We have now completed the set of rules of inference for PL and by now you should be familiar both with each individual rule and with the ways in which those simple rules can be combined in proof-construction. However, so far in our discussion, we have omitted any mention of an important distinction which may be drawn between two different kinds of rule of inference for PL.
It is precisely that distinction which I consider in the present section.
The distinction in question is that between primitive rules and derived rules. In the present context, we can simply consider our set of rules of inference for PL as a set of primitive rules. To say that a rule is primitive is just to say that the addition of that rule to the set of rules of inference for a formal system allows new sequents to become provable in that system. In contrast, a derived rule is a rule whose addition to that set of rules will not allow anything new to be proved in the formal system. In all honesty, drawing the distinction in this way turns out to imply that not all of our existing stock of rules are primitive, i.e. certain of those rules can, in fact, be derived within the system, given the remaining rules. This should not be too surprising. As I admitted earlier, any proof involving RAA can, with a
REVISION EXERCISE IV
1 Prove that the following are valid sequents of PL:
little hard work, also be proved using MT and DNE (in the not too distant future you will have the opportunity to prove that this is so).
For present purposes, however, focus on the idea of a derived rule of inference as a rule whose addition to the system does not allow anything new to be proved in that system. As such, derived rules may seem rather dull: aren’t such rules, in a sense, redundant? Indeed. But they offer important advantages none the less. As we shall see, derived rules can be used to abbreviate long, tedious proofs. As such, they constitute a logical (proof-theoretic) economy. The two particular derived rules which I consider here are designed precisely in order to maximise the possibility of exploiting that proof-theoretic economy.
The first such rule is known as theorem-introduction, or TI. Quite simply, the rule allows us to enter any theorem which we have already proved on any line of any proof with an empty set of dependency-numbers and annotated ‘TI’. The point about the dependency-numbers here reflects the familiar fact that a theorem is a logical truth, as we discussed in Chapter 2.
So, at any point in the process of proof-construction, we may enlist the help of any potentially useful theorem free of charge. Something of the usefulness of TI should be obvious already but, in fact, this rule is even more useful than might first appear. For not only can we introduce a particular proved theorem using TI but we can also introduce any formula of that form. To illustrate, not only can we introduce on an arbitrary line—call it line ‘n’ — the theorem P → P, as follows: In fact, provided the formula in question can be constructed from the original theorem simply by uniformly replacing the same original constituent formula with the same new well-formed formula, then that resulting formula may be introduced by TI. This practice of careful replacement, careful swapping, of old constituent formulas with new constituent formulas is known as uniform substitution. Simple as it may seem, uniform substitution is of crucial importance in formal logic generally and we will have cause to appeal to that notion again later.
The second derived rule we will consider here extends still further the possibilities for introducing formulas to lines of proof during the process of
proof-construction. This rule is not theorem-introduction but sequent-introduction, SI. In short, the rule of sequent-introduction allows us to exploit our existing stock of proved sequents analogously to the way in which theorem-introduction allowed us to exploit our existing stock of proved theorems. Thus, if in the process of proof-construction, you were to come across the formulas P → Q and P → ~Q on separate lines of proof, SI would enable you to write ~P on the next line of proof without having to go through the obvious RAA which would usually be required. Instead, you could simply enter ~P, annotate the new line with the line numbers of the two formulas involved and ‘SI’, and complete the line by pooling the numbers of the old lines to create the set of dependency-numbers of the new line. Again, something of the usefulness of SI is apparent and, again, via the notion of uniform substitution we can amplify the utility of SI by allowing it to apply not only to proved sequents but also to any substitution-instance of a proved sequent.
This particular pair of derived rules maximises the possibilities for making proof-theoretic economies in the process of proof-construction, and both are extremely useful in allowing us to see how we might prove new, complex sequents which we have not already proved. However, a few words of warning are apt here. First, many logicians forbid the use of both rules for exam purposes, i.e. ‘TI and SI may not be used’ is a commonplace of formal logic exam papers. Alternatively, a numbered list of legitimate theorems and/or sequents may be made explicit in the exam paper and applications of TI and/or SI may be restricted to that list (there is also no guarantee that such a list will be available prior to the exam).
These particular rules can also be the source of some cruel (if amusing) trickery on the part of logicians. In my own first logic course, for example, the logic lecturer (who shall remain nameless) pointed out that we should feel free to use TI as we pleased during the logic exam. Further, he also helpfully pointed out that our textbook contained a numbered list of theorems which we could use and, indeed, encouraged us to learn that list by rote (the course textbook was in fact Elementary Logic by Benson Mates [1972] which, in a number of respects, is a truly excellent text). When I and fellow students consulted the text we found that Mates began his list over pp. 98–9 where theorems 1–7 were stated and continued over pages 100–1.
All in all, ten theorems were stated over these pages. But we quickly found that the list continued up to p. 106 until no fewer than 100 theorems had been listed. Finally, look out for the equally amusing strategy of allowing the use of TI in an exam provided you also include a proof of the theorem in your exam paper!
In the light of these facts, I will not give any more emphasis to TI and SI in particular but will instead look very briefly at derived rules in general and the question of how to show that a rule is derivable in particular. To exhibit the derivability involved in derived rules, i.e. to show a rule to be
derivable, is really just to show a way of getting from the inputs for the rule to the outputs for the rule without actually using the rule itself. Hence, we show a rule to be derivable within a system by deriving the output of the rule from the input for the rule using only the other rules of that system, i.e.
by exploiting only the remaining stock of primitive rules. In the present context, for any derived rule, we will separate the input from the output by a line thus:
input output
Looked at in this way, showing a rule to be derivable comes to showing how to derive the output from the input. For that purpose, we need not adapt or extend our notation any further, i.e. both inputs and outputs will be given in the usual notation. Moreover, to show that a rule is derivable in PL we can simply construct a proof of the given output from the given input using only the other rules of the system. Exercise 3.10 contains some examples of derived rules. Consider each rule carefully. Many of these rules exemplify principles of inference which are honoured with traditional names and, as appropriate, I have stated the name next to the rule. In each case, construct a proof of the stated output for the rule from the stated input in the usual manner. Of course, TI and SI may not be used!
EXERCISE 3.10
1 Show the following rules to be derivable in PL: