THE MYSTERY OF LOGICAL PARADOXES SOLVED.
(1.) In general all existing analyses of logical paradoxes show only their practical shortcomings: they are meaningless, contradictive in itself for different reason and so on. If we eliminate those disadvantages we will solve the problem. One does not have to be a philosopher to uncover these shortcomings in the logical paradoxes, it is enough for a person who has a common sense. The scientific (theoretical) approach in relation to logical paradoxes begins only then when philosopher successfully explains the reasons for the existence of these shortcomings by using categories and Laws of Formal Logic. And this can’t be done by an ordinary person but by philosopher who know and understand the Laws of Formal Logic. There was attemp to solve the problem from the point of view of Laws of formal logic but it was incomplete because they ignore the law of sufficient ground. See web "Are Mysteries of the Christian Faith Against Logic? pt. 2"
(2.) According to the opinion of many philosophers the cause of these shortcomings existing in logical paradoxes, could be the problem with formal logic or problem with our language (may be something else?). For these reasons the solution of logical paradoxes was transferred from the frame of Formal Logic to the frame of mathematical logic (for example Russell) and to the frame of classical formal languages (for example Starski). To look for the solution of that problem with help of mathematical logic or with help of classical formal languages is not logically at all, because they are both based on the Formal Logic and consequently both would inherited shortcomings of formal logic if the later had ones. (About the later I doubt very much). Impossibility to solve the problem within the frame of formal logic show just one thing that philosophers don't understand the way how the formal logic works (another paradox). For this reason this article has been rejected by journal "Studia Logica" and "Notre Dame Journal of Formal Logic". Scientific explanation of existence of logical paradoxes must be done within the frame of Formal Logic and, for the first time, this is done in my article “The Mystery of Logical Paradoxes Solved”.
In this article, we will consider paradoxes in the narrow sense of the word and prove their logical groundlessness. Paradox in the narrow sense of the word is defined as two opposite, incompatible statements, for each of which there are convincing arguments at first glance. An example of paradox: "I am lying" and "This statement is false;" does the Liar tell the "truth" or a "false”? In the broad sense of the word, paradox is defined as a statement that is in conflict with conventional, universally recognized opinions, but this is not the subject of our investigation. For any science, whether it is physics, mathematics etc., there are specific laws that are discovered and supported by experiments. If, for example, a man wants to create an airplane, then it must be in agreement with the laws of aerodynamics, or else the aircraft will not fly. If a man wants to launch a satellite around Earth, he must be familiar with the law of gravity in order to overcome the gravitational force give the satellite an acceleration of about 8 km/sec. If the man fails to accelerate the satellite, it will fall down to Earth. In another example, we want to calculate the value of Y under the given conditions: 20 + 10 / 2 = x, x+15 = y. According to arithmetic rules, we first divide and then add to yield the result x = 25. We then put this value into the second equation, from which we get the value Y =40. However, if I do not follow the arithmetic rules in the first equation and add first and then divide, I will get the wrong result for X: x =15. I can not substitute this result into the second equation and expect the correct value for Y. Instead, result for the value of Y will be wrong: Y = 30.
These arguments also apply to the science of formal logic, which took on its realized form when Aristotle and Leibniz discovered its laws, as follows:
1. Law of identity.
2. Law of contradiction.
3. Law of the excluded middle.
4. Law of Sufficient Ground.
These Laws were discovered based on the practical needs of mankind. If people reasoning followed these Laws, they would not have had any problems with formal logic. Once they break at least one of these laws, the use of formal logic in their reasoning loses all sense, and the result of their reasoning becomes contradictory.
The best example of this case is the logical paradoxes that came into existence in ancient Greece and created a basis for the birth of a new paradoxes later. Before analyzing paradoxes, I would like to note that up to this time, the logical solution to that problem was still in its embryonic stage. "In the last sixty years there were hundreds of book and articles were devoted to solve these paradox, yet the results are striking poor in comparison with the efforts spent."
Let us consider an expression "This statement is false.". It can be traced back at least as far as Eubulides of Miletus, a fourth-century B.C. Greek philosopher who, according to history, was the first to create the Liar paradox. If a man says that "This statement is false," is he telling the truth? If we assume that what the man says is true, then we end in contradiction: if the man claim that he is lying is true, then he is lying, in which case what he says is false. If we assume that what the man says is false, then we are no better off: if the man claim that he is lying is false, then he is not lying, in which case what he says is true. Both answers bring us to logical contradictions; it cannot be the case that either what the man says is true or what the man says is false.
Philosophies of all times have tried to find solutions to this paradox. One of these solutions deemed these expressions meaningless,and it is this lack of meaning that leads to contradictions. "Some of the solutions to the Liar Paradox require a revision in classical logic, the formal logic in which sentences of a formal language have exactly two possible truth values (TRUE, FALSE), and in which the usual rules of inference allow one to deduce anything from an inconsistent set of assumptions." (The internet the Encyclopedia of Philosophy).
When, for the first time, we say the words "This statement," we proclaim that at this moment, there was no statement; we deal with a form that has empty content, which one can fill with many different contents. From here follows that "This statement" is not defined uniquely and therefore is in contradiction with the first law of formal logic (The Law of Identity).
When we finish pronouncing the expression "This statement( ) is false", "false" defines the form without content. For visual demonstration of this fact, I included the emptiness in parentheses. What is false? One can note that "false" defines empty content "This statement( )". In other words, the definition of "false" is also not defined uniquely and is in conflict with the Law of Identity.
As we saw, the sentence "This statement( ) is false" is meaningless when we say it for the first time. Let us fill it with definite content. "The Sun rotates around the Earth". "This statement is false". In the given example, "This statement" has, as its content, the first sentence. For this reason, "This statement is false" is transformed from a meaningless to a meaningful sentence.
Let us continue the investigation of this expression from the point of view of the Law of Sufficient Ground (i.e. in relation to what sufficient ground we define "True" and "False"). Any sentence is a form but what it states is its content. Hence, we have here two sufficient grounds in relation to which "True" and "False" are defined. When we say that this expression is "True," we define it in relation to the form (first sufficient ground); that is, to the sentence "This statement is false". On the other hand, with respect to the content of this sentence (second sufficient ground), "False" is originally defined in relation to empty content "This statement( )." However, when we say for the second time that this statement is true, the empty content disappears: "This statement ("This statement is false") is false". This statement refers to itself and the empty content in parentheses is thereby replaced by the sentence "This statement is false". Now, with respect to the content of this sentence (second sufficient ground), the "False" is defined for the sentence "This statement is false".
As a result the sentence "This statement is false" is not uniquely defined as "True" and as "False", because the first time it is defined in relation to the form and the second time it is defined in relation to the content of this form. By doing this way we break the laws of formal logic (the Law of Contradiction and the Law of Sufficient Ground). It follows that we cannot apply formal logic to the expression "This statement is false" to figure out whether a man tells the truth or lies, because the expression itself breaks the laws of formal logic.
Let us consider another expression, "I am lying". In the content of this sentence (first sufficient ground), "lying" is defined in relation to any sentence: "The sun rises from the East", "This book is interesting" etc., including the phrase "I am lying". When we say that a sentence is "True," "True" is defined in relation to the sentence (second sufficient ground). As a result, the sentence "I am lying" is not uniquely defined as "True" and as "False", because the first time it is defined through content and the second time it is defined through form. By doing this, we break the Laws of formal logic (the Law of Contradiction and the Law of Sufficient Ground). Formal logic must first of all be based only on one sufficient ground in relation to which all meanings are defined, and only under this condition can one escape the problem of breaking the other 3 laws of formal logic. Only under these conditions does formal logic not become contradictory but leads to correct results.
Another paradox that has its foundation - real or legendary - in antiquity concerns the sophist Protagoras, who lived and taught in the fifth century BC. According to the story, Protagoras made an agreement with one of his pupils Euathlus, stipulating that the pupil was to pay for his education in Law only after he had won his first case. Evatl completed his course and did not practice as a lawyer for a long time. Protagoras grew impatient and decided to sue Evatl for the amount owed him. 'For,' argued Protagoras, 'either I win this suit, or you win it. If I win, you pay me according to the judgement of the court. If you win, you pay me according to our agreement. In either case I am bound to be paid.' However, Evatl was a smart person and he replied: 'Not so. If I win, then by the judgement of the court I need not pay you. If you win, then by our agreement I need not pay you. In either case I am bound not to have to pay you.' Whose argument was right?
According to Protagoras's agreement, his pupil Euathlus had to pay him for the education, only if Evatl wins his the first case, and does not have to pay if he loses it. The condition of the pupil's payment is defined uniquely (if Euathlus will win his first case) in relation to the agreement (one sufficient ground). Hence, the definition of the pupil's payment is in agreement with the Law of Identity and the Law of Sufficient Ground. The same applies to the condition of non-payment by Evatl; it is defined uniquely (if Evatl will lose his first case) in relation to the agreement (the same sufficient ground). Hence, the definition of the pupil's non-payment is in agreement with the Law of Identity and the Law of Sufficient Ground). The situation is completely different when Protagoras filed suit against Evlat for non-payment.
1. In the situation in which Evatl loses the case against Protagoras and by the judgment of the court will be forced to pay dues, the second sufficient ground appears (judgment of the court), in relation to which Euathlus's payment to Protagoras is defined. It is not difficult to notice that this judgment contradicts the condition of Evatl's payment defined in the original agreement (if Evatl wins his first case).
2. In the situation in which Evatl wins the case against Protagoras and by the judgment of the court will not have to pay dues, the second sufficient ground appears (judgment of the court), in relation to which Evatl's non-payment is defined. Once again, this judgment contradicts the condition of Evatl's non-payment defined in the original agreement (if Evatl will lose his first case, he will not have to pay Protagoras). Consequently, in this situation there are violations of two laws of formal logic: Law of Identity, because payment and non-payment are not uniquely defined, and Law of Sufficient Ground, because payment and non-payment are defined not in relation to one sufficient ground but two (agreement and judgment of the court). Here appears a violation of the second law of formal logic (Law of Contradiction), but because this law is the consequence of the Law of Identity, I skip it for simplicity. We demonstrated in this example that this case violates the main laws of formal logic, and for this reason, formal logic is helpless to give us a true solution. Further, if we correct the agreement or if the court disregards the agreement (on the basis that the latter is illogical by violation of the Laws of formal logic), then formal logic will demonstrate its power in the judgment.
This paradox arises from the fact that Zeno in his discussion violates the 4th law of formal logic: the law of sufficient ground, which states that all discussions should be based on the true grounds on which concepts and judgments are defined unambiguously. In physics any rectilinear motion is described by the law, which expressed in the form S = vt, the path traversed by the body, is its velocity multiplied by the time that it spends for it's movement. Using this form, at any time we can determine the position of the moving body in the relation to the starting point. Zeno in his discussion trying to determine the position of the moving body, based on passage of the body certain parts of the way with no respect to the body's speed and timing of its motion, that is a clear violation of the law of rectilinear motion, which leads him to the wrong conclusion.
For a more accessible understanding, I bring another example, based on the same error, but that is obvious. Calculate the height of 50 year old man, if we assume that the average person for the year grow by 10 cm. According to the logic, this man in 50 years will be as tall as 50 years x 10 cm = 5 m, but in reality the height of this man will be 170-180 cm. Where is the mistake in our logic? Fatal error in our logic is that our calculations are built on false sufficient grounds which comes from the fact that a man grows (physically) all his life, although in reality a person grows up until 18-20 years, according to the law of biological development.
All logical paradoxes are based on the fact that they, one way or another, violate at least one of the laws of formal logic. Sometimes those violations are easy to find but sometimes not, because they are hidden. Here are general rules on which all paradoxes are based.
1. "True," "False" and other meanings are defined without relation to certain sufficient ground and therefore are not defined uniquely (violation of Law of Identity).
2. In one or more statements, there are two sufficient grounds in relation to which the same meanings are defined (violation of the Law of Sufficient Ground).
3. Hence, if we give formal logic (as input) meaningless information, the result (output) will be meaningless, as with examples in math.
4. Formal and dialectical logic it is the only weapon that people use to acquire knowledge, no matter whether it is theoretical or experimental discovery. If you don't trust logic generally you will contradict to yourself, because your arguments against it are based on the logic that you don't trust.
|
|
| Download a free web counter here. |