There are two issues at play here.The first has to do with the use of the with generic uses of noun. The second has to do with the use of the in phrasal genitive constructions.
Which unique genus?
Firstly, it's important to note that the generic use of the is actually very restricted, in that we only use it with a relatively small selection of nouns. We tend to use it with species of plant or animal. In these instances the noun must normally be in the singular:
- The Blue Whale is the heaviest animal on the planet.
- The Boabab tree is famous for looking as if it's growing the wrong way round with its roots reaching up to the sky.
We also use this with inventions in the same way:
- The mobile phone has revolutionised modern life by destroying the quality of face to face communication.
Lastly we use it with adjectives that stand in for nouns in constructions like The rich, or The French.
Notice that these uses of the are only generic in the sense that we are using the singular noun to stand in for a genus of plant animal or invention. We then use the because the genus is unique amongst other genuses of animal etc. So when we say the lion, we don't really mean lions, we mean that one unique genus of animal, the lion.
We don't tend to use singular nouns with this sense of standing in for a specific genus of thing, with any other nouns apart from animals or inventions. For example, we can't do this with prices. The following can't mean that prices have risen:
- The price has gone up every year for the last ten years. * (wrong)
With the animal example, the lion picks out a unique type of animal from amongst the entire set of types of animal. The generic use of the price here won't work because it is not clear what other entities price is being picked out from or distinguished from. Generic here doesn't mean general it means used to identify a GENUS of thing. More importantly it is used to indicate a specific genus. These Noun Phrases are not used to identify things in general. It's also important to note that it is not the definite article the here which is being used generically. It is the noun that's used generically. The the just does its normal regular job of indicating to the listener that they will be able to identify this unique thing.
Phrasal genitives
In the phrasal genitive construction Noun of Noun, we more often than not find a definite article used with the first noun. The reason for this is that the Prepositional Phrase of Y is being used restrictively, a bit like a restrictive relative clause, to explain which specific X we're talking about. The end result then is an X which, again, is uniquely identifiable by the listener. So for example, in any given conversation, if we haven't been talking about a specific CEO or a specific company, then we can't just refer to a specific CEO by saying the CEO has said that they guarantee..... By adding the Prepositional Phrase of Tesco, we restrict the possible referents of CEO to one unique and uniquely identifiable CEO:
Similarly, if we say Republic of France, we have narrowed down the possible referents of Republic to one uniquely identifiable republic, so we use the definite article to indicate that this entity is uniquely identifiable to the listener:
Price, cost, weight height
Very often we use use nouns which give some kind of measurement or numerical index, such as weight or price to indicate an average weight, or a mode weight and so forth. For example, in the phrase ...
- the weight of an adult in Australia
... we mean the notional average weight of an adult in the Australia. In order to understand how we get this reading for weight, we need to consider the genitive construction and how both of the nouns involved, X and Y are marked for definiteness or indefiniteness.
First of all, in the phrasal genitive construction, as described above, the Prepositional Phrase of Y is usually narrowing down the possible range of intended referents of the term X, to one identifiable referent (or group of referents if X is plural). This is usually underlined by the fact that we use the definite article, the. So in the phrase the price of oil, the phrase of oil tells us which price we're talking about. In other words it's not the price of beer, pencils, AK47s, baboons, ladders or clowns.
The Y part of this construction is a noun, and is therefore also marked for definiteness (read definiteness as meaning whether or not the listener should be able to identify a unique referent). If Y is definite, then we will see either a proper noun (name) or a noun marked with the. If Y isn't definite, then in the case of countable nouns, we will see a singular noun marked with the indefinite article a, or a plural or uncountable noun without any article at all.
In a construction like the price of gas, we have an X that is marked as definite. Price here denotes a numerical value. It has been marked as definite showing that we should be able to identify what price we're talking about. However, Y, in other words gas, is unmarked - there's no article here - indicating that it is indefinite. This means we are thinking about a definite price for an unspecified gas in general, not a specific amount or body of gas somewhere. This forces us into reading it as a specific price for gas in general, a notional average or common price.
The Original Question
How then do we determine whether the price of gas refers to the price of gas in a particular country, region, or globally. The answer is the same as it is for just about all nouns which are marked for definiteness. Definiteness belongs to the realm of pragmatics. If you say the book to me, what you're indicating by the is - you know which one I mean!. If you want me to identify a specific book as the intended referent, you must already have given me enough clues as to which book you want me to pick out! If you've just been talking to me about a book you bought yesterday and say the book, I know I should be able to pick out which book. The book you bought yesterday is the most prominent or salient in our conversation, so I'm going to pick out that one.
So, if we're talking about the economics of a particular country and the cost of living, then when we say the price of gas, our listeners are going to assume we mean the price of gas in that country. Similarly when talking global economics, the price of gas is the global price of gas. If the context does not make it clear, or is ambiguous, then we simply will need to modify the noun somehow. We to make it easy for our listeners to resolve the reference: the price of gas in the region, for example. Where articles and nouns are concerned, in terms of interpretation, context is king!
Best Answer
There are two different issues here. The first is to do with the way that we talk about and think about inventions. The second are the grammatical features that allow us to use the X in this way.
The Original Poster is quite correct that we use the X to refer to inventions. This is sometimes called the generic use of the definite article. It should really be called the generic use of nouns - with the definite article. In the same way that when we say the elephant we are picking out the genus elephant from the set of all other genuses of animal, when we say the typewriter we are picking out the genus typewriter from all the other genuses of invention.
However, there is a constraint on this use of the with singular nouns to represent inventions:
Uncountable nouns cannot be used in the generic sense to represent inventions. Compare ...
... with:
All of the examples above, because they are represented using uncountable nouns, can only refer to a specific amount, body or sample of lemonade, ink, hair, blood, paper or intelligence. If we want to refer to them as inventions we need to use them without an article:
Notice for example that we cannot say the paper to mean paper the invention, because it is uncountable. However, if we mean a newspaper, then newspapers are countable and this is fine:
In short, we can't use The X generically to represent inventions unless X is a countable noun!!
Aside from this very specific enquiry, the Original Poster's concept of when to use definite articles or not is spot on. We use the to tell the listener that they can readily identify the intended thing or person that the speaker is talking about. Otherwise we use a with countable nouns or zero article with plural and uncountable nouns. This indicates to the listener that they don't know which specific thing or entity the person is talking about. However, conventions for buildings, places, names, animals, inventions, diseases and so forth are very complicated. They're best studied and analysed individually - in my experience.