What sentence analysis is meant to show
The issue we are concerned with is the structure of sentences, i.e. syntax. We can represent this structure in various ways, for example trees or labelled brackets; these are representations of something more abstract but nonetheless real.
When we stipulate certain rules governing what constitutes a legitimate tree or set of brackets or what have you, we are making a claim about language; sometimes, the claims are about English specifically, but often, they are about human language in general.
For example, we assume that the linear order of words is represented in the tree, and that in a well-formed tree, branches never cross. You can easily imagine a tree with crossing branches. So why this restriction? Because we find in syntax that there are good reasons to limit the possible structures assigned to a string. One such example comes up in the contrast between (1) and (2):
(1) gullible boys and girls
(2) girls and gullible boys
Two different well-formed trees can be drawn for (1), corresponding to the two different readings (cf. Finegan pp. 150-151). But only one tree is possible for (2), if branch crossing is impossible, and this accurately reflects the fact that (2) is unambiguous. If branch crossing were possible, then we would have to come up with another reason that (2) only has one possible meaning.
We also assume that in a well-formed tree, every node has a unique mother, i.e. you dont have two branches converging on a single node. For example, a verb like show can have two NP objects, as in (3). The two NPs can refer to the same person, as in (4).
(3) Kim showed Leslie a mirror.
(4) Kim showed Leslie herself.
But this does not entail that the two objects can be the same NP, as in the tree in (5).

What kinds of arguments can be used
These general claims about possible trees are claims about language in general. The claim is that tree structures accurately represent several facts about language simultaneously; accordingly, we can use various kinds of arguments for a particular tree structure, including semantic, syntactic, and even phonological arguments. Semantic and phonological arguments are rarely as convincing, when talking about phrase structure, as syntactic ones are, but they are often the basis for our initial intuitions, which guide our hypotheses.
Thus, when Finegan writes of referring and predicating (p. 142), the intuition should be clear, even if it can be difficult to make precise the idea that e.g. uses an answering machine is predicated of She in (6), rather than an answering machine being predicated of She uses. The problem is more acute in (7).
(6) She uses an answering machine.
(7) Alex fears Kerry.
To be perfectly clear, this issue is this: (6) and (7) are analyzed into two major parts, as in (8), not as in (9).
(8) S [V O]
(9) [S V] O
And although the intuitive argument that Finegan presents may be difficult to defend for (7), there are many reasons to believe that (7) is organized into a subject and predicate just as (6) is. One such reason is that it is easier to substitute another constituent for the predicate than for the subject plus verb (cf. Finegans examples of substitution as an argument for constituency on p. 145). Consider (10-11).
(10) Toni fears Kerry. Alex does, too. (does = fears Kerry)
(11) * Toni fears Kerry. Does Alex, too. (does _ Toni fears)
This, like the arguments from predication and referring, is basically a semantic argument (though it is sometimes taken to be a syntactic one). The idea is that because a syntactic constituent usually corresponds to a coherent unit of meaning, it is likely that some short paraphrase or pronominal-type element will be able to substitute for it, whereas the same is not necessarily true of a non-constituent.
Phonological arguments can also be constructed for constituency. For example, since the phonological structure of the sentence is sensitive to its syntactic structure, a larger break in the syntactic structure may correspond to a larger break in the phonological structure. A break in the phonological structure might be a place where a pause can be inserted. So the claim that (8) is the correct analysis of (7), and (9) is not, could be supported by the fact that it is more natural to insert a pause or other intonational boundary between Alex and fears than between fears and Kerry. This is clearer with longer elements, as in (12) and (13).
(12) That the check cleared, proves that the balance was sufficient.
(13) ? That the check cleared proves, that the balance was sufficient.
Although it is considered poor form in written English to place a comma after the subject, a pause at the point of the comma is considerably more natural in (12) than in (13).
However, this test, like the predication test, is not entirely reliable. Auxiliary verbs in general and copula in particular may form a phonological unit with the subject.
(14) Andreas here.
(15) Well join Dominique.
Yet even in these cases, there is good reason to believe that there is a predicate constituent excluding the subject. Such uncertainties in the phonological and semantic tests make it very important to master the syntactic tests for constituency, which tend to be more reliable and decisive.
Syntactic constituency tests
One such constituency test is that of displacement. It is closely related to the substitution test mentioned above. Intuitively, displacement involves moving things. Thus although (16) is ambiguous, the cleft sentences in (17-18) are not (cf. Finegan p. 151).
(16) He sold the car to his brother in New York.
(17) It was to his brother in New York that he sold the car _.
(18) It was to his brother that he sold the car _ in New York.
If we think of the underlined part of the cleft construction (the focus) as having moved from the gap marked in (17-18) by an underscore, and if movement only affects constituents, then (17-18) show something about the structure of (16), namely that the two readings correspond to two different syntactic structures; i.e. (16) ambiguously either contains the constituent hisbrother in New York, or the constituent his brother followed by other material (in New York). Since (17) is unambiguous, the semantic intuition about his brother in New York referring to an entity and the syntactic test showing that it can be be displaced converge. The same holds of (18). Notice that the phonological test fits in here too, since the pronunciation of (16) with a pause after brother favors the reading where the event takes place in New York.
In addition to clefts, Finegan gives examples of topicalization or fronting (ex. 6, p. 151) and of passive (pp. 154-155) as constituency tests. Constituency tests have various limitations; for example, passive only moves direct object NPs into subject position, so it cant be used to distinguish the two readings in (16), though it can in the similar sentence in (19).
(19) They indicted his brother in New York.
(20) His brother was indicted in New York.
(21) His brother in New York was indicted.
Other important examples of limitations on constituency tests are discussed in the chapter in pp. 170-172.
Another constituency test that can be applied is that of interpolation. The basic idea is that if there is a relatively large break in a sentence, then various sorts of material should be able to appear there. Consider the subject-predicate structure again, in this light. A wide variety of adverbial elements can be inserted between a subject and the predicate, but nothing can generally be inserted between the verb and its direct object,
(22) Dani (already) met Toni.
(23) Dani met (*already) Toni.
Of course, there are other restrictions on interpolation. PPs, for example, cannot generally be inserted between the subject and the verb in English, so the results in (24-25) seem to show exactly the opposite of that expected. A comparison of (25) to (26), on the other hand, correctly suggests a closer connection between the verb and the first PP than between those elements and the second PP.
(24) Dani (*in a hushed voice) spoke to Toni in the morning.
(25) Dani spoke (?in a hushed voice) to Toni in the morning.
(26) Dani spoke to Toni (in a hushed voice) in the morning.
This closer relation can be represented in a tree by placing to Toni within the VP, and in the morning outside it, as a dependent of S. This represents the fact that to Toni is a selected PP, a complement of the verb spoke, while in the morning is a sentential adverbial, not selected by the verb but freely insertable in any sentence with the right temporal properties.
In a separate document: notes on the other half of the chapter