Copyright © James R Meyer 2012 - 2017 www.jamesrmeyer.com
Over the course of many years, various philosophers have been troubled by various aspects of natural language that they consider to be difficulties, and they have attempted to solve these difficulties by applying formal logic to various statements of natural language. Judging by the amount of material on this issue that is still being generated for various publications, the story of the attempts of various philosophers to solve these perceived difficulties is a story of dismal failure.
But it is easy to show that these perceived difficulties only arise if there is a belief that each of those statements should have a common universal “truth-value” (Footnote: Conventionally the “truth-value” of a proposition is a value according to whether that proposition is “true” or “false” (if it can have such a value). Conventionally, the terms “true” or “false” are considered not to be equivalent to the terms “provably correct” and “provably incorrect”, but see the notion of truth for why this is a Platonist assumption which lacks any logical foundation.) that is completely independent of the person who is making the statement. But if we look at natural language as a complete self-contained system, we can see that each individual person has their own unique natural language system. This means that the same statement can have a different meaning and a different “truth-value”, depending on the person who is saying it.
The approach taken by most philosophers is that natural languages are in principle fundamentally quite different to formal languages. But I think that this is a mistake, and that if we admit that:
then many of imagined problems associated with natural language disappear.
A formal language system can be said to consist of three principal parts: an alphabet, a grammar (a set of rules that determines which expressions are valid), and a set of assumptions (axioms) from which “truth-values” of expressions can be calculated and a set of rules (rules of inference) for deriving one sentence from another. (Footnote: There is a Platonist notion that the “truth-values” of propositions are somehow independent of these essential components of a formal system, but that is another matter, which will not be dealt with here - but see Platonism for more on that.)
Bertrand Russell attempted to tie down statements of natural language so that every statement of natural language would have a fixed definite “truth-value” - that is, that every natural language statement (if it is meaningful) would either be “true” or “false”. (Footnote: Russell, Bertrand. “On Denoting” Mind 14.56 (1905): 479-493.
With modern typesetting )
One of the sentences that Russell considered was ‘The present king of France is bald.’ When Russell wrote this there was no king of France, and Russell thought that this created a big problem. Because if we say that the sentence ‘The present king of France is bald’ is false, we seem to be saying that there currently is a king of France who isn’t bald. But we don’t actually mean that because we know that there isn’t a king of France at this time.
Russel’s attempt to give this sentence a definitive “truth-value” was like this. He claimed that the statement ‘The present king of France is bald’ actually includes the implicit assertion that there currently is a king of France. So his treatment is to claim that the sentence:
‘The present king of France is bald.’
is actually stating:
‘There exists a thing, and only one thing, that is the present king of France, and that thing is bald.’
Given the second sentence, since there does not exist a thing that is the king of France, Russel concludes that the “truth-value” of the sentence (B) is “false”, and therefore the “truth-value” of the sentence (A), which is:
‘The present king of France is bald.’
is also “false”.
But whether one accepts Russell’s treatment depends completely on whether you agree that the two sentences are interchangeable. Certainly Russell’s treatment is not the only possible one, and I cannot agree with Russell’s treatment at all. Consider the sentence:
‘The hexagonal table that I had designed for my living room but was never made, is something that I still have a memory of.’
The main assertion is that the table is something that I have a memory of, the part about being designed but never made is a description of the table, as well as the physical description that it is hexagonal. If we follow Russell’s analysis, then the sentence is equivalent to the sentence:
‘There exists a thing, and only one thing, and that thing is a hexagonal table that I had designed for my living room but was never made, and it is something that I still have a memory of.’
Now, the table does not physically exist any more than the King of France exists, but the sentence (C) is perfectly valid and has a valid “truth-value” of “true” whenever I state it. But if I were to say the sentence (D), that sentence has a “truth-value” of “false”, since the table does not exist. Clearly Russell’s simplistic analysis does not capture the essence of what is happening, and the two sentences are not equivalent.
The clue to resolving the problem is to perceive that the sentence (C) does not have a simple singular fixed “truth-value” - its “truth-value” is actually completely dependent on whoever is stating the sentence. If the person stating the sentence did have a hexagonal table designed, and if they still remember it, then the “truth-value” of the sentence is “true”. On the other hand, if the person stating the sentence never had a hexagonal table designed, or if they did have, but have totally forgotten it, then the “truth-value” of the sentence is “false”.
For both sentences:
‘The present king of France is bald.’
‘The hexagonal table that I had designed for my living room but was never made is something that I still have a memory of.’
there is an assertion about a specific entity. The mistake that most logicians/philosophers make is to concentrate on what the physical thing is that they think might correspond to that specific entity of the sentence. But their notion of that correspondence is a red herring, and it leads them to ignore what the person’s conception of that specific entity is. If that specific entity does not in fact exist as a valid entity within that person’s language, then the expression would not be logically valid in that person’s language and would not have a true or false value on that account - it would not be a meaningful sentence in that person’s language. In principle this is no different to the case where I am working in a purely arithmetical mathematical language, and then I write “The chair > 7”. Here “The chair” is not an entity that exists in arithmetic, and so the correct response is that the phrase “The chair > 7” is not a valid expression of arithmetic since “The chair” is not an entity of arithmetic. And so the expression is not a meaningful expression of that language, and so it does not have a “truth-value” in that language.
In the same way, ‘The present king of France is bald’ is not a valid expression of a natural language if the King of France is not a valid entity of that particular language system. For a language system that corresponds to reality, the entity “The King of France” does not exist, and so the expression is not valid in that language system.
But why would anyone think that the sentence ‘The present king of France is bald’ must necessarily belong to a language system in which there is no entity which is the king of France? It is absurdly easy, for example, to conceive of a play in which there is a king of France, and so the language system of an actor in the play can include an entity that is a king of France, and in fact, the language system must include that entity.
The common error is to make the assumption that any language system in which expression ‘The present king of France is bald’ occurs must correspond to physical reality. No cognizance is made as to whether the expression has an innate “truth-value” in the language system it actually belongs to. In the case of fully defined languages, such as a formal mathematical language, the “truth-values” of an expression depend only on the complete definition of the language, and not on any physical properties of the objects of the language.
Let us now suppose a person who has no knowledge at all of French monarchy is told the sentence: ‘The present king of France is bald.’ That person could then have a mental concept of a King of France who is bald. That person believes what he has been told, and that it is an accurate picture of reality. Now, if one is to ask that person the question as to what is the “truth-value” of ‘The present king of France is bald’, it is illogical to expect that that person can provide a “truth-value” that has any meaningful relationship to the physical real world. The sentence is an expression of the language system of that one particular individual, whose underlying assumptions may not coincide with those of others. Therefore, for that particular individual and his particular language system with its attendant underlying assumptions, the statement ‘The present king of France is bald’ is a true statement – it is “true” within his own language system with its own realm of underlying assumptions. Of course, I use the term “true” here as signifying that, given that person’s set of what might be termed axiomatic assumptions of his language, that sentence can be given a “truth-value” by deduction from those assumptions.
Of course, that person’s language system does not correspond precisely to reality, and the term “true” as used here does not signify any correspondence to physical reality. But, then again, who does have a language system that corresponds precisely to reality? We all make assumptions about reality. With natural language, there is a myriad of different language systems, each person having their own language system. Of course, unlike a mathematical language, such language systems are constantly changing over time, and the assumptions that we hold change over time. But that does not affect the underlying analysis. And, for a sentence such as ‘The hexagonal table that I had designed for my living room but was never made, is something that I still have a memory of’, the interpretation of the sentence is completely dependent on the language system of whoever is stating the sentence.
Another favorite subject for philosophers to get themselves twisted into knots over is the question of whether a person believes that a single thing is actually two completely separate things. A common example that is used is Lois Lane, who believes (at least initially) that Superman and Clark Kent are two different people. But once you see that Lois has her own language system in which her underlying assumptions are that Superman and Clark Kent are two different entities, then there simply is no problem, and no difficulty to discuss. The fact that she does not have the knowledge of the real situation, and that her world picture does not correspond precisely to reality is incidental - it is a triviality.
Of course, Lois Lane is a fictional character, and here we are contrasting her “world-view”(the beliefs that she has) and the fictional “reality” that she lives in. But the principles are the same, in that in our reality (the real world that we live in), people can have “world-views” that may not correspond to that reality, and so the truth-values of their statements depend entirely on that “world-view”, and there is no reason why we should expect that their statements should correspond to the real world situation.
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There is now a new page on Lebesgue measure theory and how it is contradictory.
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It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
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There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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