Short working

Imagine that ‘P’ ⇒ T, ‘Q’ ⇒ T, ‘R’ ⇒ F. The wff would then evaluate in both these ways:

‘P’⇒ T ‘Q’⇒ T ‘Q’⇒ T ‘R’⇒ F

‘P∨Q’_{⇒ T} ‘R’_{⇒ F} ‘P’_{⇒ T} ‘Q∧R’_{⇒ F}

‘P∨Q∧R’⇒ F ‘P∨Q∧R’⇒ T

Here the final valuations differ because, on the left, the wff is being seen as con- joining a truth and a falsehood (‘∧’ has wider scope than ‘∨’), and on the right as disjoining a different truth and falsehood (‘∨’ has wider scope). The syntactic structural ambiguity leads to a semantic ambiguity: fixing the truth-values of the atoms does not uniquely fix the truth-value of the whole unbracketed expression ‘P∨Q∧R’.

Moral: leaving out brackets from PL formulae introduces the risk of semantic ambiguity. Don’t do it!

9.7 Short working

And that’s it as far as the basic semantics of PL is concerned. This final section merely introduces a way of presenting truth-value calculations rather more snappily.

Go back to example A again, and consider the following table of calculations:

On the left, we display the initial assignment of values to the atomic subformu- lae. On the right, we’ve copied the same values directly under the corresponding atoms and then, as we calculate the values of more and more complex parts of the final formula, we write the value under the connective whose impact is being

calculated at each stage. For example, when we are evaluating ‘(_¬Q ∨R)’ by considering how ‘∨’ operates on the truth-values of ‘_{¬Q’ and ‘R’, we write the} result of evaluating that disjunction under the ‘∨’. (To aid comparison with the calculation in §9.5, we have numbered the stages at which values are calculated by the corresponding line number. For clarity, the final result is underlined.)

Here, in the same short form is the second calculation from §9.5:

This time, the numbers correspond to the lines of the derivation B. For a third example, take the wff

((S∧¬(Q ∧¬P))∨ (R∨¬¬Q))

and suppose ‘P’_{⇒ T, ‘Q’ ⇒ F, ‘R’ ⇒ F, ‘S’ ⇒ F. This time, let’s do the working} directly in short form. Looking at the wff, and bracket counting, we see that the main connective is the first ‘∨’; so evaluating this wff involves evaluating the two

P Q R ¬ (P ∧ ¬ (¬Q ∨ R)) T F T T T F F T F T T 8 6 7 5 2 1 4 3 P Q R S ((P ∨ (R ∧ ¬ S)) ∧ ¬(Q ∧ ¬ P)) F F T F F T T T T F T T F F T F 5 6 1 4 3 2 11 10 7 9 8 5

80 The semantics of PL

disjuncts ‘(S∧¬(Q ∧¬P))’ and ‘(R∨¬¬Q)’. It doesn’t matter, of course, which

half of the disjunction you tackle first, and the numbers here just indicate the order of steps in one way of doing the calculation (and we’ve used different shapes of brackets to highlight the structure of wff):

For a fourth example, let’s evaluate the same wff for ‘P’⇒ F, ‘Q’⇒ T, ‘R’⇒ T, ‘S’⇒ T (and henceforth let’s not repeat the values of atoms on the right; and we’ll just number off the remaining steps of the calculation):

Finally, let’s take the same valuation of atoms but this time evaluate the new dis- played wff (again, we’ve not repeated the values of the atoms on the right):

Once more, we work from the inside outwards, evaluating the two subformulae in curly brackets, ‘{P ∧ (¬S ∨ Q)}’ (step 3), and ‘{¬R ∧ ¬P}’ (step 6) which

immediately yields the value of ‘¬{¬R ∧¬P}’ (step 7). That gives us the value of

the square-bracketed conjunction (step 8), and so the value of the whole wff. We will introduce some more ways of cutting down working in Chapter 11: but we’ve explained enough to enable you to tackle some first examples. Of course, you don’t actually need to write down the step numbers: that’s inessen- tial commentary. Practice quickly makes perfect.

9.8 Summary

• Atomic wffs in PL get interpreted, on a particular occasion of use, by being directly assigned meanings (we do this for as many atoms as we need for the purposes at hand). The connectives get meanings – expressing bare conjunction, inclusive disjunction, and strict negation – which remain con- stant across different uses of PL.

• The factual meaning of an atom fixes its ‘truth-conditions’ (i.e. tells us which situation has to obtain for the atom to be true). The factual meaning and the state of the world combine to fix the truth-value of (interpreted) atoms. Or at least, they do so given that we are setting aside cases of vague- ness, paradoxical sentences, etc.

• The standard interpretation of the connectives means that fixing the truth or falsity of each of the wffs A and B will also fix the truth or falsity of the whole conjunction (A∧B), the disjunction (A∨B), and the negation _¬A.

P Q R S ({S ∧ ¬ (Q ¬ P)}∧ ∨ {R ∨ ¬¬Q}) T F F F F F T F F F T F F F F T F 1 7 6 2 5 4 3 13 8 12 1110 9 P Q R S ({S ∧ ¬ (Q ¬ P)}∧ ∨ {R ∨ ¬¬Q}) F T T T F F T T T T T F 4 3 2 1 8 7 6 5 P Q R S ¬[{P ∧ (¬S ∨ Q)} ∧ ¬{ ¬R ¬P}]∧ F T T T T F F T F T F F T 9 3 1 2 8 7 4 6 5

Exercises 9 81

• This means that a valuation of the atoms in a wff of PL (i.e. an assignment of the value true or false to each of the relevant atoms) will in fact determine a valuation for the whole wff, however complex it is; and this valuation will be unique because of the uniqueness of its constructional history.

Exercises 9

A Suppose ‘P’ means Plato is a great philosopher; ‘Q’ means Quine is a great

philosopher; ‘R’ means Russell is a great philosopher. Translate the follow-

ing sentences into PL.

1. Either Quine is a great philosopher or Russell is. 2. Neither Plato nor Quine is a great philosopher. 3. Plato, Russell and Quine are great philosophers. 4. Not both Quine and Russell are great philosophers. 5. Quine is a great philosopher and Russell isn’t.

6. Either Quine and Russell are great philosophers, or Plato is. 7. It isn’t the case the Quine is and Russell isn’t a great philosopher.

B Suppose ‘P’ means Fred is a fool; ‘Q’ means Fred knows some logic; ‘R’ means Fred is a rocket scientist. Translate the following sentences into PL as best you can. (What do you think is lost in the translations, given that

PL only has the ‘colourless’ connectives ‘∧’, ‘∨’ and ‘¬’?) 1. Fred is a rocket scientist, but he knows no logic. 2. Fred’s a fool, even though he knows some logic.

3. Although Fred’s a rocket scientist, he’s a fool and even knows no logic.

4. Fred’s a fool, yet he’s a rocket scientist who knows some logic. 5. Fred is not a rocket scientist who knows some logic.

6. Fred is a fool despite the fact that he knows some logic. 7. Fred knows some logic unless he is a fool.

C Confirm that all the following strings are wffs by producing construction trees. Suppose that ‘P’ and ‘R’ are both true and ‘Q’ false. Evaluate the wffs by working down the trees. Then do the working again in the short form.

1. ¬(P∧R) 2. ¬¬¬¬(Q∨¬R) 3. (¬(P∨¬R)∧Q) 4. ((R∨¬Q)∧ (Q∨P)) 5. ¬(P∨ ((Q∧¬P)∨R)) 6. ¬(¬P∨¬(Q∧¬R)) 7. (¬(P∧¬Q)∧¬¬R) 8. (((P∨¬Q)∧ (Q∨R))_{∨ ¬¬(Q}∨¬R)) 9. (¬(¬P∨¬(Q∧¬R))∨ ¬¬(Q∨¬P)) 10. ¬((¬(P∧¬Q)∧¬¬R)∧¬(¬(P∨¬R)∧Q))

Before putting PL to use in the following chapters, we should pause to make it absolutely clear how the ‘A’s and ‘B’s, the ‘P’s and ‘Q’s, are being used, and why it is very important to differentiate between them. We should also explain the rules governing the use of quotation marks. By all means, skip this chapter for the moment if you are impatient to get to see our work on PL being put to use: but return later, because it is important to get clear on these points.