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COSC462 Lecture 4 More Metatheory COSC462 Lecture 4 More Metatheory Willem Labuschagne University of Otago Abstract We look at some more nice properties of propositional lan- guages and classical entailment, including compactness, and de- scribe a very crude automated reasoning algorithm. 1 Model-theoretic properties As in the previous lecture, we avoid complications by assuming that the ontology for every language LA is (S; V ) with S = WA and V the identity function on WA. The Ine¤ability Theorem showed us a way in which propositional lan- guages LA with in…nite A are di¤erent from those with …nite A. However, there are more similarities than di¤erences. For example, regardless of the size of A, the set of models of a sentence can be constructed from the models of the atoms in by set-theoretic operations like union, intersection, and complementation. Theorem 1 For all '; LA it is the case that 2 ( ') = (') M : M (' ) = (') ( ) M ^ M \M (' ) = (') ( ) M _ M [M Proof. To see that ( ') = (') it su¢ ces to note that s satis…es ' i¤ s fails to satisfyM'. : M : To see that (' ) = (') ( ), it is su¢ cient to note that s satis…es ' Mi¤ s^satis…esM' and\Ms satis…es . To see that^ (' ) = (') ( ), note that s satis…es ' i¤ s satis…es ' Mor s satis…es_ M . [M _ 1 There is another way in which all propositional languages behave very similarly, regardless of whether A is …nite or in…nite. This is the compactness property, which is a kind of …niteness property having to do with satis…ability. De…nition 2 (Satis…ability) A set LA of sentences is satis…able if there is at least one state s S that satis…es every . 2 2 So a set of sentences is satis…able if its set of models () is nonempty. (In the case of a single sentence , is satis…able if M( ) = M 6 ?). A contradiction like p0 p0 is an unsatis…able sentence, while a ^ : set like = p0; p1; (p1 p0) is an example of an unsatis…able set of sentences (asf you are invited!: tog verify). Theorem 3 (Compactness) A set of sentences is satis…able i¤ every …nite subset of is satis…able. Proof. One direction of the proof is easy. If is satis…able, then there is a state s satisfying all the sentences in , and so every …nite subset of is satis…ed by s. The converse direction is more of a challenge. Suppose every …nite subset of is satis…able. We shall build a bigger set , such that , and show that is satis…able. The reason we build instead of just working with is that is going to be big enough to help us de…ne a way to allocate truth values to all sentences, in other words we use to build a state (valuation) which turns out to satisfy the whole . First of all, let us imagine the sentences of LA written down in order as 1; 2;::: It’s not hard to dream up an algorithm to do this — we might use a grammar for the language to generate longer and longer strings, and write them down in the order in which they are generated. Now let us de…ne a sequence of sets that gradually add sentences to . Let 0 = , and for every n let n+1 = n n+1 if every …nite subset [f g of n n+1 is satis…able, or let n+1 = n n+1 otherwise. There[ f areg a couple of things to notice. The[ construction f: g of the se- quence is something we can imagine being carried out, but we may not ourselves be able to do it in practice, because it not only goes on forever but it may be very hard work to decide whether every …nite subset of n n+1 is satis…able. Nevertheless, if we had unlimited time we could[ usef truthg tables and do it. So it is possible in principle. The second thing to notice is that every i has the property, by virtue of our construction, that all its …nite subsets are satis…able. The reason is this. The initial set 0 has the property, since 0 = . Suppose n has the property that all its …nite subsets are satis…able, but neither n n+1 nor n n+1 has the property. Then there must be [ f g [ f: g 2 …nite subsets X and Y of n such that X n+1 and Y n+1 are both unsatis…able. But what about X Y[? f Anyg valuation[ satisfying f: g [ X Y must satisfy either n+1 or n+1. So either we contradict the [ : unsatis…ability of X n+1 and Y n+1 , or else the …nite subset [ f g [ f: g X Y of n is unsatis…able, which is also a contradiction, since we [ assumed all …nite subsets of n to be satis…able. Let’s return to the construction. Take to be the union of all the n, in other words take to be the set of all sentences which belong to at least one of the n. This set has three interesting properties. Obviously , since = 0. Also, for every sentence LA, either or else . And …nally, every …nite subset of is2 satis…able. Why?2 Well, because: 2 that …nite subset is, for some n, a …nite subset of n and we know that the …nite subsets of every n are all satis…able. Now consider the valuation v such that, for every p A, v(p) = 1 i¤ 2 p . We claim that for each sentence LA, the state v satis…es i¤2 . 2 The2 proof uses induction. The shortest sentences are the atoms, and if is an atom then by construction v satis…es i¤ . Assume that this holds for all sentences shorter than k (the Induction2 Hypothesis). Let be of length k. There are various cases: If = ' then ' is shorter than k and so we may argue that if v sat- is…es then: v does not satisfy ', so ' = by the Induction Hypothesis, and so . Conversely, if then2 ' = , so by the Induction Hypothesis2v does not satisfy ', and2 so v satis…es2 . If = ' , then both ' and are shorter than k and so we may argue that^ if v satis…es then v satis…es both ' and , so by the Induction Hypothesis ' and , and now we are left with two possibilities: either ' 2 or '2 = . We can eliminate the latter, for if ' = ^ then2 (' ^) 2, and now the …nite subset '; ; (' ) ^of 2is unsatis…able.: ^ So2' . Conversely, suppose f :. Now^ g both ' and , for^ if2 not we again get a …nite subset2 of that is unsatis…able.2 For2 example, if ' = , then ' and so '; ' is an unsatis…able …nite subset of2 . But we: know2 that thef: …nite subsets^ g of are all satis…able. And so v satis…es both ' and , whence v satis…es . The cases = ' , = ' , and = ' are similar and left for the exercises._ ! $ The induction now follows, so that for sentences of all lengths it is the case that they are satis…ed by v i¤ they belong to . But since v satis…es all the sentences in , v certainly satis…es all the sentences in . So is satis…able. The Compactness Theorem illustrates a technique that logicians of- 3 ten use to prove results about logic, namely the idea of a maximal sat- is…able set . Versions of the Compactness Theorem could be proved for di¤erent kinds of logic, but since our concern is with applied logic rather than logic as a part of mathematics, we shall not do so. In order to fully appreciate the importance of the Compactness The- orem we shall pause to examine some of its consequences. Recall that i¤ () ( ):For example, ; because every valua- tion satisfyingM as M well as is of coursef a valuationg satisfying . Corollary 4 If then there is some …nite subset 0 such that 0 . Proof. First we establish a connection between entailment and un- satis…ability, namely that i¤ is unsatis…able. [ f: g Suppose . Then every valuation satisfying all the sentences in also satis…es . So no valuation can satisfy all the sentences in as well as satisfying . So the set is unsatis…able. Conversely, if : is unsatis…able,[ f: g then every valuation satis- [ f: g fying all the sentences of must also satisfy , so that . Now we use the connection between entailment and unsatis…ability. Suppose . Then is unsatis…able. [ f: g So 0 is unsatis…able for some …nite 0 (otherwise by compactness[ f: g would have to be satis…able). [ f: g So 0 for some …nite 0 . Isn’tthis remarkable? Use as many sentences as you like to build a set . Look at the set of models of . Pick any sentence which is true in all those models. Then there is a …nite subset of , which is really just another way of saying that there is a single sentence (since we could take the conjunction of all the sentences in the …nite subset) such that . On the other hand, maybe it’snot so remarkable. The set expresses some information (has some nonmodels). If then this just means expresses some of the information in . And is a …nitely long string. So we would expect to …nd some …nite part of that expresses all the information in (and perhaps even more). So the corollary to the Compactness Theorem is a vindication of our intuition about information, not a surprise! My treatment of the Compactness Theorem is loosely based on that in Enderton H: A Mathematical Introduction to Logic (2nd edition), Har- court/Academic Press 2001, which I would describe as one of the better logic textbooks out there.
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