Language and Logic
Every argument, every logical proof, is stated in some language. Every language includes variables – terms that are general terms which indicate a range of things, rather than one single thing. For example, the word ‘city’ does not represent any one particular city; when we talk about ‘a city’, for example, in the sentence ‘John is travelling to a city today’, we know that we are not referring to any one specific city, but we have a pretty good idea of what sort of place John is travelling to. If we find out that it is actually Paris, then we could say, ‘John is travelling to Paris today’. When we use the words ‘a city’, we can replace those words in a sentence like the one above by the name of a city, such as Paris, or London, or Lisbon. All the things that the general term ‘city’ refers to are cities - this is called the domain of the variable - the domain of a variable is the set of all values that the variable can take; so the domain of ‘city’ includes every city in the world.
Many errors of logic arise from a misuse of variables, as we shall see below. Some sentences do not refer to anything else other than cities. Such sentences make statements where the only things that are referred to are cities. For example, we could have the sentence, ‘Paris is bigger than Oslo’. This sentence expresses a relationship between two things, the two things being cities. If we replace ‘Paris’ by ‘The city where the Eiffel Tower is located’, we get another perfectly sensible sentence: ‘The city where the Eiffel Tower is located is bigger than Oslo’.
This new sentence, of course, still refers to two cities; but it refers to Paris indirectly, by way of reference to a thing that is not a city – the Eiffel Tower. So, as well as this sentence referring directly to one city – Oslo – and to one city indirectly – Paris – it refers directly to a thing that is not a city – the Eiffel Tower.
Now, if you claimed, simply because ‘The city where the Eiffel Tower is located is bigger than Oslo’ refers to two cities, that the sentence refers only to cities and not to any other thing, other people would question your intellect. Yet that is the sort of claim that occurs in many proofs that are claimed to be completely logical proofs - as will be seen below.
How some logically invalid proofs arrive at their conclusion
A key point in such proofs is the fact that certain expressions that the proof refers are expressions about numbers, and only about numbers, and not about anything else. In other words, the things that these expressions refer to are all numbers, and they never refer to any other things. These expressions can refer to numbers in two ways – either by referring to a specific number, such as 89, or 4976 – or by referring to numbers in a general way, by using variables that have the domain of numbers.
Now, when the proof includes some such expression that refers only to numbers, that expression must belong to some language. That language can include things that are not numbers, and in that language there can be expressions that are not expressions that are only about numbers, but expressions that refer to other things besides numbers.
The proof will refer at some point to an expression that is defined as an expression that only refers to numbers, and which has a variable whose domain is only numbers. In the proof that variable is replaced by a string of symbols.
This gives a new expression. Now, in the proof, according to the rules of the language, the replacement string of symbols evaluates as a number. But that string of symbols is an expression that refers to things that are not numbers.
And that means that the new expression, with the replacement string of symbols within it, is an expression that refers to things that are not numbers. That should be obvious, since the replacement string of symbols refers to things that are not numbers.
But despite this patently obvious fact, the proof continues - and claims that the new expression is an expression that refers only to numbers.
As remarked above, if anyone tried to claim that the sentence, ‘The city where the Eiffel Tower is located is bigger than Oslo’ is a sentence that does not refer to anything other than cities, they would be deemed of questionable intelligence. When the equivalent claim is made about an expression, that it is an expression that is purely about numbers, everyone nods sagely, and congratulates the creator of the expression for their depth of logical insight.
(You can read more about this type of error Errors in Incompleteness Proofs: here and there are detailed papers on several such illogical proofs, see the papers on Analysis of Smith’s proof: Smith, Analysis of Kleene’s proof: Kleene, Analysis of Harrison’s proof: Harrison, and Analysis of O’Connor’s proof: O’Connor)
Misunderstandings in Logic
The above is just one indication that there are fundamental deep-seated problems in our understanding and use of logic. These problems started to arise about 150 years ago. Before that, little consideration had been given such matters, but then various people attempted to make a logical analysis of language. It was a perfectly reasonable aspiration, but almost everyone has set off down a path that has led to confusion rather than understanding. Much of the blame for the failures to perceive the illogicality of such arguments as the example above can be attributed to a baseless belief in the ‘real’ existence of non-physical things, a belief which is called Platonism.
Platonism and its influence on mathematics, logic and philosophy
Platonism is essentially the notion that the things we write down on paper (or elsewhere) as mathematical symbols represent real things that exist in some way in some non-physical way - how this might be the case is not explained. For many years Platonists have dictated the path of progress in mathematics and logic, which has had a profound knock-on effect on the entire foundations of mathematics and logic, and a similar effect also on philosophy in general. In attempts to justify their beliefs, Platonists have engaged in some very risible arguments, such as the Indispensability Argument.
Because Platonists believe that whenever they are talking about numbers (and similar things), they are referring to ‘real’ things, they tend to ignore the salient fact that, regardless of their belief in these ‘real’ things, they are using symbols of some language. In a logical argument, whatever language it might be in, if the manipulation within that language is not logical, then the result will not be logical, regardless of any belief in the ‘real’ non-physical existence of whatever is referred to by that language.
How logical is mathematics?
It is commonly held that mathematics holds a special place in the world of logic, and that mathematical proofs are the most perfect examples of precisely logical argument. That may be the perception, but the reality is quite different. This is discussed in Pseudo-mathematics and Mathematical Proofs.
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