4 We can make further use of our metalinguistic variables to represent semantic equivalences between types of compound formulas in PL quite generally. Show that I have correctly identified the following pairs as semantically equivalent.
X
Truth-Trees
You will be pleased to learn that the final semantic method which we will consider together in this chapter is a deceptively simple one which is
remarkably easy to use. The method also has enormous potential which we will be able to exploit to our advantage both in this section and beyond. As we shall see, when the new method is combined with certain insights which we have already gained in the course of this chapter we will have arrived at a semantic method which we will still be able to exploit in the final chapter of this book.
The new method provides a straightforward way of testing formulas for consistency. The idea of consistency between sentences in a language is of fundamental importance in logic. In formal logic, however, the term
‘consistency’ is used in a number of importantly different senses. In the present context, the consistency of a set of well-formed formulas just implies that each and every well-formed formula in that set can be true at one and the same time, i.e.:
A set of well-formed formulas of PL is consistent if and only if every member of that set can be true simultaneously.
It follows that a proof of consistency in this sense is a proof of the existence of a true interpretation of all the members of that set. Earlier in this chapter, we defined a true interpretation as a model. So, in the present context, we can equally well say that a proof of consistency is a proof of the existence of a model. The new method is precisely a test for consistency in that sense, i.e. it provides an answer to the question: could the formulas comprising a given set of formulas all be true together? In fact, the new method guarantees us an answer to just that question. If there is a true interpretation for all the formulas in the set then the new method will find it. In other words, the new method is an effective procedure for model-detection in PL.
The way in which the method proceeds to answer questions about consistency is by constructing what is known as a consistency-tree for a set of formulas. The tree format used is not entirely dissimilar to the format we exploited for the syntactical trees which we considered in Chapter 2. Like syntactical trees, consistency-trees are upside-down trees which break complex formulas down into their simple constituents. Indeed, consistency-trees can be more tree-like than their syntactical counterparts just because, when we construct a consistency-tree, we first list each formula on a separate line, one underneath the other. This procedure reflects the most obvious way in which the formulas might all be true together and gives a nice, trunk-like aspect to the inverted tree. Having listed the complex formulas we are interested in we then begin to break down or ‘develop’ each complex formula in terms of certain development rules. As ever, the development rules exploit the type of formula involved, i.e. conjunction, disjunction, etc.
The rule for conjunction simply requires us to write each conjunct on a separate line one below the other, like so:
In contrast, the rule for disjunction requires us to create two new branches, placing each disjunct on the end of a separate branch. This produces a pleasing, tree-like, branching effect:
If we always develop every conjunction before we develop any disjunction then we will preserve the trunk-like effect for as long as possible and we will ensure that all the branching occurs at the top of the tree. Just as you would expect, that is exactly how we do proceed.
Having developed each formula in terms of that procedure we then carefully study the formulas in each branch, reading up from the tip of the branch back to the very beginning of the trunk. What we are looking for are contradictions among the formulas lying on that branch. If a branch does not contain any contradiction that branch is live and we mark that fact by writing a ‘ ’ under that branch. However, if a branch does contain a contradiction then that branch is dead and we record that fact by writing an
‘ ’ under it. If every branch dies then the tree is dead.
Although consistency-trees are easy to construct and produce pleasant-looking structures we should never lose sight of the fact that we use the trees to test for consistency. Indeed, each branch represents a different possible way in which all the formulas involved might be true together. So, for example, in the case of disjunction, our splitting the branch represents two possible ways in which the disjunction might be true, i.e. if either disjunct is true. But because there is only one way in which a conjunction can be true, namely, when both conjuncts are true, we do not split the branch. This represents the fact that there is only one possibility involved here.
In fact, each branch represents an attempt to assign truth-values to the component sentence-letters of the formulas so as to bring those formulas out as true simultaneously. Hence, each branch represents an interpretation.
Moreover, because each branch represents an interpretation of the components of each formula, we know that when a branch dies that particular interpretation results in a contradiction, i.e. an inconsistency. If every branch dies then there is no interpretation which does not result in inconsistency. But, if that is so, then there is no way in which all the formulas being tested could all be true together. Therefore, that set of formulas is inconsistent. Conversely, if there is even one live branch then there is one
way in which all the formulas being tested could be true together, i.e. a true interpretation or model. So, that set is consistent.
Let’s consider some examples:
Reading up the right-hand branch, we have a contradiction between Q on line 6 and ~Q on line 5. So, the right-hand branch is dead. However, there is no contradiction in the left-hand branch. So, we have a live branch on the left. Remember: each branch represents an attempt to assign truth-values to the sentence-letters composing the formulas in a way which makes all those formulas true. And a live branch represents an assignment which does not result in inconsistency. So, in this case there is an assignment which does not result in inconsistency and therefore this set is a consistent set.
Consider another case:
This time, all three branches are dead. So there is no interpretation under which all the original formulas could all be true together. It follows that this set is inconsistent.
Note carefully how each line containing one of the original formulas is developed as the tree progresses. In each case, I have annotated each new line with the number of the line developed at that point. Although this is a useful device for making explicit exactly which line is developed at which point, you may well find that you are not required so to annotate your trees in your particular course in formal logic. The same point holds for subscripting each contradiction with the line numbers of formulas which contradict one another. None the less, in the first few cases you attempt on your own, you may well find that these devices help you to keep track.
Before we move on, Exercise 4.6 gives a few examples for you to try yourself.
EXERCISE 4.6
1 Test the following sets of formulas for consistency using consistency-trees: