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Chapter 4

 

Sentential Connectives

 

 

 

1   LOGICAL CONNECTIVES AND PROPOSITIONAL LOGIC

 

There are certain expressions in logic the meaning of which is constant and can be precisely specified. Among such expressions are logical connectives that perform logical operations, such as conjunction (approximately corresponding to ‘and’), disjunction (‘or’), implication (‘if … then’), equivalence (‘if and only if’) and, perhaps surprisingly, also negation (‘not’, ‘it is not the case that’).

       There are many more connectives in natural languages than in propositional logic. In addition to the above, there are also, for example, ‘but’, ‘therefore’, ‘because’, ‘since’, ‘after all’, ‘moreover’, ‘so’, ‘before’, ‘as’, ‘even though’. These are lesser interest to logicians, who are not directly interested in the intricacies of natural languages but rather in finding a perfect artificial language to capture the rules of inference. In addition, there are also discourse connectives (Blakemore 1987, 1992, 2000) that belong to a larger category of discourse markers (Schiffrin 1987; Fraser 1990a, 1999): expressions that are commonly used in the initial position of an utterance and are syntactically detachable from a sentence, such as in (1) and (2).

 

(1)  Tom is very   handsome. And   he is rich too.

(2)  This library building is impressive. After all , it was designed by Christopher Wren.

 

They are said to signal procedures for interpreting the concerning sentence. In other words, their meaning is procedural rather than conceptual; they do not name a concept .

       We have established that in order to state the meaning of a sentence we have to provide its logical form. If the sentence is composed out of two or more simple sentences, the first step is to break the sentence down into simple sentences joined by sentential connectives. There are five connectives that are of interest to propositional logic and they are represented by the symbols in the right column below. The existence of two equivalent symbols in the case of some connectives is the result of the acceptance of various notational variants in the literature:

 

            conjunction (and)                                  &, ʌ

            disjunction   (or)                                       v

            implication  (if … then)                          --->

            equivalence (if and only if)                      ↔, ≡

negation (not)                                  ¬, ~    

 

Negation is included as a logical connective  because it operates on the whole proposition, or, so to speak, connects to a proposition, and hence belongs to propositional logic. These connectives are truth-functional, that is they have constant meaning in logic. This makes  them different from their English counterparts, whose meaning is not so rigidly fixed.

 

 

2   CONJUNCTION

The meaning of ‘and’ is dictated by propositional logic. It is constant, it is a logical constant. The compound sentence is true when both of its conjuncts are true, as summarized in the truth table below (table 1), where ‘t’ stands for ‘true’ and ‘f’ for ‘false’.

 

 

Table 1

p          q          p &  q

t           t                t

t           f                f

f           t                f

f           f                f

 

for example, (3) is true only if both (3a) and (3b) are true. In other words, p & q is true iff p is true and  q   is true.

 

(3)     Bill got up and went to school.              (p & q)

(3a)   Bill got up.                                             (p)

(3b)   Bill went to school.                                (q)

 

On the widest of approximations, logical conjunction corresponds to the English connective and . However, other connectives, such as but , are also translated as ‘&’. On the other hand, ‘and’ can have meaning that is richer than that of its logical counterpart. For example, in (4) it introduces an ambiguity between the collective and distributive readings.

 

(4)   Tom and Peter own a car.                        (one each or jointly)

 

In (5), ‘and’ is very likely to include the specification of the order of the events and mean ‘and then’. The truth-conditional relevance of the sequential order is visible in examples such as (6).

 

(5)     They got married and had a baby.

(6)   If they got married and had a baby, their parents would be pleased, but if they had a baby and got married, their parents would be upset.

 

In (7), the causal relation between the conjuncts is the standard interpretation.

 

(7)   I dropped the camera and it broke.

 

This enrichment of the logical conjunction signifies that there are aspects of meaning of sentential connectives that are not accounted for by propositional logic. Ambiguous sentences, such as (8) and (9), have two logical forms for each and two distinct sets of truth conditions.

 

(8)   Mary went to the bank.    (river bank or financial institution)

(9)   Little girls and boys were playing.            (little girls and little, or any, boys)

 

This is not the case with (5) and (7) where the temporal and causal interpretations respectively have logical forms that are developments of the logical form of the interpretation with the pure logical conjunction.  The consequential and need not presume the temporal and , states can be fully overlapping and yet stand in causal relation to each other as in (10):

 

(10)   Susan is under age and can’t drink.

 

So, sentential conjunction is not uncontroversially truth-functional. The propositional form requires a great deal of adjustment in order to fit in the mould of unitary semantics, that is, to reject the ambiguity view.

       Further, in (11), the conjunction inside the simple sentence is not easily rendered in terms of propositional logic. The connective can only join together full propositions but when we try to amend (11) to satisfy this criterion, the meaning of the amended sentence (11a) differs from that of (11), at least on its standard interpretation.

 

(11)    Sue and Bill are divorced.

(11a) Sue is divorced and bill is divorced.

Finally, and can have a meaning that is closer to implication than conjunction, as in (12).

 

(12)   Touch me and I will hit you.      (=If you touch me, then I   will hit you.)

 

 

3   DISJUNCTION

The disjunction is false when both the simple sentences (disjuncts) are false. If at least one disjunct is true, the disjunction is true. These properties of disjunction are summarized in the truth table below (Table 2).

 

Table 2

p          q          p v  q

t           t                t

t           f                t

f           t                t

f           f                f

 

logical disjunction allows for both disjuncts to be true. It is called inclusive disjunction and it differs from the disjunction in English which is normally exclusive : it is implicit in English or that one or the other statement holds but not both .  The properties of exclusive disjunction, which is standardly   written by means of the symbol v or  (v) , are stated in Table 3.

 

Table 3

p          q          p (v)    q

t           t                f

t           f                t

f           t                t

f           f                f

 

The exclusive nature of English or is obvious in examples (13) and (14).

 

(13)   He likes either red or white wine.

(14)   Your money or your life!

 

However, (15) is naturally interpreted as an inclusive disjunction, while (16) tends to be inclusive but the interpretation is likely to be context-dependent of left unresolved in context.

 

(15)  Every citizen or permanent resident is eligible for unemployment benefits.

(16)   She is either happy or rich.

 

Sentence (17) tends to be exclusive in English but it is also restricted to some specific contexts because normally the speaker knows which state of affairs is the case.

 

(17)   It is snowing or raining.

 

A further property of natural language disjunction is that it can apply to the style rather than to two different propositions. In (18), disjunction serves the purpose of a stylistic self-correction.

 

(18)   Ten firemen – or should I say firefighters – tried to save the children from the flames.

 

It can be concluded from these examples that there is more to linguistic communication than the truth-functional properties of logical words and than the logical form in general. There are also rules of communication.

 

3              IMPLICATION

An implication, also called material implication, expresses a causal connection between an antecedent and a consequent. In logic, an implication is false only if the antecedent is true and the consequent is false. In other cases it is true. In particular, rather counterintuitively, it is true when the antecedent is false and the consequent is true. These properties of implication are summarized in the truth table below (Table 4).

 

Table 4.

p          q          p à   q

t           t                t

t           f                f

f           t                t

f           f                t

 

in English, implication is rendered by such expressions as ‘if’, ‘if … then’, ‘provided’, ‘whenever’ or ‘unless’. For example, (19) expresses an implication, also known as a conditional .

 

(19)      If it is raining, then it will be wet.

 

It is worth nothing that p is a sufficient condition for q but not a necessary one. If it is raining, then the streets will be wet, but they can also be wet for other reasons, such as being washed by a sprinkle.

       It is important that the antecedent and the consequent must be meaningfully (causally) tied together. Sentence (20) is pragmatically ill-formed.

 

(20)      ˀ If penguins are birds then semantics is a study of meaning

 

The antecedent is normally important for the meaning of the implication and should be true, unlike in (21). It should also be fulfilled, for example in (22).

 

(21)      ˀ If penguins are mammals then they have wings.

(22)      If you cook the main course, I will make the dessert

 

In fact, we rarely use implication in English. We normally use ‘if’ to mean a stronger relation, namely ‘if and only’, called an equivalence or biconditional. This strengthening came to be known in the literature as conditional perfection . For example, (23) is normally used to mean (24):

 

(23)      If you mow the lawn, I’ll give you five dollars.

(24)      If and only if you mow the lawn will I give you five dollars.

 

Sentence (24) is more commonly expressed as (25).

 

(25)      I’ll give you five dollars only if you mow the lawn.

 

In other words, (23) invites an inference to (26).

 

(26)      If you don’t mow the lawn, I won’t give you five dollars.

 

The strengthening of if to iff does not mean that if is ambiguous in English. Instead, it means that it can be pragmatically strengthened. This strengthening is obvious when we group expressions into scales arranged from the strongest to the weakest. For example, all and some constitute such a scale: when the speaker utters (27), can be inferred that (28) is not the case.

 

(27)   Some of the invited guests came to the party.

(28)   All of the invited guests came to the party.

 

Similarly, (23) implies that a stronger (29) does not hold. It is stronger because   it can   be formalized as ((p  ---> q) & (r  ---> q)   &   (s  ---> q) …), as opposed to the weaker (p  --->  q).

 

(29)      Whatever may be the case, I will give you five dollars.

 

Conditionals are used for a variety of purposes. There are epistemic conditionals, such as (30), and speech act conditionals, such as (31), where conditionals do not refer to states of affairs:

 

(30)      If she’s divorced, she’s been married.

(31)      John has left, if you haven’t heard.

 

They can also be used for stylistic reasons as in (32) and (33).

 

(32)      If you are thirsty, there is some beer in the fridge.

(33)      If I may say so, you look tired.

 

Now, in a cognitive-semantic framework, instead of conditionals with reference to mental representations (or mental spaces) in the following way. In (23), the payment of five dollars is conditionally predicted .

       Finally, it is worth observing that implication is equivalent to conjunction of the negated antecedent and the consequent, as represented in (34) and exemplified in (35a) and (35b).

 

(34)      ---> q =  ¬ p v q  

 

(35a)    If I’m right, I owe you £10.                 (p à q)           

(35b)    Either I am wrong or I owe you £10.  ( ¬ p  v   q)

 

All in all, material implication in logic does not quite agree with our intuitions and does not fit all the uses of if in English.

 

 

5   EQUIVALENCE

Equivalence, or biconditional, is normally expressed by ‘if and only if’, ‘exactly when’, ‘only if’. As was observed in our analysis of implication, in English, if can be difficult to distinguish from if and only if. For example, sentence (36) strongly suggests the if and only if interpretation.

 

(36)   I will help you only if you are too busy to do it yourself.

 

The equivalence is true only  when both sentences have the same truth value, as demonstrated in the truth table (table 5).

 

 

Table 4.5

p          q          p   q

t           t                t

t           f                f

f           t                f

f           f                t

 

 

Sentence q is the necessary condition for p, for example in (37).

 

(37)      Tom will help you if and only if you asked him.

 

The name ‘biconditional’ becomes diaphanous when we consider its property of a two-way conditional, stated in (38).

 

(38)      p q = (p à q) & (q à p)

 

6   NEGATION

Unlike the connectives discussed so far, negation does not link two sentences; instead, it attaches itself to a sentence to form another sentence. For this reason, it is included in the category of sentential operators. The truth value of the negated sentence is opposite to the original sentence, as is demonstrated in the truth table (table 6).

 

Table 4.6

    p            - p

    t                f

    f               t

 

Negation takes various forms in English, such as ‘not’, ‘it is false that’, ‘it is not the case that’, ‘it is incorrect that’, ‘it is not true that’, ‘it is wrong that’ and many others. Like other sentential connectives in English, it does not fit  very well the mould of truth-functional negation. For example, (39) contains negation of a constituent of the sentence, which cannot be rendered by a simple logical form ¬ p in propositional logic.

 

(39)      Non-students are allowed.

 

In (40), although on the surface negation seems to apply to the whole sentence, it is normally interpreted as applying to the subordinate clause.

 

(40)      I don’t think that Tom will win.

 

It is so because people have a tendency to assign a  lower-clause reading to a higher-clause  negation. In other words, (40) is normally taken to mean (40a) rather than (40b).

 

(40a)    I think that Tom won’t win.

(40b)    It is not the case that I think that Tom will win.

 

Next, intonation can also affect the meaning of negation, as the comparison of  (41a) and (41b) demonstrates.

 

(41a)    Fido didn’t eat your cake.

(41b)    Fido didn’t eat your cake .

 

Further, we can negate aspects of the utterance other than the propositional content.  For example, in (42) – (44), the style is negated.

 

(42)      Grandma isn’t feeling lousy’, she is indisposed.

(43)      Fido didn’t ‘shit the rug’, he pooped on the carpet.

(44)      The glass isn’t half full – it is half empty.

 

In (45), the grammar is negated:

 

(45)      I didn’t catch mongeese – I caught mongooses.

 

We can also negate a presupposition of the sentence. For example, (46) presupposes (47).

 

(46)      The farmer hasn’t stopped beating his donkey.

(47)      The farmer was beating his donkey

 

Preposition differs from entailment in the following way. Entailment is a relation between sentences where the truth of the second sentence (S2) necessarily follows from the truth of the first (S1), as in the table below. So, in a sense, entailment is a weaker relation than presupposition in that the falsity of S1 does not guarantee the truth of S2 (Table 7)

 

 

Table 7

S1                    S2

t        --->         t

f         <---         f

f        --->          t  v  f

 

For example,  (48) entails (49).

 

(48)   I bought some tulips.

(49)   I bought some flowers.

 

       The negation of a presupposition is exemplified in (50b) as contrasted with standard propositional negation in (50a).

 

(50a)    The British Prime Minister is not bald.

(50b)    The king of France is not bald; there isn’t any king of France.

Finally, implicatures of the utterance can be negated. Implicatures are inferences that go beyond the logical form of the sentence. They can be general, context-independent or can be triggered by the context. sentences (51) – (52)   are instances of negating a context-independent implicature rather than the propositional content. Sentence (53) is said to involve negation of the implicature associated with the word ‘manage’:

 

(51)      It isn’t warm, it’s hot.

(52)      Some men aren’t chauvinists – all men are chauvinists.

(53)      John didn’t manage to solve the problem – it was quite

            easy for him to solve.

 

Sentence (54) is an instance of such an implicature negation, as opposed to (55) which is a standard, descriptive negation:

 

(54)      Max doesn’t have three children, he has four.

(55)      Tom doesn’t have three children, he has two.

 

The classification of (54) as implicature negation is founded on the assumption that, semantically, the numeral ‘three’ means ‘at least three’ rather than ‘exactly three’. In other words, when Max  has four children, it is also true to say that he has three. What is negated in (54) is pragmatically strengthened meaning, ‘exactly three’. The literature  on this topic in vast. Instead of the ‘at least’ semantics, it has been proposed that numerals are under determined as to their sense, which would accommodate examples such as (56) and (57). In (57), the punctual ‘exactly three’ meaning is necessary; the ‘at least three’ renders a non-sensical interpretation:

 

(56)      She can have 2000 calories without putting on weight.

(57)      More than three people came.

 

According to the most reliable of the tests, called the Identify Test, conjunction reduction should be possible only when the conjoined constituents have matching understanding, as in (60) which is the result of conjoining (58) and (59):

 

(58)      They saw her duck.                 (bird or physical action)

(59)      They saw her swallow             (bird or eating)

(60)      They saw her duck and (her) swallow.

 

Mon, 16 May 2011 @11:40

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