Problem of a Paradoxical Sentence. There is a sentence which is stated as follow: N= NO ONE KNOWETH THAT N IS TRUE. This Assertive sentence is paradoxical since if it is true that No one knoweth that it is true then it becometh false since those who know that it is true prove that it is false. If it is false then it implieth that its negation is true. But its negation implieth that the original Declarative Assertive Sentence is True. A sentence which is true in all its values is true. So the question is that if it is false then it is true. If it is true then a number of persons do know that it is true. This implieth that it is false. Now the problem must be analysed according to the Laws Of Thought. 1] Law of Contradiction or Law of Non Contradiction says that A Sentence is either true or false but not both. If a sentence is true then its Negation is false, Ithe Sentence is false then its negation is true. It may be seen that its negation is false. Proof: Suppose that its negation is true. Then its negation is Not N= Some do know that N is True. But this implieth that N is true. So if Not N is True then N is True. This implieth that its Negation is false. Since Not N violateth the Law of Non Contradiction. So Negation is false since Any Sencetence which implieth its own Negation is false. For Example “A is B” Implieth “A is Not B” then “A is B” is false. Similarly if “A is Not B” implieth “A is B” THEN “A is Not B” is false. So Not N is false. Now the question is whether N is false or true. It is a Paradoxical Sentence. Its Truth Implieth its Falsehood. If a Sentence is false then no one knoweth its truth. So either this sentence is true or it is not true.
If this statement is true then its truth is known by some knowers, and this implieth that it is false. If it is false then it is implied that it is true. So the statement is paradoxical. Since no one knoweth the that a false sentence is true. So the sentence is false. Since if it is true then it is implied that there are some Knowers who do know that it is true. This means that it is not true. If it is not true then it is true since then it is true that know one knoweth that it is true. To such sentences one must know that a false sentence is one that is not true. But the meaning of truth and falsehood may be redefined. There are two meanings of Truth and Falsehood. 1] Paradoxical Truth or Paradoxical Falsehood. 2] Non Paradoxical Truth or Non Paradoxical Falsehood. Paradoxical Truth. If Truth of a sentence directly or indirectly implieth its falsehood then it is false. For example truth of a sentence implieth some thing which implieth the falsehood of the sentence then the sentence is false. Consider the case of this very sentence itself. It is said as follow: If N is false then Not N is True. If Not N is true then Not N implieth that N is True. A sentence which hath all true values in all cases is Logically True. A logically True sentence is True since Logically True is a special case of True. But as it is said that this sentence is Logically True hence True, its truth is known and exposed. So any one who knoweth that it is true in all cases knoweht it is Logically True and hence true. This implieth that it is false. So this sentence is Paradoxically True and Paradoxically false. Additionally its negation is false. It is some thing of the form that a sentence is Paradoxically True and Paradoxically false whether its negation is true or false or Paradoxically True and Paradoxically False.
S sentence may be self contradicting while its negation ,may not be so. But it is also the case that both the sentence and its negation are self contradicting. Real True and False. A sentence is true if it is not Paradoxically True and Not False. A Real False is one that is not real true. So the case that sentences like “Negation of this very sentence is True” , “This sentence is true if and only if it is false” etc. are paradoxically true. One may rename them as Pseudo True and Pseudo Falsehood. In such cases the law of Non Contradiction implieth that it is applicalble to Real True and Real False sentences and not to Complex True or Complex False Sentences like follow: “Now one knoweth that some Self Contradicting Sentences are True” Its Negation is : Some do know that some self contradiction sentences are true. But they both are equally false.