User:Banno/scratch
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Some of my jokes:
http://en.wikipedia.org/w/index.php?title=Knowledge_management&diff=4625877&oldid=4625423
Falsification in formal logic
The explication of falsification by Popper perhaps is responsible for the formulation of the two main objections to empirical positivism. Falsification of some theory occurs through Modus Tollens, via some observation. Suppose some theory T, implies an observation O:
- <math>T \rightarrow O<math>
But the required observation is not made, so
- <math>\sim O<math>
So by Modus Tollens,
- <math>\sim T<math>
Theory-dependence of observation
But, all observations make use of prior assumptions; so
- <math>O \equiv \left( p_1 \wedge p_2 \wedge p_3 \cdots p_n \right) <math>
and so,
- <math>\sim O \equiv \sim \left( p_1 \wedge p_2 \wedge p_3 \cdots \wedge p_n \right)<math>
but
- <math>\sim \left( p_1 \wedge p_2 \wedge p_3 \cdots \wedge p_n \right) \equiv \sim \left( \sim p_1 \vee \sim p_2 \vee \sim p_3 \cdots \vee \sim p_n \right)<math>
by De Morgan's law.
In words, failure to make some observation only implies the failure of at least one of the prior assumptions that went into making the observation. It is always possible to reject an apparently falsifying observation by claiming that some one of its underlying assumptions is false; since there are an indeterminate number of such assumptions, any observation can potentially be made compatible with any theory. So it is quite valid to use a theory to reject an observation.
Indeterminacy of theory by evidence
Similarly, a theory consists of some indeterminate conjunction of hypotheses,
- <math>T \equiv \left( h_1 \wedge h_2 \wedge h_3 \cdots h_n \right) <math>
and so,
- <math>\sim T \equiv \sim \left( h_1 \wedge h_2 \wedge h_3 \cdots \wedge h_n \right)<math>
but
- <math>\sim \left( h_1 \wedge h_2 \wedge h_3 \cdots \wedge h_n \right) \equiv \sim \left( \sim h_1 \vee \sim h_2 \vee \sim h_3 \cdots \vee \sim h_n \right)<math>
In words: failure of some theory implies only failure of at least one of the underlying hypotheses of the theory. It is always possible to resurrect a falsified theory by claiming that some one of its underlying hypotheses is false; again, since there are in indeterminate number of such hypotheses, any theory can potentially be made compatible with any particular observation. So it is not possible to determine the falsity of a theory by reference to the evidence.