By Bhagawandas P. Lathi
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Extra info for An Introduction to Random Signals and Communications Theory
But while knowledge is cooptive—according to KxKyp ͉- Kxp—ignorance is not. For if KxϳKyt entailed ϳKxt, then Kxt would entail ϳKxϳKyt, and nobody would ever know that something he knows is unknown to another individual. Keeping a secret from someone would become, in principle, impossible. Accordingly, one can sometimes know that a certain truth known to oneself is not known to any other individual. Knowing (∃t)(Kit & Ki ϳ(∃x)[x ϶ i & Kxt]) is unproblematic. And in fact the complexity of truth and the diversity of people’s access to it means that no two knowers will share the same conjunctive totality of knowledge.
There can be little doubt about the capacity of finite knowers to know universal truths: “x knows that all humans are mortal” is an unproblematic instance of knowing a collective fact. But the corresponding distributive case is something else again. Maintaining that “x knows of every human that he or she is mortal” is simply not practicable with finite knowers, seeing that there is simply no possible way for such a knower x to get around to considering all the relevant instances. Aspects of Knower Limitedness The knowers that are at issue in our present deliberations are not only finite but also limited knowers.
Kxϳp ∨ ϳKx[ϳp͉- ϳp]) 5. ϳKxϳp assumption from (1) by definition A from (2) from (3) substituting ϳp/q from (4) and ϳp ʈ- ϳp And it is also possible to establish the converse of this theorem: ϳKxϳp ͉- Axp or equivalently ϳAxp ͉- Kxϳp 25 26 For Aught That Someone Knows For by the definition of Axp this amounts to (∃q)(Kxq & Kx[q ͉- ϳp]) ͉- Kxϳp. And this follows at once from the thesis at hand by the deductivity principle. Accordingly, we have it that Axp iff ϳKxϳp. ” Notwithstanding any seeming difference, these two contentions are demonstrably equivalent.
An Introduction to Random Signals and Communications Theory by Bhagawandas P. Lathi