phi_270_text_2_1

Here are a few example analyses written out in full as models for the exercises to this section. In each case a few comments follow the actual analysis.

Roses are red and violets are blue
Roses are red ∧ violets are blue

R ∧ B
both R and B

R: roses are red; B: violets are blue

As a last step here, unanalyzed have been abbreviated components with capital letters in order to highlight logical forms. The final form is stated both symbolically and using English notation, something that will be done also in the examples to follow.

The next example is worked out in two steps, first analyzing the whole sentence as a conjunction and then analyzing one of its components.

It’s cool even though it’s bright and sunny
It’s cool ∧ it’s bright and sunny
It’s cool ∧ (it’s bright ∧ it’s sunny)

C ∧ (B ∧ S)
both C and both B and S

C: it’s cool; B: it’s bright; S: it’s sunny

The parentheses in the final result correspond to the grouping of bright and sunny together in the predicate of the second clause of the original sentence.

In the following example, it would not be wrong to use parentheses (or grouping with both), but that would be an artifact of our analysis and correspond to nothing in the English.

He was cool, calm, and collected
He was cool ∧ he was calm ∧ he was collected

C ∧ M ∧ T
C and M and T

C: he was cool; M: he was calm; T: he was collected

Accordingly, the analysis uses a run-on conjunction in the symbolic version, and use of both is similarly suppressed in the English statement of the form. If grouping were used here, either way conjunction might be assigned widest scope.

Finally, there can be cases where some grouping reflects the structure of the English, but other grouping does not.

It is a two-story brick building with a slate roof
It is a two-story brick building ∧ it has a slate roof
(it is a building ∧ it is made of brick ∧ it has two stories) ∧ it has a slate roof

(B ∧ R ∧ T) ∧ S
(B and R and T) and S

B: it is a building; R: it is made of brick; S: it has a slate roof; T: it has two stories

No grouping is used within the first three components because it is not obvious that any is imposed by the phrase two-story brick building. The English notation employs parentheses because there is no good way of indicating the combination of run-on conjunction with ordinary conjunction using both.

As in the last example, there would be nothing wrong with imposing a grouping here. If we were to group the first three components to the left, we would end up with the following in symbols and English:

((B ∧ R) ∧ T) ∧ S
both both both B and R and T and S

In the English notation, each of the boths tells us that a certain component is a conjunction—first the whole sentence, then its first component, and finally the first component of this component—and this settles the scope of the ands that follow.

The value of English notation does not lie in the possibility of making such a calculation but rather in our ability to understand the significance both automatically; however, that ability is limited to fairly simple forms, and a row of three boths is hard to follow without reflection. (To cite a standard example of a similar limitation in the case of a different sort of grouping, it is just possible to understand Bears bears fight fight to say what is said by Bears that bears fight fight—i.e., so that the first bears is modified by a relative clause bears fight and is the subject of the second fight; but it is virtually impossible to understand Bears bears bears fight fight fight as anything other than a cheer, even though it is grammatically possible for it to say something that might be expressed by Bears (which bears (that bears fight) fight) fight.)