## Everything that is True is Known to be True

Posted by allzermalmer on April 26, 2013

(i) Assume that *If “proposition p is true” then it is possible that it is known that “proposition p is true”*. p →◊Kp

(ii) Assume that *Both “proposition p is true” & it is not known that “proposition p is true”*. p & ~Kp

(iii) Assume that *it is known that “proposition p is true” if and only if “proposition p is true”*. Kp ≡ p

(iv) Assume *it is known that both “proposition p is true” & “proposition q is true” if and only if it is known that “proposition p is true” & it is known that “proposition q is true”*. K(p & q) ≡ Kp & Kq

*Substitute the p in (i) with (ii), and we get [(p & ~Kp) → ◊Kp]. This means (i*) *If both “proposition p is true” & it is not known that “proposition p is true”, then it is possible that it is known that “proposition p is true”*.

(v) *It is possible that it is known that both “proposition p is true” & it is not known that “proposition p is true”*. ◊K(p & ~Kp) [Logical Consequence of (i*) & (ii) by Modus Ponens]

(vi) Assume that *It is known that both “proposition p is true” & it is not known that “proposition p is true”*. K(p & ~Kp)

(vii) *It is known that “proposition p is true” & it is known that it is not known that “proposition p is true”*. Kp & K~Kp [Logical Consequence of (vi) and (iv)]

(viii) *It is known that “proposition p is true” & It is not known that “proposition p is true”*. Kp & ~Kp [Logical Consequence of

(ix) *It is not known that both “proposition p is true” & it is not known that “proposition p is true”*.

(x) proven “proposition p is true” if and only if it is necessary that “proposition p is true”.

(xi)* It is necessary that it is not known that both “proposition p is true” & it is not known that “proposition p is true”*.

(xii)* It is necessary that “proposition p is not true” if and only if it is not possible that “proposition p is true”*.

(xiii)* It is not possible that it is known that both “proposition p is true” & it is not known that “proposition p is true”*.

(xiv) It is necessary that it is not known that both “proposition p is true” & it is not known that “proposition p is true”.

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