Learn English – finite verbs in these sentences

Snailboat's comment points to the source of this confusion.

According to the analysis presented in The Cambridge Grammar of the English Language, when the plain form of a verb (such as come or be) is used in a subjunctive construction, it is finite; when it is used in an infinitival construction it is non-finite, regardless of whether or not to is present.

Until about the middle of the 20th century, the terms used to talk about grammar were those inherited from the ‘classical’ grammarians, scholars of the Greek and Latin languages. Those languages (and indeed most modern European languages) have complex morphological verbal systems: each verb has scores of different forms which expressed fairly precise differences of number, person, time, voice and mode. In Greek or Latin (and to an only slightly lesser extent in, say, French or German) it is very easy to distinguish a ‘finite’ verb—one which is ‘limited’ to a particular person and number and tense and voice and mood—from a ‘non-finite’ verb, which has fewer such limitations, or none.

English, however, is different. Old English had most of the morphological distinctions we find in Greek and Latin; but in about the 11th century it began to lose most of the affixes and vowel changes which expressed differences of person, number, tense and the rest. By about 1700, almost all the morphological distinctions had disappeared. In Modern English, no verb except be has more than five distinct forms, and even be has only eight. Some have only one or two.

Consequently, every Modern English verb form has to serve many different purposes, each of which would be served by a distinct form in Latin.

So while the old Latin-derived terminology —finite, infinitive, past tense, subjunctive and the like—are still in use, the uses can be very misleading. In English these terms generally refer not to forms, as they did in Latin, but to uses.

A “finite verb”, for instance, is not a specific form or collection of forms. It is rather a constraint imposed in specific syntactic contexts. The term designates the rule that in a matrix clause, or in certain sorts of subordinate clause, the verb which ‘heads’ the clause must take the form which ‘agrees’ with its subject in number and person and with its intended sense in tense and mood to the extent that it is capable of doing so. The term finite verb in English means nothing more than this, and has no meaning beyond this.

For instance, take your second example: “They demand that she be present at the meeting.” Traditional (Latin-derived) grammar analyzed be as the 3rd person singular present subjunctive form of the verb BE; it understood be as a finite verb because it is appropriately conjugated for person, number, tense, mood and voice. Modern grammars simply observe that two rules operate here: 1) that clauses require a finite verb, as defined in the last paragraph, and 2) the verb demand requires its complement to take the ‘plain’ form (which it may or not call a ‘subjunctive’, depending on the particular grammarian’s methodology). It understands be as a finite verb because it satisfies rule #1.

Note, however, that neither of these analyses takes the form be to be in itself a finite verb. Both traditional and modern grammars understand quite well that be appears in many contexts as a non-finite verb, an infinitive:

They want her to be present tomorrow.
She is to be present tomorrow.
She will be present tomorrow.

This is not a problem for sophisticated users—experts like Profs. Huddleston and Pullum, the authors of CGEL, who are heirs to several centuries of practice using technical terms to mean different things in different contexts. When Prof. Huddleston talks to Prof. Pullum, he may use the term finite verb in full confidence that Prof. Pullum will understand that he is speaking of a particular English use, not a particular English form. And when Profs. Huddleston and Pullum write for an educated lay audience they may take care to make their current meaning clear and precise, as in the passage snailboat paraphrases.

But when the same terminology trickles down to the harried and hurried English teacher or blogger, who in many cases is not trained in these subtle distinctions, the terms lose their precision. It is never explained to the Learner that it is uses rather than forms which are being discussed. This is particularly confusing when the Learner is not a native speaker and lacks that long immersion in the language which might provide him an intuitive understanding of context.

As a non-native Learner, you are confronted first of all by forms, and it is quite natural that you try to understand as much as possible from the forms themselves. Sometimes this is quite sufficient. If you encounter the word comes, you may be sure that you are dealing with a finite verb, because the ending -s only appears when a third person singular non-past tense is called for—a finite use. If you encounter the word came, you may be sure that you are dealing with a finite verb, because this form only appears when a past tense or irrealis mood is called for—both finite uses. But if you encounter come you must look beyond the form to the syntactic context, because come may be either an infinitive or a simple or subjunctive non-past; only the context can tell you which it is.

To found is correct: to set up or establish on a firm basis or for enduring existence.

But you should use the infinitive:

I want to buy a book
I want to found a company

Not

I want to buying a book
I want to founding a company

(Edit:) As KRyan notes in the comment below there is a third possibility:

I want to be founding a company.

There is a slight difference in the meaning between the two. Using want to found implies that this is a thing you want to do at some point, want to be founding implies a particular point in time that you want to be involved in this process (e.g. "What do you see yourself doing in five years?" "I want to be founding a company.)"