Some simple examples

Suppose we adopt the following translation key: ‘P’ means Jack loves Jill.

‘Q’ means Jill loves Jack. ‘R’ means Jo loves Jill. ‘S’ means Jack is wise.

Let’s render the following into PL (even with our very quick explanations so far, these should not cause too much trouble: so try your hand at the translations before reading on).

1. Jack doesn’t love Jill. 2. Jack is wise and he loves Jill. 3. Either Jack loves Jill or Jo does. 4. Jack and Jill love each other. 5. Neither Jack loves Jill nor does Jo.

6. It is not the case that Jack loves Jill and it is not the case the Jill loves Jack.

7. Either Jack is not wise or both he and Jo love Jill.

8. It isn’t the case that either Jack loves Jill or Jill loves Jack. The first four are indeed done very easily:

1_{′. ¬P} 2′. (S∧P) 3′. (P∨R) 4′. (P∧Q)

7.5 Some simple examples 61

Just three comments. First remember that we are insisting that whenever we introduce an occurrence of ‘∧’ or ‘∨’, there needs to be a pair of matching brack- ets to indicate the scope of the connectives. To be sure, the brackets in (2′) to (4′) are strictly speaking redundant; if there is only one connective in a sentence, there is no possibility of a scope ambiguity needing to be resolved by bracketing. No matter, our bracketing policy will be strict. Second, note that rendering the English as best we can into PL isn’t a matter of mere phrase-for-phrase transliter- ation or mechanical coding. For example, in rendering (2) we have to assume that the ‘he’ refers to Jack; likewise we need to read (3) as saying the same as ‘either Jack loves Jill or Jo loves Jill’ – and assume too that the disjunction here is inclusive. Third, note that it is customary not to conclude sentences of our mini- language PL by full stops.

To continue. We don’t have a ‘neither …, nor …’ connective in PL. But we can translate (5) in two equivalent ways:

5′. ¬(P∨R) 5_′′. (_¬P∧¬R)

(Convince yourself that these renditions do come to the same.)

The natural reading of (6), as remarked in §7.2, treats it as the conjunction of ‘It is not the case that Jack loves Jill’ and ‘It is not the case the Jill loves Jack’; so translating into PL we get:

6′. (¬P∧¬Q)

And the next proposition is to be rendered as follows: 7′. (¬S∨ (P∧R))

(Check you see why the brackets need to be placed as they are.) Finally, we have 8′. ¬(P∨Q)

So far, so simple. But now we’ve got going, we can translate ever more compli- cated sentences. For example, consider

9. Either Jack and Jill love each other or it isn’t the case that either Jack loves Jill or Jill loves Jack.

This is naturally read as the disjunction of (4) and (8); so we translate it by dis- joining the translations of (4) and (8), thus:

9_′. ((P∧Q)∨¬(P∨Q))

If we want to translate the negation of (9), we write 10′. ¬((P∧Q)∨¬(P∨Q))

And so it goes.

In the next chapter we will begin characterizing PL rather more carefully. This won’t involve any major new ideas; the aim will be to be maximally clear and careful about what is going on. But before plunging into those details, let’s just summarize again why we are engaging in this enterprise.

62 Three propositional connectives

7.6 Summary

• There is a class of inferences whose validity (or invalidity) depends on the way conjunction, disjunction, and negation feature in the premisses and conclusion.

• ‘And’, ‘or’ and ‘not’ in English are used to express conjunction, disjunction, and negation; but there are many quirks and ambiguities in the way ordi- nary language expresses these logical operations.

• So to avoid the quirks of the vernacular, we are going to use a special-pur- pose language PL to express arguments whose relevant logical materials are the connectives ‘and’, ‘or’ and ‘not’. The design brief for PL is for the sim- plest language to do this without any ambiguities or obscurities.

Exercises 7

A The strict negation of a proposition can usually be expressed by prefixing the operator ‘It is not the case that’. Can you think of any exceptions in addition to the kind described in §7.2? Can you think of further cases which suggest that ‘and’ in English might sometimes mean more than bare conjunction?

B What is the most natural way in English of expressing the strict negations of the following?

1. No one loves Jack.

2. Only unmarried men love Jill. 3. Everyone who loves Jack admires Jill. 4. Someone loves both Jack and Jill. 5. Jill always arrives on time.

6. Whoever did that ought to be prosecuted. 7. Whenever it rains, it pours.

8. No one may smoke.

C Two propositions are contraries if they cannot be true together; they are

contradictories if one is true exactly when the other is false. (Example: ‘All

philosophers are wise’ and ‘No philosophers are wise’ are contraries – they can’t both be true. But maybe they are both false, so they are not contradic- tories.) Give examples of propositions which are contraries but not contra- dictories of B 1–7.

D Using the same translation manual as in §7.5, render the following into PL: 1. Jack is unwise and loves Jill.

2. It isn’t true that Jack doesn’t love Jill. 3. Jack loves Jill and Jo doesn’t.

4. Jack doesn’t love Jill, neither is he wise. 5. Either Jack loves Jill or Jill loves Jack.

6. Either Jack loves Jill or Jill loves Jack, but not both. 7. Either Jack is unwise or he loves Jill and Jo loves Jill.

In the previous chapter, we explained why it is advantageous to introduce an artificial language PL in which to express arguments involving conjunction, disjunction and negation. In this chapter, we’ll describe the structure of our new language much more carefully.