The concepts of omniscience can be defined as follows (using the notation of modal logic):
x is omniscient =def In words: x is omniscient =def For all propositions p: if p (is true) and p is (logically) knowable, then x knows [/can know] that p (is true) The latter definition is necessary, because there are logically true but logically unknowable propositions such as "Nobody knows that this sentence is true": N = "Nobody knows that N is true" If N is true, then nobody knows that N is true; and if N is false, then it is not the case that nobody knows that N is true, which means that somebody knows that N is true. And if somebody knows that N is true, then N is true; therefore, N is true in any case. But if N is true in any case, then it is logically true and nobody knows it. What is more, the logically true N is not only not known to be true but also impossibly known to be true, for what is logically true is impossibly false. Sentence N is a logical counter-example to the unqualified definition of "omniscience", but it does not undermine the qualified one. ANSWER This is an incorerct argument.. according to laws of contradiction and non con-tradiction if a statement is true then its nagation is false. If a statement is supposed such that it does imply that the laws of contrdiction and non contradiction both are in valid then the statement is false and there is no such true statement such that the truth of its negation does imply the truth of the given statement. now consider the statement:N1 = "Nobody knows that N is true ................................................. In the case N1 = N THERE IS A PROBLEN THAT TRUTH OF NEGATION OF N THAT IS ~ N IMPLIES TRUTH OF N. THIS IS NOT POSSIBLE RATHER ABSURD. IF ~ IS TRUE THEN P NECESSARILY FALSE AND IF P IS TRUE THEN ~P IS NECESARILY FALSE. THERE IS NO TRUE STATEMENT SUCH THAT TRUTH OF ITS NEGATION IMPLIES IMPLIES THE TRUTH OF IT.
SO IN THIS CASE
N1 =/= "Nobody knows that N is true IS A STATEMENT.
N1 = "Nobody knows that N is true , N1=/=N IS A STATEMNT. WHAT IF TRUTH OF ANY STATEMENT SAY P IMPLIES TRUTH OF ITS NEGATION OR TRUTH OF ITS NEGATION IMPLIES THE TRUTH OF THE STATEMENT. IN GENERAL IF P----> Q AND P IS TRUE AND Q IS FALSE THEN P---->Q IS FALSE. IF P IS FALSE AND Q IS TRUE THEN P---->Q IS TRUE. . IF BOTH P AND Q ARE TRUE THEN P--->Q IS TRUE AND IF BOTH ARE FALSE THEN P----------->Q IS TRUE. BUT WE KNOW THAT P=/=~Q. WHAT IF P=~Q THEN ~P---->P OR P----->~P IN THIS CASE WE CAN NOT ASSUME THE CASES BOTH P AND Q ARE TRUE AND BOTH AND Q ARE FALSE. SO IF ~P IS TRUE THEN P IS FALSE. THEN IF ~P--->P THEN ~P---->P IS FALSE. IF ~P FALSE AND P IS TRUE THEN ~P---->P IS TRUE. BUT IF IT IS SUPPOSED THAT TRUTH OF ~P IMPLIES TRUTH OF P THEN THERE IS SO SUCH STATEMENT AND IF THIS IS A TYE OF STATEMENT THEN SOME NEW LAWS MAY BE MADE FOR THEM. IF ~P IMPLIES P THAT IS TRUTH OF ~P IMPLIES TRUTH OF P OR TRUTH OF P IMPLIES TRUTH OF ~P THEN BOTH P AND ~P ARE FALSE. THERE FORE THESE SENTENCES HAVE NO TRUTH VALUES AND THEY MAY BE TERMED AS STATEMENT-OIDS INSTEAD OF SENTENCES. A SENTENCE MAY BE CALLED AN STATEMENT IF IT SATISFIES THE FOLLOIWING AXIOMS. 1] THE TRUTH OF NEGATION OF IT DOES NOT IMPLY THE TRUTH OF THE STATEMENT. 2] THE TRUTH OF THE STATEMENT DOES NOT IMPLY THE TRUTH OF ITS NEGATION. 3] IF THE STATEMENT IS TRUE THEN ITS NEGATION IS FALSE. 4] IF THE NEGATION OF THE STATEMENT IS TRUE THEN THE STATEMENT IS FALSE.
THERE FORE N IS NOT A LOGICAL TRUTH SINCE IT VIOLATES THE LAWS OF NON-CONTRADICTION AND CONTRDICTION. WHAT IS A LOGICAL TRUTH ANDW HAT IS A TOTAULOGY IS A PROBLEM AND A NUMBER OF DIFFERENT DEFINATIONS MAY BE SUPPOSED. IN GENERAL A TOTAULOGY IS A LOGICAL TRUTH BUT NOT ALL LOGICAL TRUTHS ARE TOTAULOGIES . BUT THE GIVEN STATEMENT IS NOT A OGICAL TRUTH SINCE TRUTH OF ITS NEGATION DOES IMPLY THE TRUTH OF ITS AFFIRMATION AND THE CONDITION 2 IS VIOLATED. IF N1= (ﬠN) IS A STATEMENT SUCH THAT IT DOES IMPLY ~N1--à N1 THEN IT IS NOT A LOGICAL TRUTH AND HENCE NOT A TOTAULOGY. IN OTHER WORDS LET ~N1 =(ﬠX) [SUPPOSITION] ~N1 --àN ASBURD/ CONTRDICTION. THEN N1=/=(ﬠX) ONE MAY USE LOGIC OF EXCEPTION TO SHEW THE FAALCY IN THE ARGUMENT. THAT IS ‘‘ N1 = "Nobody knows that N is true EXCEPT G-D’’ AN OTHER EXAMPLE CONSIDER THE NON ATOMIC STATEMENT PΛ~P IT HAS NO TRUTH VALUE. BUT IF ~P--àP THEN IT HAS A TRUTH VALUE. THIS VIOLATES THAT TRUTH OF NEGATION OF P DOES NOT IMPLY P. A FUNDAMENTAL RULE..
Unfortunately, there are further logical examples that seem to undermine even this restricted definition, such as the following one (called "The Strengthened Divine Liar"): B = "God does not believe that B is true" If B is true, then God (or any other person) does not believe that B is true and thus doesn't
\forall p((p \land \Diamond Kxp) \Rightarrow Kxp)
know that B is true. Therefore, if B is true, then there is a truth (viz. "B is true") which God doesn't know. And if B is not true (= false), then God falsely believes that B is true. But to believe the falsity that B is true is to believe the truth that B is not true. Therefore, if B is not true, then there is a truth (viz. "B is not true") which God doesn't know. So, in any case there is a truth that God does not and cannot know, for knowledge implies true belief. While sentence N is a non-knower-relative unknowability, B is a knower-relative unknowability, which means that our concept of omniscience apparently needs to be redefined again: RESPONSE:THE BELEAVING IS A PROPERTY OF A HUMAN BEING AND NOT OF GOD. IT IS OMNISCIENCE THAT IS THE PROPERTY OF G-D. BUT THE STATED ABOVE ARGUMRENT MAY BE USED TO REFUTHE THIS ARGUMENT AGAINST OMNISCIENCE OF G-D AS WELL. IT IS NOT THAT THE CONCEPT OF OMNISCIENCE IS TO BE REDEFINED BUT IT DOES MEAN THAT THE CONCEPT OF A STATEMENT MUST NOT VIOLATE THE LAWS OF CONTRADICTION AND NON CONTRADICTION. LAW OF CONTRADICTION= IT IS ABSURD /IMPOSSIBLE A STATEMENT AND ITS NEGATION BOTH ARE TRUE . LAW OF NON CONTRADICTION:- IF ONE OF THE TWO CONTRADICTORIES IS TRUE THEN THE OTHER ONE IS FALSE4.
"MORE RESURCH SHALL BE GIVEN LATTER. INSHAA ALLAH