PL trees vindicated
19.3 A corollary and a further result
First, we’ll quickly note a corollary of the Basic Result: as announced in §16.4,
the order in which the tree-building rules are applied (when we have a choice) can’t matter. Why?
Well, note that the Basic Result entails that if an argument is tautologically valid, then any properly built tree will close, irrespective of the order in which we apply the relevant tree-building rules. Likewise, if an argument is invalid, any appropriate tree will stay open, again irrespective of the order in which tree-
19.3 A corollary and a further result 183
building rules are applied.
Secondly, let’s note a result which slightly extends our argument for (C). Let’s say that a set Σ of PL wffs is saturated if the following conditions obtain:
(a) if a wff of the form ¬¬A is in Σ, then so is A;
(b) if a wff of the form (A∧B) is in Σ, then so are both A and B;
(c) if a wff of the form ¬(A∨B) is in Σ, then so are both ¬A and ¬B;
(d) if a wff of the form (A∨B) is in Σ, then so is at least one of A and B;
(e) if a wff of the form ¬(A∧B) is in Σ, then so is at least one of ¬A and ¬B.
Likewise a set of PLC wffsΣ is saturated if it satisfies (a) to (e) plus the conditions (f) if a wff of the form (A⊃ B) is in Σ, then so is at least one of ¬A and B; (g) if a wff of the form ¬(A ⊃ B) is in Σ, then so are both A and ¬B; (h) if a wff of the form (A B) is in Σ, then so are either both A and B
or both ¬A and ¬B;
(i) if a wff of the form ¬(A B) is in Σ, then so are either both A and
¬B or both ¬A and B.
(Why ‘saturated’? Because such a set is saturated with truth-makers, containing a truth-maker for every non-primitive wff in it.)
We’ll also say that Σ is syntactically consistent if it contains no pairs of wffs of the form A, ¬A. (Compare the semantic definition of logical consistency in
§2.1.)
The conditions (a) to (i) will have an entirely familiar look. They correspond exactly to the rules for building up a path on a PL (PLC) tree; so it is immediate that the wffs on a completed open path form a syntactically consistent saturated set. The set will be consistent, since an open path contains no contradictions, and saturated because checking off every non-primitive wff on the path in order to complete the tree ensures that (a) to (i) are fulfilled on the path. So our earlier result that a completed open path is satisfiable is just a version of the rather more general result that every syntactically consistent saturated set Σ of PL
(PLC) wffs is satisfiable – i.e. there is a valuation which makes all the wffs in Σ
true.
The proof of this result goes exactly as before. First, we define the chosen val- uation for Σ: if the atom A appears naked in Σ, assign it the value T; otherwise assign that atom the value F. It is simple to show – as in (C1) – that this chosen valuation makes all the primitives in Σ true, and then – as in (C2) – that a valua- tion which makes all the primitives in Σ true will make every wff in Σ true.
The advantage of talking about ‘consistent saturated sets of wffs’ as opposed to ‘sets of wffs on a completed open branch of a tree’ is that it emphasizes that the arrangement of wffs onto a tree isn’t of the essence here. It’s the structure described by (a) to (i) above that ensures that the set, if consistent, is satisfiable. And, if we abstract from the case where the wffs are to be written out on a common-or-garden finite tree, then we can even allow the case where the con- sistent saturated set Σ contains infinitely many wffs. The proof that consistent saturated sets are satisfiable doesn’t depend on any assumption of finitude. Of
184 PL trees vindicated
course, in the case of propositional logic, the consideration of infinite sets of wffs is mostly of ‘technical’ interest. But matters are different, as we will see in Chap- ter 30, when we turn to thinking about more complex logical systems.
19.4 Summary
• We have shown that the tree test is sound: i.e. if an argument is judged valid by the test – i.e. the appropriate tree closes – then it is indeed tauto- logically valid.
• We have shown that the tree test is complete: i.e. if an argument is tauto- logically valid, then the tree test will reveal it as such by the appropriate tree closing. Equivalently, if an appropriate tree stays open, the argument under test is invalid.
• The proof of these results shows that it doesn’t matter in which order we apply the tree-building rules (when we have a choice).
Exercises 19
A Show how to extend the following sets of wffs into consistent saturated sets (hint: think of the given sets as ‘trunks’ of trees; you want to extend the trunks to some compete open path).
1. ¬Q, (P ⊃ Q), (¬¬R ⊃ (P∧R))
2. ¬(Q∧ (P′∨R)),¬(¬Q∧P), ((P′∧ S)∨ (R∧S))
3. (Q⊃ (P∨R)),¬(P ⊃ S), ((P∨ S)∧ (¬Q ⊃ S)), ((S∨R)⊃ ¬P) In each case, what is the ‘chosen valuation’ for the set? Check that it does indeed make true all the wffs in the set.
B Rewrite the arguments (C1) and (C2) to turn them into explicit arguments for the thesis that every syntactically consistent saturated set Σ of PL (PLC) wffs is satisfiable. Confirm that these arguments don’t depend on whether Σ is finite.
The unsigned trees that we have been exploring were introduced as shorthand for signed trees, a way of setting out arguments in English about PL inferences. But, as we will now explain, we can also think of them as proofs in PL.
20.1 Choices, choices …
Outside the logic classroom, when we want to convince ourselves that an infer- ence is valid, we don’t use techniques like the truth-table test. As we pointed out way back in Chapter 5, we usually seek proofs: i.e. we try to show that the desired conclusion follows from the premisses by chaining together a sequence of obviously acceptable steps into a multi-step argument.
Suppose, then, that we want to develop ways of warranting PL inferences by means of proof-reasoning inside PL. How should we proceed? (By the way, take talk of ‘PL’ in this chapter to cover PLC as well.)
It is very important to stress that there is no single ‘right’ way of setting out proofs. Indeed, in this book we have already touched on no less than three dif- ferent ways of setting out multi-step arguments.
(1) In Chapter 5, we introduced proofs set out linearly, one step below another (though we also added an indentation device to signal when sup- positions were being made temporarily, for the sake of argument). (2) Then in Chapter 8 (see §8.2), we noted that some linear arguments can
very naturally be rearranged into tree structures, with the initial assump- tions appearing at the tips of branches, and the branches joining as we bring together propositions to derive a consequence. We work down the tree until we are left with the final conclusion of the proof at the bottom of the trunk. (Our examples happened to feature arguments in augmented English about which strings of PL symbols are wffs.)
(3) In Chapters 16–18, we found that some other arguments are naturally set out as inverted trees. We start with some assumptions at the top of the tree, and work down trying to find contradictions, splitting branches at various choice points as we descend the downwards-branching tree. (Our