Syllogistic Logic and its Extensions

Transcription

1 1/31 Syllogistic Logic and its Extensions Larry Moss, Indiana University NASSLLI 2014

2 2/31 Logic and Language: Traditional Syllogisms All men are mortal. Socrates is a man. Socrates is mortal. Some men are mortal. Socrates is a man. Socrates is mortal. All frogs are reptiles. All reptiles are animals. All frogs are animals.

3 3/31 More examples All frogs are reptiles. All frogs are animals. All reptiles are animals. All sagatricians are maltnomans. All sagatricians are aikims. All maltnomans are aikims. There is an exact definition of validity for arguments. The form is as important, even more important, than the particular words.

4 4/31 More examples All X are Y. All Y are Z. All W are X. All W are Z. So valid arguments can have more than two premises.

5 5/31 Syntax and Semantics Probably the key point of logic is that there is a distinction between syntax and semantics. The idea is that syntax is the raw symbols. The semantics is where we get the meaning. So in our examples, we need some context or model to give a meaning. In our examples, the syntax will start with some variables p, q, n, n 1,.... Then our sentences are expressions of the form All p are q.

6 Semantics To say whether All sagatricians (s) are maltnomans (m). is true or not needs a context. This is given by a few things: First, a set U called the universe. Second, for the words sagatrician and maltnoman, we need sets [[sagatrician]] U and [[maltnomanan]] U. Given all of this, we say that All s are m is true in the context if [[s]] [[m]]. Otherwise, All s are m is false in the context. What should we say about Some s are m and No s are m? 6/31

7 7/31 Semantics, again Syntax: All p are q, Some p are q, No p are q Semantics: A model M is a set M, and for each variable p we have an interpretation [[p]] M. M = All p are q iff [[p]] [[q]] The symbols M = ϕ is read as M satisfies ϕ. A statement like M = All p are q could also be read as All p are q is true in M

8 8/31 One fine point on the definition is that if [[X]] is the empty set, then our sentence All X are Y is true! So in this room now, All people in the room over 7 feet tall are standing is (on this definition) true. A Quirk This strange point will lead us to various issues over the next few lectures. For now, it might be best to say that it s true because there are no exceptions. But we again admit that the semantics of All that we are giving is not what most people would agree to in cases where [[X]] =.

9 Validity of Arguments At this point, we know how to give the semantics of single sentences. We say that a sentence ϕ follows from sentences A 1,..., A n if every model that makes all of the As true also makes ϕ true. We write this as A 1,..., A n = ϕ and we also say that the A s semantically imply ϕ. To argue that A 1,..., A n = ϕ we need some reasoning. Usually, we do this in English and in an informal way, just as one would do ordinary reasoning. But to argue that A 1,..., A n = ϕ we can produce a counterexample. In all of this work, the main thing is that we have a rigorous definition. 9/31

10 10/31 A small note on notation We use letters like Γ (Greek letter Gamma) for sets of sentences. And then we would write Γ = ϕ to mean that every model of all the sentences in Γ is also a model of ϕ. However, if Γ is a set that we have listed out, say Γ = {A 1, A 2,..., A 104 }. then usually we would write Γ = ϕ as rather than as A 1, A 2,..., A 104 = ϕ {A 1, A 2,..., A 104 } = ϕ. That is, we drop the set braces on the left of the = symbol. We do this to make things a little more readable.

11 11/31 Validity: the idea premises {}}{ A 1, A 2,..., A n = conclusion {}}{ S The intuition is that A 1, A 2,..., A n = S means that any circumstance in which the premises A 1, A 2,..., A n are all true is also a circumstance in which the conclusion S is true

12 12/31 Some points Note the difference between syntax and semantics. = is intended to mean follows by general-purpose reasoning. We can check whether our definitions match with our intuitions. In the case of our very simple fragment, this mostly is right. The main exception is that people usually wouldn t say All X are Y in a context where they know that there are no X.

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14 14/31 Formal proofs: a preliminary point A formal proof is like a caricature of the reasoning that a person would go through in showing that some premises lead to a conclusion. It is very common in introductory logic classes to present one or another kind of formal proof systems. (There are probably hundreds of them.) Working with a formal proof system is usually a tedious and boring experience.

15 15/31 Let Γ be a set of sentences {A 1,..., A n }. A proof tree over Γ is a tree following properties: Proof Trees 1 The leaves are either labeled with sentences from Γ, or with sentences of the form All X are X. 2 The interior leaves match one of the rules of our system (see the next slide). The trees are drawn with the root at the bottom and the leaves at the top. If there is a proof tree over Γ whose root is labeled ϕ, we write Γ ϕ. We say that ϕ is provable from Γ in our system.

16 16/31 The rules for building trees All p are p All p are n All n are q All p are q

17 17/31 Example Let Γ be the set {All A are B, All Q are A, All B are D, All C are D, All A are Q} Let ϕ be All Q are D. Here is a proof tree showing that Γ S: All A are B All B are D All Q are A All A are D All Q are D All of the leaves belong to Γ. Note also that some elements of Γ are not used as leaves. This is permitted according to our definition. The proof tree above shows that Γ S.

18 What are we doing here? The idea is that proof trees are our model of basic reasoning using the words all, some, no. A proof tree is like a caricature of a real proof. It can be examined (and even constructed) by a person or computer who has no understanding of anything but the rules! There are several hopes about this work: The whole thing will scale up to include many more words. (This would call on linguistic semantics to provide the correct notion of context.) The formal relation should have something to do with = (logic) The proof system should have something to do with actual human reasoning (psychology) A computer should be able to work with without understanding anything. 18/31

19 19/31 Soundness A computer could check whether a purported tree actually satisfies our definition, even if it didn t understand All. So one important question is: what is the relation between Γ A and Γ = A? Soundness If Γ S, then Γ = S. This means that proof trees do not lead us astray: if Γ ϕ, then in any context where the sentences of Γ all hold, ϕ too must hold. Our proof system will not lead us to believe that bogus syllogisms are in fact valid.

20 20/31 Soundness Here is the basic idea of why the Soundness Lemma holds. The two most basic facts about are: 1 X X for all sets X. 2 For all sets X, Y, and Z: if X Y and Y Z, then X Z. (Probably the third basic fact would be that X for all X.)

21 Let s go back to our example proof tree. Soundness Sketch, Continued All A are B All B are D All Q are A All A are D All Q are D Take any model, say M. Assume that in M, [[A]] [[B]], etc. We have to show that in this same model M, [[Q]] [[D]]. The idea is to use our proof tree and read it as talking about subsets of this one model M: [[A]] [[B]] [[B]] [[D]] [[Q]] [[A]] [[A]] [[D]] [[Q]] [[D]] And then going downward mirrors intuitively valid reasoning in the model. Since the model M was arbitrary (had no special features), the conclusion Γ = S holds. 21/31

22 22/31 Review: Γ ϕ and Γ = ϕ Γ = ϕ means that every model of all sentences in Γ is again a model of ϕ. Γ ϕ means that there is a formal proof in our system using the sentences in Γ as assumptions and ϕ as the conclusion. It s important to see that these two concepts Γ = ϕ and Γ ϕ are different. It will turn out that Γ = ϕ if and only if Γ ϕ but this should not be obvious!

23 Completeness At this point, we know that our system is sound: If Γ S, then Γ = S. Perhaps more important is the converse of this: Completeness If Γ = S, then Γ S. Before we turn to the proof, it is important to see what this says. Soundness says that the proof system will not lead us astray. Completeness tells us that if Γ semantically implies A, then we can find one of our (semantics-free!) proof trees showing Γ A. 23/31

24 24/31 A proof tree in this system Γ = We see that Γ All B are G. Do you think that Γ All D are E? All A are B, All A are C, All B are C, All C are B, All C are D, All B are E, All D are G, All F are G, All G are F

25 25/31 Preorders Definition A preorder is a pair (P, ), where P is a set and is a relation on it with the following properties: reflexive p p transitive If p q and q r, then p r. We need not have the following property: anti-symmetric if p q and q p, then p = q. An anti-symmetric preorder is a partially ordered set (poset).

26 26/31 A picture of a preorder Γ = All A are B, All A are C, All B are C, All C are B, All C are D, All B are E, All D are G, All F are G, All G are F F, G D B, C E The set P here is {A,..., G}. The order go up (counting a node s mate, if any). A

27 27/31 Downsets in preorders In a preorder, p = {x : x p}. F, G F = G = {A,..., G} D E D = {A, B, C, D}, E = {A, B, C, E} B, C B = {A, B, C} = C A A = {A} is monotone: if p q, then p q.

28 28/31 Proof of completeness Suppose that Γ = All X are Y. At this point, we re going to make up a special model. Let M be the set of variables. Define A B to mean that Γ All A are B. Check that this is reflexive and transitive, using the logic. We get to interpret the nouns any way we like. We use the downsets: By transitivity, M = Γ. [[A]] = A = {B : B A}

29 28/31 Proof of completeness Suppose that Γ = All X are Y. At this point, we re going to make up a special model. Let M be the set of variables. Define A B to mean that Γ All A are B. Check that this is reflexive and transitive, using the logic. We get to interpret the nouns any way we like. We use the downsets: [[A]] = A = {B : B A} By transitivity, M = Γ. In more detail, suppose Γ contains All C are D. Then if W C, then also W D.

30 Proof of completeness Suppose that Γ = All X are Y. At this point, we re going to make up a special model. Let M be the set of variables. Define A B to mean that Γ All A are B. Check that this is reflexive and transitive, using the logic. We get to interpret the nouns any way we like. We use the downsets: [[A]] = A = {B : B A} By transitivity, M = Γ. So [[X ]] [[Y ]]. But by reflexivity X [[X ]]. And so X [[X ]]; this means that X Y. 28/31

31 29/31 Proof of Completeness I am going to show one piece of the completeness proof. Suppose that Γ = All X are Y We need to show that Γ All X are Y.

32 30/31 Proof of Completeness Suppose that Γ = All X are Y We need to show that Γ All X are Y. Define a model M by: M = all variables [[U]] = {V : Γ All V are U}

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