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2. Conjunctions 2.1.
And
: adding content
2.1.0. Overview In this chapter, we will study the logical properties of the English word
and and certain related expressions. Along the way, we will encounter some gen
eral ways of approaching the study of logical properties that will serve us in later chapters, too. For the next several chapters, the logical forms we consider will reflect ways that sentences are formed from other sentences; the operators which form such sentences are known as connectives
2.1.1. A connective We begin with conjunction , a connective that enables us to combine the con
tent of a pair of sentences. 2.1.2. A truth function The meaning of conjunction can be given by specifying the truth value of a conjunction in terms of the truth values of the sentences that were com
bined. 2.1.3. Conjunction in English Although conjunction is most closely associated with the word and , there is a variety of ways of expressing it in English. 2.1.4. Limits on analysis On the other hand, even the appearance of and is not a sure sign that a sen
tence may be analyzed as a conjunction. 2.1.5. Multiple conjunction The operation of forming a sentence from sentences can be repeated. We will look at this sort of iteration in the case of conjunction. 2.1.6. Some sample analyses We will then apply these ideas to analyze several examples. 2.1.7. Logical forms And we will look in more general terms at the relation of logical forms to actual sentences. 2.1.8. Interpretations Finally, we will introduce some ways of talking about the relation between abstract logical forms and the meanings of sentences. Glen Helman 01 Aug 2013
2.1.1. A connective We are interested in logical forms as a way of stating general laws of entail
ment. Let us begin by looking at cases of entailment that seem to involve the word and
. Here is an example: That bear is large and edgy
That bear is large In attempting to understand any fact, it is useful to collect related facts. One way to search for related facts about entailment is to look for cases involving sentences similar in grammatical form to those above. If we follow this route, we run into entailments like this: That car is cheap and reliable
That car is cheap And we will eventually hit upon a general pattern like this: a^ is
P
and
Q
a is
P.
Although we will soon move on to more general patterns than this, any pattern that abstracts from particular words makes the label
formal logic
”^
appropri
ate. If we look a little farther afield in our search for related facts, we also find examples like It was hot and there was a storm before dark
It was hot
which follows the pattern φ and ψ
φ
This pattern can be seen to operate also in examples of the first group if we paraphrase them, transforming That bear is large and edgy
for instance, into That bear is large and that bear is edgy
When we apply a pattern by first paraphrasing, as we have done here, we treat a sentence as having a form that is hidden by its surface appearance. Much of our analysis of logical form will involve this sort of transition. Both of the patterns above give us general laws of inference. But, in the sec
ond more general pattern, it is especially clear that the word and plays a key role. If we look at what this role involves, we see that and marks a particular sort of compound sentence formed of component sentences, one that we will label a conjunction
. So the word and is a sign for an operator that forms con
junctions. We will call an operator that forms compound sentences out of com
ponent sentences a connective , and we will refer to the connective we are con
sidering here as conjunction , marking it with the sign
(one of whose names is logical and ). (The use of the term conjunction for both the operation of con
joining, the operator that performs this operation, and the compound that re
sults from it may seem confusing, but it follows a pattern that is used fairly of
ten in English—as when the word distribution is used both for the act of dis
tributing and for its result.) It will often be convenient to employ a further re
lated term and refer to the components of a conjunction as its conjuncts
Using these ideas, we can express our analysis of That bear is large and edgy as That bear is large
that bear is edgy
and we can express our principle of entailment as φ^
ψ
φ
This symbolic notation can save space, but it is often convenient to use English to mark conjunction. When we do this, we will use the construction both
and … and write it (as done here) using a special type. So the principle above could be stated as both φ and ψ
φ
(The reason for using the particle both in addition to and will be discussed later.) At this point, we have reached a stage like that reached by a physicist who recognizes pressure, temperature, and volume as physical quantities and has formulated a law relating them but who does not know why the law holds. That is, we have a generalization about entailment that we can apply in special cases, but we cannot say why this generalization is true. So let’s go on to ask what it is about conjunction that makes this sort of entailment work. We can find an answer by again scaring up some more facts. Notice, for ex
ample, that the entailment that got us started is matched by a second. That bear is large and edgy
That bear is edgy
Moreover, we can see not only that the sentence That bear is large and edgy entails each of the two sentences That bear is large and That bear is edgy but also that it is entailed in turn by the two taken together. If we abbreviate the longer sentence by B and the two shorter sentences as L and E, respectively, we have collected the following facts:
B
L
B
E
L, E
B
And checking other cases of conjunction would show us that these are in
stances of three general laws. φ
ψ
φ φ
ψ
ψ φ ,^ ψ
φ
ψ
Or, using English to express the forms, both φ and ψ
φ both φ and ψ
ψ φ ,^ ψ
both φ and ψ
So far, all we have done is to accumulate more general laws, but sometimes a larger number of facts is easier to understand because a pattern can begin to emerge. Glen Helman 01 Aug 2013
logical form will come through laws governing its role as the conclusion of an entailment or as one among its possibly many premises, specific characteristics of a logical form can often be highlighted most clear by its significance for re
lations between pairs of sentences, especially the positive relations of implica
tion and equivalence. The following principles are some of the more important examples of this in the case of conjunction:
C
OMMUTATIVITY
.^ The order of conjuncts in a conjunction does not affect the content. That is, φ
ψ
ψ
φ
A
SSOCIATIVITY
.^
When a conjunction is a conjunct of a larger conjunction, the way components are grouped does not affect the content. That is, φ
ψ
χ
)^
φ^
ψ
)^
χ
IDEMPOTENCE
.^
Conjoining a sentence to itself does not change the content. That is, φ
φ
φ
C
OVARIANCE
.^
A conjunction implies the result of replacing a component with anything that component implies. That is, if ψ
χ, then φ
ψ
φ
χ and ψ
φ
χ
φ
The names of these principles are terms used for analogous principles in other contexts. For example, you may have encountered the first two as names of principles for addition and multiplication since order and grouping do not mat
ter for these operators. Conjunction shares the third property with numerical operators that produce the maximum or minimum of a pair of numbers, and this is not surprising since, if we think of truth values as being ordered so that falsity comes below truth, the truth value of a conjunction is just the minimum of the truth values of its components. The last property, covariance, says roughly that the content of a conjunction varies in the same direction as the content of its components. An analogous property holds for addition and the maximum and minimum operators (e.g., if y^
z then min(x, y)
min(x, z)) but it doesn’t hold for multiplication when negative numbers are considered (e.g.,
×
×
4 even though 3
4). We cannot say that an increase or decrease in the content of one component will produce an actual increase or decrease, respectively, in the conjunction since information added or lost in a change to one component may be provided in any case by the other component. For example, although The sign had red letters on a blue background says more than does The sign had red let
ters , the conjunction The sign had red letters on a blue background, and the background was light blue is equivalent to The sign had red letters, and the background was light blue
. (This is analogous to the fact that, min(2, 3) = min(2, 4) even though 3 < 4.) What can be said is that, if the con
tent of one component of conjunction increases, the content of the conjunction must increase if it changes at all. One consequence of covariance is the following principle: C OMPOSITIONALITY
.^
Conjunctions are equivalent if their corresponding components are equivalent. That is, if φ
φ′ and ψ
ψ′
,^
then φ
ψ
φ′
ψ′
Although this follows from covariance (since equivalent components imply each other), it can hold when covariance does not. And compositionality is so fundamental that, if conjunction did not satisfy it, we might hesitate even to count it as a logical form. Since sentences are logically equivalent when they express the same proposition, this principle says that conjunctions cannot ex
press different propositions unless there is some difference in the propositions expressed by their components. Understanding the meanings of sentences to be the propositions expressed, the principle of compositionality tell us that the meaning of a conjunction is composed out of the meanings of its components in the particular way we label
conjunction.
Glen Helman 01 Aug 2013
2.1.3. Conjunction in English Conjunction is most often marked by the word
and , but there are English sen
tences without this word that also may be analyzed as conjunctions. First of all, there are quite a number of expressions—such as also
,^
in addition , and moreover —that serve as stylistic variants of and
. But conjunctions also may employ another group of words that are not simple stylistic variants of and
The principal example is the word but. This may be a surprise. Although a sharp ear might detect a slight difference in meaning between and and moreover , the difference between and and but is unmistakable. Consider, for example, the following two sentences, which dif
fer only in the use of these two words: Adams spoke forcefully to the committee, and they agreed to the expenditure Adams spoke forcefully to the committee, but they agreed to the expenditure. These sentences would be used under different circumstances, and it may seem odd to count them as logically equivalent, which is what we must do if we are to analyze both as conjunctions of the same two components. This is the first of several points at which we must recall the distinctions be
tween truth and appropriateness and between implication and implicature. As was noted in
, our concern is with only the first concept in each pair and thus with only certain aspects of meaning. Specifically, we count two sen
tences as equivalent if they have the same truth conditions. Any differences between their meanings that have no effect on their truth and falsity are irrele
vant for our purposes. So we must look more closely at the nature of the difference in meaning be
tween and and but
. It is clear that the second sentence above carries a sugges
tion of contrast between the two components—perhaps Adams spoke against the expenditure or the committee usually rejected Adams’s advice—and it is also clear that the suggestion of contrast is absent in the first sentence. Now, suppose that the second sentence was used in a context where the suggested contrast is not present—perhaps the expenditure was approved because Adams spoke for it. The assertion of the second sentence would then be inappropriate, but would it be false? Let us use the test of a yes-no question. Imagine that you attended a meet
ing were Adams persuaded a committee to agree to a certain expenditure and that later someone who had heard rumors of the proceedings asked you the question Is it true that Adams spoke forcefully to the committee, but they agreed to the expenditure?
. How would you reply? This is something you must decide for yourself; but, for my own part, I would say something like,
Yes, but he spoke for the expenditure, not against it.
”^
That is, I would give a yes-but answer, reacting to the sentence whose truth was asked about as one whose assertion would be true but would be misleading. This suggests that the difference between but and and is a difference in conditions of appro
priateness rather than conditions of truth, and it is for this reason that I will suggest we analyze sentences formed using but and other similar words—such as however ,^ though , and nonetheless —as conjunctions. These words are not merely signs of conjunction; but their differences from and lie outside their ef
fect on truth conditions. There are cases of other sorts where analysis by conjunction is legitimate though not obvious. Sometimes, for example, there is no word at all marking the conjunction. The operation of conjoining produces a compound sentence that commits us to the truth of both its components, and there are linguistic de
vices other than the use of particular words that enable us to roll two claims up into one in this way. For example, the sentence It was a hot, windy day is equivalent to It was a hot and windy day and can be analyzed as the con
junction It was a hot day
it was a windy day. An analysis of a sentence might even break a modifier off from the expres
sion it modifies. One common case of this is provided by adjectives used at ‐ tributively —i.e., applied directly to the noun they modify. For example, we may treat Sam’s car is a green Chevy as if it were Sam’s car is a Chevy, and it’s green. But it is important to note that, for reasons discussed in the next section, these analyses work only because the adjectives appear in a pred
icate nominative employing the indefinite article—i.e., in the form represented by X is a … Y or represented by a similar form with a different tense. However, this is a very common pattern so there will be many occasions to apply this sort of analysis. Another rather specific but important case of breaking off modifiers con
cerns relative clauses. There are really two cases here. The first is non-restric
tive relative clauses—that is, ones marked off by commas. These can usually be analyzed as conjunctions. For example, Ann, who you met yesterday, called this morning can be understood as a conjunction of You met Ann yes
to be wary of duplicating other quantifier phrases—in, for example, the subject of the sentence—when you make the analysis. And this is true even for com
pound predicate adjectives: Sam’s car was cheap and reliable is equivalent to Sam’s car was cheap
Sam’s car was reliable but One model is cheap and reliable is not equivalent to One model is cheap
one model is reli
able
Even when quantifier words are not involved, analyses by conjunction can
not always be used to break modifiers off from the words they modify. For ex
ample, it would be wrong to analyze Tristram is a large flea as Tristram is a flea
Tristram is large because a sentence with this analysis entails that at least one flea is to be found among the large things of the world. The prob
lem in this case is that an adjective modifying a noun has its meaning deter
mined in part by the noun it is applied to; large indicates a different range of sizes when it is applied to fleas than when it is applied to elephants. This is an example of a phenomenon discussed in
: vague terms have their meaning determined in part by their context of use. A noun can contribute to the context in which an adjective is used when the adjective is applied to the noun directly and also when the adjective follows the noun in a stream of discourse. This means that it also would be wrong to analyze Tristram is a flea and Tris
tram is large as Tristram is a flea
Tristram is large , for the adjective large acquires part of its meaning from the noun flea in the English sentence. (But the way a noun affects the meaning of a vague adjective is not simple. Although the sentence No fleas are large speaks about fleas, the range of sizes indicated by large in this sentence is different from the range indicated by its use in Tristram is a flea and Tristram is large
But why does the same thing not happen with the conjunction Tristram is a flea
Tristram is large ? Although the symbol
is closely related to the English conjunction and , it is not a simple abbreviation; and we do not assume that their contribution to the meaning of a sentence is exactly the same. The symbol
(and the construction both
and … that we use as an alternative notation for it) are signs for the operator conjunction. The conjunction of two sentences is a sentence that, in any context, has truth conditions that are related to those of its two components in the way shown by the table we considered earlier. And the stipulation that this is so in any context is a crucial one here; in particular, it need not be part of that context that either component has been as
serted. So in the conjunction, we cannot assume that the meaning of the second component Tristram is large will be influenced by the meaning of the first component. In certain sorts of context, Tristram is large will have the same meaning as Tristram is large for a flea
. But it is only in such contexts that Tristram is a flea
Tristram is large has the same truth conditions as Tristram is a flea and Tristram is large , and our analyses should not de
pend on equivalences that hold only for certain contexts. This indicates a further difference between our model of the operation of language and the way things work in English. Everything that is said in English has the potential of affecting the context of what follows it and, to a more limited extent, what precedes it. But when we analyze sentences, we treat their components as independent and as each understood in the same context. Our excuse for this limitation of our model is the same as that for many others: a model that was more accurate in this respect would require significant com
plications—and complications that no one yet understands very well. Of course, we can analyze Tristram is a large flea as a conjunction after all if we modify the second component to remove its dependence on the con
text established by the assertion of the first. One way of doing that was sug
gested in passing above: we may use the conjunction Tristram is a flea
Tristram is large for a flea
. Here we have modified the second component to replace the implicit effect of the context with a more explicit indication of the range of sizes in question. Though generalizations about such matters are risky, something like this device can be applied in many cases where adjec
tives acquire part of their meaning from the surrounding context. There are still other factors that can prevent breaking attributive adjectives off from the nouns they modify. We could be guilty of slander if we were to analyze Alfred is an alleged murderer as Alfred is a murderer
Alfred is alleged to be a murderer
. The difference between this and the example above is that the attributive adjective alleged modifies the meaning of a noun in a different way from an adjective like large . Adjectives like large narrow down the class of things marked out by the noun by adding a further property; in contrast, alleged shifts the membership of this class by adding as well as dropping members. The class of alleged murderers is not included in the class of murderers in the way the class of large fleas is included in the class of fleas. As a result, no analysis as a conjunction is possible. While the issues of contextual dependence can also affect our ability to break relative clauses off from the nouns they modify, this latter problem does not occur for them. If we say Alfred is a murderer who is alleged to be one we already imply that Alfred is a murderer, and analysis as a conjunction is possible. This means that one initial test for cases where we may break an attributive adjective off from the noun it modifies is to see if restatement using
a relative clause changes the meaning. While That’s an unknown Rembrandt is equivalent to That’s a Rembrandt that is unknown and can be analyzed as a conjunction, That’s a fake Rembrandt is not equivalent to That’s a Rem
brandt that is fake and cannot be analyzed as a conjunction. But, in the end, the test that an analysis must pass is that the conjunction we use to represent a sentence really has the same truth conditions. Since the truth table for conjunction is directly tied to the laws of entailment discussed in 2.1.1, one way to apply this test is to check whether the original sentence re
ally entails both components of the analysis (when these are considered as in
dependent sentences) and whether they, taken together, entail it. And we have used this test in the discussion of examples above; for example, because Al
fred is an alleged murderer does not entail Alfred is a murderer , we can
not analyze the premise as conjunction with the conclusion as one of its con
juncts. Due to the problems associated with the contextual dependence of meaning, we must be careful, when applying this test, not to fill out the mean
ings of terms in one of the sentences we are considering by a surreptitious ref
erence to another sentence. Glen Helman 01 Aug 2013
2.1.5. Multiple conjunction Although conjunction can compound sentences only two at a time, the word and
in English can be used with any series of more than two items. To analyze the serial conjunction He went to Gary, South Bend, and Fort Wayne , we need to regard the sentence as the result of two uses of conjunction, first to join two of the components and then to tack on a third. There are two ways of doing this and, although the associativity of conjunction noted in
tells that they have the same content, they arrive at their common meaning in dif
ferent ways. We can represent this difference in our symbolic notation by using parenthe
ses: He went to Gary ∧ ( he went to South Bend ∧ he went to Fort Wayne ) (He went to Gary ∧ he went to South Bend )^ ∧^ he went to Fort Wayne . There are a number of ways of describing the difference displayed here. We can say, first, that in each case a different one of the two uses of conjunction is the main connective or the one at the top level
. The main or top-level connec
tive is the operator that would be used last in forming the sentence, and it marks the place the sentence would be broken first when it is decomposed. In the first sentence above, it is the first use of conjunction that is the main con
nective or the one at top level while, in the second sentence, it is the second use. Another way of describing the difference between the two analyses is to speak of the scope of a connective, the part of the whole sentence that is made up of the connective and the components it applies to. Thus the scope of the first
in the first of the sentences above is the whole sentence while the scope of the second
is the portion in parentheses. This situation is reversed in the second sentence; there, the scope of the first
is limited by the parentheses and is included in the scope of the second
He went to Gary ∧ (he went to South Bend ∧ he went to Fort Wayne ) (He went to Gary ∧ he went to South Bend )^ ∧ he went to Fort Wayne So we say that the two examples differ in the relative scope of the two uses of conjunction. In one, the first use has wider scope; in the other, the second has wider scope. These two ideas are depicted together in Figure 2.1.5-1. The main connec
tive of each analysis appears quite literally at the top level, and the scope of each connective is the portion of the analysis that branches out from under it.
cation of the right end of its scope: is the second component only ψ , or is it the whole of ψ and χ ? If we supply a both for every and , we can write ( φ^
ψ
)^
χ as both both φ and ψ and χ
This is hardly elegant prose, but it does make the grouping definite; finding a second both immediately following the first, we know the first component of the main conjunction is itself a conjunction. Of course, we could also mark scope using parentheses. It may seem odd to do this if we are using English notation, too; but it is possible to mix the two forms, and it can sometimes be helpful to indicate a logical form by combining the word and with grouping marked by parentheses. Although scope distinctions can be made in English in these ways, the English and is often applied to a series of items that are all on the same level. It would be possible to treat conjunction as an operator that was similar to the English and in this respect, but it would cost us the trouble of more complex accounts of the properties of conjunction without yielding much greater in
sight. Still, we can (and often will) mimic the way addition and multiplication are usually treated in algebra and drop parentheses when they make no differ
ence in the value of an expression. This introduces no real complications but it has limitations. Since our principles concerning conjunction will be stated only for 2-component conjunctions, we can apply them to a run-on conjunction like φ
ψ
χ —or, in English notation, φ and ψ and χ —only after we have chosen one of the two conjunction symbols as marking the main operator. And, al
though we could regard either the first or last of the three components as a component of the top-level conjunction, the middle one, ψ , always ends up as a component of the lower level conjunction, so we really have not put the three components on the same level. Glen Helman 01 Aug 2013
2.1.6. Some sample analyses 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 components have been abbreviated with capital letters in order to highlight logical forms. The final form is stated both symbol
ically and using English notation, something that will be done also in the ex
amples to follow. The next example is worked out in two steps, first analyzing the whole sen
tence 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 corre
spond 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 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 no
tation 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 both s 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 and s 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 auto
matically; however, that ability is limited to fairly simple forms, and a row of three both s 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 themselves 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 vir
tually impossible to understand Bears bears bears fight fight fight as any
thing other than a cheer, even though it is grammatically possible for it to say something that might be expressed by Bears which are fought by bears that bears fight themselves fight
Glen Helman 01 Aug 2013
2.1.7. Logical forms We will conclude this first look at analysis by considering its results in more general terms. The aim of analysis is to uncover logical form. While it is natu
ral to speak of the result of an analysis as the logical form of the sentence that was analyzed, a sentence will usually have many logical forms of differing complexity. Many of these may be displayed as we carry out an analysis step by step. Consider, for example, the following analysis of a fairly complex sen
tence: He went to Gary, South Bend, and Fort Wayne, leaving at dawn and returning after dark He went to Gary, South Bend, and Fort Wayne
he left at dawn and returned after dark (he went to Gary and South Bend
he went to Fort Wayne
)^
he left at dawn and returned after dark (( he went to Gary
he went to South Bend
)^ ∧
he went to Fort Wayne
)^
he left at dawn and returned after dark (( he went to Gary
he went to South Bend
)^ ∧
he went to Fort Wayne
)^
he left at dawn
he returned after dark
((G
S)
F)
(L
R)
both both both
G
and
S
and
F
and both
L
and
R
F:
he went to Fort Wayne
; G:
he went to Gary
; L:
he left at dawn
R:
he returned after dark
; S:
he went to South Bend The first line exhibits the sentence without further analysis, the second shows it as a conjunction, the third as a conjunction whose first component is a con
junction, and so on. (The first component might have been analyzed as a run-on conjunction; but, for the purposes of this example, we need a fully specified structure.) Each line ascribes a form to the sentence, and if we ignore the identity of unanalyzed components, this is a form that the sentence shares with many other sentences. These abstract forms are indicated below (in the order in which they appear in the analysis) with symbolic notation on the left and a de
scription of the form on the right:
tences that express them, intensional interpretations will be specified by as
signing sentences to variables. (This assumes were are working with a fixed context of use, so sentences express propositions.) This assignment is the exact inverse of the process of abbreviating ultimate components by capital letters, and we will use the same notation for the association of letters and sentences in both. For example, we can give an intensional interpretation of the form (A ∧
B)
C by making the following assignment of sentences to the variables that mark its ultimate components.
A:
I got it apart
B:
I don’t know how I got it apart
C:
I couldn’t get it together again Since the sentences assigned to variables serve only to specify propositions, we will not be concerned about their logical forms; they may be as simple or complex as we wish. Especially in later chapters, the proposition assigned to a compound sen
tence by an intensional interpretation may not be apparent until we find an id
iomatic English sentence that expresses the same proposition. This can be done by a step-by-step process of synthesizing English that reverses the process of analysis. For the example above, this might proceed as follows:
(A
B)
C
(I got it apart
I don’t know how I got it apart
)^ ∧
I couldn’t get it together again I got it apart but I don’t know how
I couldn’t get it together again I got it apart but I don’t know how, and I couldn’t get it together again Of course, other wording is possible here, and the process of synthesizing English will rarely have a unique correct result. Extensional interpretations are easier to manage and will often provide all the information we need. We will adapt the tabular notation used for truth ta
bles.
A
B
C
(A
C)
(B
C)
T
F^
T
T
F
Variables are listed at the left with the assigned value under each of them. The whole form we are interested in is displayed to their right. The values of the larger components may be calculated by using the truth table for conjunction just as a multiplication table may be used to calculate the numerical value of a product: we find the values of the smallest components first and use these to calculate the values of larger components. The truth value calculated for each compound component is displayed below the main connective of that compo
nent. The value for the sentence as a whole is shown circled. Our interest will generally be only in this final value, but examples in this text will usually also display the intermediate values in order to show how the final value was reached. Glen Helman 01 Aug 2013
φ ψ φ
ψ T
T
T
T
F
F
F
T
F
F
F
F
2.1.s. Summary 3 4 5
The prime role of the logical word and is to mark the use of a connective
called conjunction , that serves to form a compound sentence (also called a conjunction) from component sentences that may be referred to as its con
juncts
. The process of interpreting a sentence as a conjunction is analysis
We use the sign
logical and ) as symbolic notation for the operator con
junction, marking the scope of a conjunction by parentheses. Alternatively, we can write a conjunction φ
ψ as both φ and ψ , where both plays the role of a left parenthesis. The two forms can be mixed using and to mark con
junction and parentheses to mark scope. We will use capital letters to stand for unanalyzed components as we use lower case Greek to stand for any sentences, analyzed or not. The effect of conjunction on the truth conditions of the compounds formed using it may be described in a truth ta
ble showing the compound to be true if and only if both components are true. The truth table specifies a truth func
tion , so conjunction can be said to have a truth function as its meaning. Some of the properties conjunction has in virtue of its meaning have standard names. It is commutative
,^
associative
and idempotent (i.e., the order, grouping, and number of conjuncts does not affect the content of a sentence formed using conjunction, perhaps repeat
edly); and it is covariant (adding or reducing the content of a component makes the content of the conjunction vary in an analogous way). Conjunction is marked in English by stylistic variants of and as well as by but and similar words. Conjunction also can appear without explicit indica
tion, particularly through the use of modifiers like attributive adjectives and relative clauses.^ Care is needed to be sure that such modifications can be captured by con
junction and to identify components that make independent contributions to the compound. The presence of quantifier words can preclude analysis as a conjunction even when the word and is present. Since conjunction is used to combine only two components, uses of con
junction to combine more than two in a multiple conjunction will involve two or more connectives of differing scope , the one with widest scope counting as the main connective of the sentence. Such differences in scope can be marked in several ways in English but such markings may be absent in a serial conjunction
. Some of the effect of serial conjunction without
scope distinctions can be achieved by run-on conjunctions, such as φ
ψ
χ, which suppress parentheses. In all but the simplest cases, the analysis of conjunctions will find compo
nents that are themselves conjunctions. The result of an analysis will exhibit this structure using symbolic and English notation. Although it is never wrong to mark the scope of conjunction within serial conjunctions, the re
sulting differences in the scopes of connectives are more significant in some cases than in others. The analysis of the logical form of a sentence can occur in stages in which we identify the immediate components of a compound, any immediate components of these, and so on. The last components arrived at are the ulti
mate components of the analysis; the full class of components includes them as well as all other sentences that could appear in the course of analy
sis (including the analyzed sentence itself). A sentence will usually have^ many logical forms representing different partial analyses of it. We can specify a proposition or a truth value for a logical form by means of an intensional or extensional interpretation , assigning truth values or sen
tences, respectively, to its ultimate components. A sentence expressing the proposition provided by an intensional interpretation can be found by carry
ing out a process of synthesis that reverses the process of analysis. The truth value provided by an extensional interpretation can be found by calcu
lation using the truth table for conjunction. The tabular notation used to write the truth table of conjunction may be used also to describe extensional interpretations and the values that they give to compound forms. Glen Helman 01 Aug 2013
2.1.xa. Exercise answers 1.
a. Mike visited London
Mike visited Paris L
P
both
L
and
P
L:
Mike visited London
; P:
Mike visited Paris b. Ann wanted white wine
Bill and Carol wanted red wine Ann wanted white wine
Bill wanted red wine
Carol wanted red wine
A
(B
C)
both
A
and both
B
and
C
A:
Ann wanted white wine
; B:
Bill wanted red wine
; C:
Carol wanted red wine c. It will rain and clear off
it will rain (it will rain
it will clear off
)^
it will rain (R
C)
R
both both
R
and
C
and
R
C:
it will clear off
; R:
it will rain d. That is a market
that is new relative to other markets but growing That is a market
(that is new relative to other markets
that is growing
M
(N
G)
both
M
and both
N
and
G
G:
that is growing
; M:
that is a market
; N:
that is new relative to other markets e. Confucius is affable but dignified
Confucius is austere but not harsh
Confucius is polite but completely at ease (Confucius is affable
Confucius is dignified
)^
Confucius is austere
Confucius is not harsh
)^
Confucius is polite
Confucius is completely at ease
(A
D)
(S
H)
(P
E)
(both
A
and
D)
and
both
S
and
H)
and
both
P
and
E)
A:
Confucius is affable
; D:
Confucius is dignified
; E:
Confucius is completely at ease
; H:
Confucius is not harsh
; P:
Confucius is polite
; S:
Confucius is austere f. Tim lost his glasses and his wallet
Tim’s glasses and wallet were each returned (Tim lost his glasses
Tim lost his wallet
)^
Tim’s glasses were returned
Tim’s wallet was returned
(G
W)
(R
T)
both both
G
and
W
and both
R
and
T
G:
Tim lost his glasses
; R:
Tim’s glasses were returned
T:
Tim’s wallet was returned
; W:
Tim lost his wallet g. Tim lost his glasses and his wallet
one person found both Tim’s glasses and his wallet (Tim lost his glasses
Tim lost his wallet
)^
one person found both Tim’s glasses and his wallet
(G
W)
O
both both
G
and
W
and
O
G:
Tim lost his glasses
; O:
one person found both Tim’s glasses and his wallet
; W:
Tim lost his wallet Note
:^
One person found both Tim’s glasses and his wallet cannot be analyzed further because One person found Tim’s glasses
one person found Tim’s wallet does not imply that the same person found both. 2. a.
A
B
C
b.
(A
B
)^ ∧
C
c. both both
A
and
B
and both
C
and
D
d. both
A
and both both
B
and
C
and
D
e. both both
A
and both
B
and
C
and
D
f.
A
B
)^
C
)^
D
a. (Fred visited Venice
Fred visited Florence
)^
Fred spent a week in Rome Fred visited Venice and Florence
Fred spent a week in Rome Fred visited Venice and Florence, and he spent a week in Rome
b. (he was a judge
(he was stern
he was fair
he had an ex
cellent knowledge of the law (he was a judge
he was stern but fair
)^
he had an excellent knowledge of the law He was a stern but fair judge who had an excellent knowledge of the law c. (we arrived cold
we arrived tired
we arrived hungry
)^
we left warm
we left stuffed
we left sleepy
We arrived cold, tired, and hungry
we left warm, stuffed, and sleepy We arrived cold, tired, and hungry; and we left warm, stuffed, and sleepy d. Old King Cole was a merry old soul
Old King Cole was a merry old soul Old King Cole was a merry old soul, and a merry old soul was he 4. Numbers below the tables indicate the order in which values were com
puted a.
A
B
C
A
(B
C)
T
T
F^
F
2 1 b.
A
B
C
D
((A
D)
C)
(B
A)
T
T
F^
T
T
F^
T
1 2 3 1 Glen Helman 01 Aug 2013
