Many may consider one of the primary distinguishing features that makes humanity unique within the kingdom animalia is our use of language. It’s capability to offer an infinite variety with a finite set of sounds and words makes it quite the marvel from evolution and human capabilities.
However, within it and us there are two conflicting goals that cannot be reconciled.
- Precision of language, where we want to supply precise, exact, unambiguous information to our listener.
- Economy of language, where we want to minimize the number of syllables, sounds and everything while providing information.
To have complete one means sacrificing the other completely, we humans make a compromise where we take a bit of both because in most instances when ambiguity is introduced, the context itself is enough to salvage the precise meaning. The few instances when it is not, and we made the erroneous assumption that it is enough, the other bloke will ask a clarification question, answer is given and the conversation continuous on.
In normal speech it is adequate for virtually all cases, in text however, we tend to lean more toward Precision of language. This is because, while there is a certain level of context given by the text, it is not nearly as much as if we had been discussing with each other face to face but more importantly, when an ambiguity arises there is no way for the reader to ask for clarification and they are stuck attempting to work through multiple interpretation of what the author might mean and hope they find the correct one.
And still, no matter how much one might try, the precision is never enough to eliminate this issue because a text that was that precise in normal natural language would be unwieldy,difficult to read and quite frankly, it’d be exceptionally boring to read cause it would not feel like the product of a human being as it goes against everything we know how a human would express themselves even in the most formal manner. The way natural language works is so ingrained into our minds that we essentially require it to express ourselves and be comfortable.
In most cases, this ambiguity is not an issue, after all the text written by ancient Greeks were highly ambiguous, much more than anything we do today and took five to ten times reading a text before understanding it. However, when tiny nuance changes have fundamental effects on the meaning and further processes, this miniscule ambiguity will cause issues. In mathematics this the exceptionally important due to subclauses. A classical example is
One morning I shot an elephant in my pyjamas
Of course by context of knowing what an elephant is, we know it is not the elephant wearing my pyjamas, but if you are not familiar with what an elephant is, the pyjama might very well be on the elephant cause there is no reason why it should not be able to wear it when you do not know what it refers to. In mathematics, we often use the same type of objects within the same discussion so this kind of subclauses is highly ambiguous, but again if we do not use them in a natural manner, it will sound like it is not human and feel unreadable.
This is what formal languages exists to solve. Formal languages are not languages per say because they contain nothing we humans would recognise as a language, no sounds, nothing to speak or the likes and they lack semantic meaning in them. In ordinary language, words meaning is dependent on each other in a circular sense and meaning is assigned through usage but a formal language lacks all forms of meaning as we recognise it.
Why they are called a language though is that it shares some properties with a language.
- Collection of symbols – This can be said to be equivalent to a natural languages word but unlike a natural language, these have no meaning and are just symbols which can be represented as symbols on a paper, binary sequences, strings of sound or whatnot.
- Rules of composition – A sequence of symbols may not be of any kind, most sequences of symbols are forbidden and there are only some that is allowed. This can be said to be equivalent to natural language of grammar. Unlike a natural language, cause it assigns no meaning, any string that does not violate these rules is valid. These rules are strict and clear and all sequences can be determined to be valid or not.
The last one is important because in English we have for example
Colourless green ideas sleep furiously.
which is a legitimate string under English grammar and words, Rules of composition and Finite set of symbols, but anyone knowing English would reject this sentence outright because it is semantically non-sense. This restriction might be seen and considered a rule to add into our Rules of Composition, but because it is semantically grounded, this rule cannot be precise. As speakers of English, or any language with many dialects, can affirm, what is semantic non-sense in one dialect might be semantically valid in another, despite the grammatical rules remaining identical, so we get ambiguity in the rules as everyone uses different rules of composition.
One important thing to notice with all of this is that, a formal language is ultimately just symbols with some rules on how to put them together. We humans however still utilizes natural languages and as such, we often assign words and ways to say things in the formal language. That is however not a property OF the formal language, but a necessity of us humans so we can EXPRESS a formal language in a way that is befitting of us. Most formal languages are so rigid in rules that even a computer is capable to determine its validity. We humans might use words that are ambiguous to represent the symbols and order and everything within, but these words we choose to use and their inherent meaning have no affect or impact on the formal language. We do often choose words to fit the general property of the symbol within the formal language and how we would call it in our natural language, but that choice is ultimately entirely arbitrary.
More advanced formal languages add additional rules and properties to the language but those two properties are fundamentally what defines them.