Are languages logical?

Languages usually consist of a vocabulary and grammar in its simplest form. Natural languages, in particular, have grown over years and many people with different mindsets contributed to the language. Three examples:

As a result, natural language often incorporates various structures the native speaker might not even be familiar with.

Does a language without structure exist?

Mathematics and computer science define various kinds of logic. Fuzzy logic, temporal logic and first-order logic shall be given for reference. They do not necessarily have anything in common. The foundation (for most of mathematics) is the axiom set by Zermelo-Fraenkel in combination with the the controversial Axiom of Choice (⇒ ZFC). Because formally ZFC is a one-sorted theory in first-order logic, most mathematical theorems rely upon first-order logic.

However, in computer science, you will often find that propositional logic is used as basis (due to computability issues). This subset of first-order logic is also, what people are more familiar with.

What characterizes logic? Logic is defined by a set of axioms that allow to infer new knowledge. In propositional logic, every statement is true or false. If we negate “A and B”, then we get “(not A) or (not B)”. This is stated by De Morgan’s laws. Propositional logic defines the axioms (true/false, and/or/not operators) and the inference rules are to be discovered. In the end, which axioms you define, is up to you. I have not yet discovered a math paper which defines something “for fun”, but because it is “useful” to discover new structures and relations.

We can derive a simple rule:

From a logicians’s perspective, “want coffee or tea” is true if and only if “want coffee” or “want tea” (any of the two or both) is true. This is different from “want coffee and tea” being true if and only if “want coffee” and “want tea” is true (both of the two). If my friend asks me the question upon visit, I will understand the question the same way. I will chose tea, coffee, or in an unusual case both. So from a logical point of view (POV), there is a difference but from a linguistic POW, there is none.

This is ambiguity. This is non-consistency. But is this a counter-example for structure? I think so, since the choice of word is significant in other context:

In this case, the speaker certainly wants to convey that he/she is going to pick one of them, but not both. Exchanging “or” with “and” would specify a sequence of actions.

There are exceptions in every natural language. You might have a harder time finding one in the vocabulary, because few people are into etymology, but everyone knows exceptions in pronunciation of words (think of loanwords, for example).

Let’s have a look at our previous rule “If ‘not’ occurs in a compound word, ‘not’ can be replaced by ‘n’t’”. This will be a contrived example, but it establishes an exception. The word notable could be considered as compound word (“not” and “able”). However, if we apply the rule, it yields “n’table” and “notable” as admissible words. But the former is just wrong. We could rephrase our rule (“If ‘not’ occurs in a compound word and refers to semantic negation, ‘not’ can be replaced by ‘n’t’”) or we could add an exception:

What might surprise people is that this is valid, not only from a linguistic POV, but also from a logical POV. Exceptions do not violate logic. In many mathematical theorem, you will find statements like “any, but not the same” and “greater but not equal” which model exceptions.