not even meaningful to say that "A is true if and only if every component of A is true." And if it were meaningful, it would be wrong. Often even the person positing A isn't suggesting that every component of A is true - only that some component of A is true, or even that A is true in some limited sense.
For all these reasons (and more that aren't relevant enough to mention), it's an outrageous error to treat a normal English statement such that it must be either true or false!