# Truth & Falsehood of symbol sequences

Every theorem of mathematics is True within the scope of a higher level ‘True’ concept. That higher True concept is the concept of the meaning embedded in the symbols used to define that Theorem. For e.g.

‘Parallel lines do not meet in Euclidean plane’

We interpret the meaning embedded in the statement above to be true without giving it a second thought. If the symbols in the above statement are not from the English language then the sentence may mean something very different. And THAT NEW meaning may be True or False. So, that means the meaning of a statement depends on ‘fixing’ the meaning of symbols of a particular language and depending on the ‘fixation’ the statement may mean Multiple things!

Now, let’s consider the meaning of the above statement. If you imagine the ‘Parallel lines’, Euclidean plane to be what they mean as per our understanding with our English language meaning set of symbols, then the lines themselves are some sort of ‘symbols’ - like infinitely long lines- in an expanded meaning of the word Symbol. Call that 'SYM1' set to represent the symbols( the lines, planes, etc., which are infinite by nature) at one level above the current meaning set in English! Now, when we say that the statement above is true in English then we are essentially describing how to draw or write two symbols from SYM1 - two continuous lines with equal distance between them at all the time. To a SYM1 reader( say with MEAN1 meaning set) it looks like 11 written beautifully. The number 11 is the meaning set in English or Math in our world so we cannot apply it for SYM1. Or two bars. But what does that mean when you look at finite parts of it from above? The meaning associated depends on an observer’s meaning set- like MEAN1 - if it exists - for those lines. But what we with our statement assert is that the existence of those lines is True, provided, we apply the meaning of the word ‘parallel’ from our English world. The meaning of the word ‘Parallel’ is an axiom. We are ‘fixing’ the meanings from our world & defining higherlevel Theorems arising out of those meaning set.

What happens when we fix a different meaning to the same symbols?

Many might turn out to spill out Untrue statements.. some might turn out to be true on the new meaning set. In some sense then every ‘thing’ in our world is kind of a symbol / Theorem arising out of lower level particle interaction axioms..whose meaning is embedded possibly at the quantum level.

Doesn’t it convey mathematically, that there is No absolute Truth / Theorems & all of the so called Theorems are based on ‘Fixing’ a meaning set for symbols at lower & lower levels? Then doesn't Godel's Incompleteness apply to more than the Formal Systems?

• Yes, formal language is made of symbols that we can interpret in different ways. An Intepretation of a formal system is mathematically defined and is a way to give meaning to symbols and formulas (statements). The truth value of a formula may be different in different interpretations. Commented Feb 3, 2022 at 6:54
• For natural language it is not so: the meaning of words is more or less fixed: shared meaning is necessary for communication. Commented Feb 3, 2022 at 6:55
• Nothing to do with Gödel’s Incompleteness Theorems Commented Feb 3, 2022 at 6:55
• Thanks...Then we can theoretically temporarily change the meaning set for symbol sequences, compute new Theorems in the new meaning set & reinstate original meaning set to look at resulting statements & see if they are valid in original meaning set..much like Godel did correct? And, are there theorems describing properties of such "reparametrization" of meaning set?...Thanks a lot Commented Feb 3, 2022 at 17:11

The question has an affinity with a philosophical debate revolving around the proper characterisation of logical consequence. The technical basis of the debate is the two options identified to evaluate formal expressions.

Keeping Alfred Tarski's The Concept of Truth in Formalised Languages and On the Concept of Logical Consequence in view, the two options can be described as follows:

• Interpretational semantics picks out for each non-logical term $$x$$ the semantic value it would be admissible to have in the actual world $$w_0$$ in accordance with its possible meaning in a language $$\mathcal{L}$$. Hence, the occurrence of $$x$$ is interpretational.

• Representational semantics picks out for each non-logical term $$x$$ the semantic value it would be admissible to have in some possible world $$w_i$$ in accordance with its actual meaning in a language $$\mathcal{L}$$. Hence, the occurrence of $$x$$ is representational.

To illustrate the discrepancy, I borrow an example from Stephen Read's Formal and Material Consequence. Read gives an argument that can be formalised as

$$\forall x(Fx\rightarrow Gx), \exists x(Gx\wedge Hx)\vdash\exists x(Fx \wedge Hx):$$

All cats are animals.

Some animals have tails.

So some cats have tails.

The premisses and the conclusion are true, however, the conclusion does not logically follow from the premisses. We can demonstrate this by reinterpreting $$Hx$$ as '$$x$$ is a dog' instead of '$$x$$ has a tail':

Cats are animals.

Some animals are dogs.

So some cats are dogs.

Hence, on the interpretational reading, $$x$$ guides us to a class of interpretations (of the sentence).

We can reach the same judgement by evaluating the statements in a different possible world also. As Read's example goes, we may consider a world $$w_i$$ in which the word 'cat' represents cats which do not have tails. On the representational reading, $$x$$ guides us to a certain representation (of a situation).

Both options have their pros and cons. Model-theoretically speaking, both of them are methods of seeking satisfiability of formulas. A noteworthy point is that satisfiability is an absolute property of a formula; a property it would have, once a model of it existed, whereas truth is a relative property, which can be judged in reference to an interpretation (recall that a formula $$\phi$$ is a logical truth if and only if it is true under all interpretations).

The debate has interesting issues on its own. Relevant to the present context, what it makes manifest is that we have a broader than, I would say epitheoretical (in contrast to object-theoretical and metatheoretical), understanding of meaning and truth, beyond semantics supplied (by whatever method) to formalisms, that decides on what is acceptable. That is, the locus of the judge is not in the atomic constituents, but much higher above.