# Lecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view.

Save this PDF as:

Size: px
Start display at page:

Download "Lecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view."

## Transcription

1 V. Borschev and B. Partee, October 17-19, 2006 p. 1 Lecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view. CONTENTS 0. Syntax and semantics; proof theory and model theory Syntactic provability and semantic entailment Semantic entailment and validity Soundness and completeness of a logic Consistency, completeness, independence, and other notions Some syntactic concepts Some semantic concepts Soundness and completeness again Independence Categoricity An algebraic view on provability and on the notions discussed above Closure systems Galois connection Morphisms for models; categoricity... 8 Appendix: An axiomatization of statement logic... 9 Appendix: Note on model-theoretic vs. proof-theoretic syntax... 9 Homework 9, due Oct Reading: Chapter 5, Chapter 8: , of PtMW, pp , Good supplementary reading: Fred Landman (1991) Structures for Semantics. Chapter 1, Sections 1.1 and 1.3., which we draw on heavily here. Another nice resource (thanks to Luis Alonso-Ovalle for the suggestion) is Gary Hardegree s online in-progress textbook Introduction to Metalogic : (Ch 5: the semantic characterization of logic, with notions of consistency, etc.; Ch 14: the semantics of classical first-order logic.) The notation is slightly different from ours and it may be hard to read those chapters in isolation, especially Ch 14, since the book is cumulative. Anyone who has a chance to take Hardegree s Mathematical Logic course would get a thorough grounding in all the notions we are discussing here and more. 0. Syntax and semantics; proof theory and model theory Starting out informally (see Chapter 5 of PtMW), we can say that the distinction between syntax and semantics in logic and formal systems is the distinction between talking about properties of expressions of the logic or formal system itself, such as its primitives, axioms, rules of inference or rewrite rules, and theorems, vs. talking about relations between the system and its models or interpretations. Examples: Syntactic activity: Constructing a proof from premises or axioms according to specified rules of inference or rewrite rules. (Proof theory is about this.) Semantic activity: Demonstrating that a certain set of axioms is consistent by showing that it has a model (see Section 2 below, or Ch. 8 in PtMW.) (Model theory is about such things.) Syntactic notions: Well-formedness, well-formedness rules, derivations, proofs, other notions definable in terms of the forms of expressions.

2 V. Borschev and B. Partee, October 17-19, 2006 p. 2 Semantic notions: truth, reference, valuation, satisfaction, assignment function, semantic entailment, and various other properties that relate expressions to models or interpretations. The program of studying only the syntax of a system without making any appeal, explicit or tacit, to its meaning constitutes the formalist research program, which is known as Hilbert s program in the foundations of mathematics, and, stretching the concept perhaps, as Chomsky s program of studying syntax autonomously in the theory of generative grammar. (PtMW p. 92) Irony: The term formal semantics refers to the model-theoretic semantics tradition that grew in part out of logician s model-theoretic approach to the study of the semantics of the formal languages of logic; that tradition is not formalist in Hilbert s sense, but is simply formal rather than informal. (The book of Montague s collected works is called Formal Philosophy; and cf. the use of formal in linguistics also to signal something like theoretical/explicit as opposed to practical/informal/descriptive.) We have already looked at the syntax and the semantics of statement logic and first-order predicate logic. Note that the syntax is in each case autonomous in the sense that it is fully described independently of the semantics, and the semantics is inherently relational, involving a compositional mapping from syntactic expressions to objects that belong to a model structure. 1. Syntactic provability and semantic entailment Proof theory: When we presented first order logic in earlier lectures, we specified only the syntax of well-formed formulas and their semantics. We did not give any proof theory, simply because we (unlike logicians) have not had proofs at the center of our interests. Let us add the bare basics of proof theory now. Specify for our first-order logic a set of axioms 1 and Rules of Inference, such as those described in Chapter 8.6 of PtMW or Chapter 1 of Landman. (We are not going to specify a set of axioms and rules of inference for first-order logic here, but see the Appendix for a complete axiomatization of statement logic, which is, of course, much simpler than that needed for predicate logic.) Then we can specify the notion of a proof of a formula ϕ from premises ϕ 1,..., ϕ n. A proof of a formula ϕ from premises ϕ 1,..., ϕ n is a finite sequence of formulas <ψ 1,..., ψ m > such that ψ m = ϕ, and each ψ i is either (a) an axiom, (b) a premise, or (c) inferred by means of one of the Rules of Inference from earlier formulas in the sequence. Call the resulting system of logic L0. An important aspect of the rules of inference is that they are strictly formal, i.e. syntactic : they apply when expressions are of the right form, with no need to know anything about their semantics. Corresponding notion of syntactic derivability or provability: (Landman p 8) Let be a set of formulas and φ a formula. We write φ, meaning φ is provable from, φ is derivable from, iff there is an L0-proof of φ from premises δ 1,..., δ n. The symbol is pronounced turnstile. (In Russian it is called shtopor, i.e. corkscrew.) 1 Here we consider axioms for first order predicate logic itself (PtMW 8.6), which we should distinguish from axioms for theories describing axiomatic classes of models. The former are tautologies which are true in every model. The latter are contingencies which are true in some models and false in other.

3 V. Borschev and B. Partee, October 17-19, 2006 p. 3 We write φ, φ is provable, φ is [syntactically] a tautology, for φ, i.e. φ is provable from the axioms and inference rules of the logic without further assumptions. φ is [syntactically] a contradiction if φ. It is common to use a special symbol ( bottom ) to stand for an arbitrary contradiction. Note that here we have introduced a syntactic notion of tautologies and contradictions. Our earlier definitions of tautologies in terms of truth-tables (for propositional logic) or truth in every model (for predicate logic) were semantic notions of tautology, and below we also consider corresponding semantic notions Semantic entailment and validity. Instead of writing [[φ ]] M,g = 1, it is customary when discussing model theory to write M φ[g], meaning φ is true in M relative to g. We write M φ (and say that φ is true in M) iff M φ[g] for all g. Other locutions: M satisfies φ, M is a model for φ. Note that if φ is a sentence (i.e., φ has no free variables), then for every model M, φ is true in M or φ is false in M. In general we are mostly interested in sentences, so we will often restrict ourselves to them. Let be a set of sentences and φ a sentence. Then we define φ, entails φ, iff for every model M it holds that: If M δ for every δ, then M φ. In other words, entails φ if φ is true in every model in which all the premises in are true. We write φ for φ. We say φ is valid, or logically valid, or a semantic tautology in that case. φ holds iff for every M, M φ. Validity means truth in all models Soundness and completeness of a logic. We have defined syntactic notions and semantic notions separately and independently; but we would of course be very dissatisfied with a logic which failed to provide a good correspondence between syntactic provability and semantic entailment. (We can make some such requirements a condition for any logic; see Hardegree s discussion of possible logics in his Chapter 7.) The notions of soundness and completeness relate to these correspondences. Soundness (of a logic): If L0 φ, then φ. I.e. anything you can prove is semantically valid. Our system of proofs doesn t give us anything bad. Completeness (of a logic): If φ, then L0 φ. Every argument that is semantically valid can be derived with the L0 rules. [Note: there are various different senses of complete ; this is one. On this sense, it is a LOGIC that is complete, sometimes called (strongly) semantically complete. We will see other senses in which a theory may be semantically or syntactically complete.]

4 V. Borschev and B. Partee, October 17-19, 2006 p. 4 Note: First-order predicate logic on any of its standard axiomatizations is sound and complete. Higher-order logics such as Montague s typed intensional logic are sound but often not complete. Soundness is an essential requirement; completeness isn t always possible. Completeness is one of the properties that makes first-order logic nice. 2. Consistency, completeness, independence, and other notions. (PtMW 8.5.2, Landman Chapter 1) Let L be a first order language. (I.e. we specify a particular set of individual constants and predicate constants (of various arities) for L; the rest of the specification of L follows from the definition of well-formed formulas of predicate logic.) A theory in L is a set of sentences of L. [This is Landman s usage; PtMW is slightly different; below, discussing the algebraic approach, we also will use this notion in a slightly different way] So a first order theory is a set of first order sentences Some syntactic concepts. Let be a theory. is inconsistent if. is consistent if /- (i.e. NOT ( ) ) I.e. a theory is inconsistent if you can derive a contradiction from it, consistent if you can t. It s often hard to prove that you can t derive a contradiction that requires a metalevel proof about possible proofs. We ll come to an easier semantic way of showing consistency in a minute. is deductively closed iff: if ϕ, then ϕ. Everything you can derive from is already in. is maximally consistent iff is consistent and there is no such that and and is consistent. c, the deductive closure of, is the set {ϕ : ϕ}. So c is the result of adding to every sentence that can be derived from. A theory is (formally) complete iff c, the deductive closure of, is maximally consistent. Equivalently, is formally complete if for every sentence ϕ in the language, either ϕ or ϕ. (This is the syntactic notion of completeness alluded to above.) Axioms and theories in the syntactic sense: is a set of axioms for Γ iff c = Γ c. Γ is finitely axiomatizable iff there is a finite set of axioms for Γ. Some facts (Landman p.11) FACT 1: If is consistent, then c is consistent. FACT 2: If is maximally consistent then is deductively closed. (Optional exercise: prove FACT 2.) FACT 3: (Deduction theorem) If {ϕ} ψ, then ( ψ ϕ). FACT 5: (Lindenbaum s Lemma) Any consistent theory can be extended to a maximally consistent theory.

5 V. Borschev and B. Partee, October 17-19, 2006 p Some semantic concepts. is satisfiable, equivalently has a model, iff there is a model M such that for all δ : M δ. is closed under logical consequence (under entailment) iff: if ϕ then ϕ. If ϕ is a sentence (i.e. has no free variables) and M ϕ, then we say that M is a model for ϕ, or ϕ holds in (or on) M. Similarly, we say that M is a model for theory, and we write M, if M is a model for every δ. Landman writes MOD(ϕ) for the class of all models for ϕ, and MOD( ) for the class of all models for. A set of axioms is semantically complete with respect to a model M, or weakly semantically complete, if every sentence which holds in M is derivable from. Three notions of completeness. We have now seen three notions of completeness: (i) a logic may be complete: everything which should be a theorem in the semantic sense, i.e. every sentence which is valid is indeed a theorem in the syntactic sense, i.e. is derivable, provable. (ii) Given a logic and a particular first-order language, a set of axioms is formally complete if the deductive closure of is maximally consistent: for every sentence, either it or its negation is provable from. (iii) A set of axioms is (weakly) semantically complete with respect to model M if every sentence which holds in M is derivable from. What do all these notions have in common? They all say that your logic or your axioms are sufficient to derive everything that meets a certain criterion; what varies is the criterion Soundness and completeness again. Another formulation of soundness and completeness for a logic, provably equivalent to the earlier one, is the following. Soundness (of a logic): If has a model, then is consistent. Completeness (of a logic): If is consistent, then has a model. Because first-order logic is sound and complete, we can freely choose whether to give a semantic or syntactic argument of consistency or inconsistency. Suppose you are asked to show whether some set of sentences is consistent or not. Usually if the answer is YES, the easiest way to show it is to show that has a model (by giving a model and showing, if it isn t obvious, that all of the sentences in hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that. See homework problems Independence. The notion of independence is less crucial than some of the other notions we have studied; it concerns the question of whether a given axiom within an axiom system is superfluous. There is nothing logically wrong with having some superfluous axioms; in fact they may make the system easier to work with (just as we typically work with two quantifiers and five connectives even though we know we could define some in terms of others.) But it is often an interesting issue, as

6 V. Borschev and B. Partee, October 17-19, 2006 p. 6 in the case of the discovery of non-euclidean geometries, which resulted from the quest to derive the fifth Euclidean postulate from the other four. The fifth postulate turned out to be independent of the others, and that was demonstrated by producing models (non-euclidean geometries) in which the first four postulates were true and the fifth one false. In general: if an axiom is not independent, you can prove it from the remaining axioms, and that is the standard way to prove non-independence. If an axiom is independent, the easiest way to show it is to produce a model that satisfies the remaining axioms but does not satisfy the one in question. See homework questions 2,3,4, Categoricity. An axiom system is categorical if all of its models are isomorphic. See more in Section 4 below, and see homework questions 3,6,8,10. Question 3 connects the notions of independence and categoricity. 3. An algebraic view on provability and on the notions discussed above [Here we use Universal Algebra by P.M. Cohn (1965), Harper & Row, New York, Evanston, and London] We will begin with extremely abstract (but simple) notions such as closure systems and Galois Connections. We will see how they clarify the notions we have just discussed Closure systems Let A be any set and (A) its power set, i.e. set of all its subsets. We wish to consider certain subsets of (A), or as we shall say, systems of subsets of A. A system R of subsets of A is said to be a closure system if R is closed under intersections, i.e. For any subsystem I R, we have I R In particular, taking I =, we see that A is always belong to R [if you don t see it don t despair, it is not always easy to think about intersection of empty set of sets 2 ; just take it that A always belongs to any closure system R of subsets of A by definition]. Note. (if you have heard of closure in topology): The algebraic notion of closure is similar to the topological one but weaker. Examples 1. The old one. The system of all subalgebras of a given algebra is a closure system. [Think why, it s is a good exercise] 2. Let S be a set of all sentences of some first order language. Then a) the set R D of all deductively closed theories is a closure system; b) the set R E of all theories closed under logical consequence (under entailment) is a closure system; c) R D = R E. [Think why 2 is true] 2 BHP doesn t see it, for instance. But VB does and will try to explain it if asked. Aha, wait, Dr. Math has a clear explanation: see The fact that the intersection of the empty family of sets is the universal set is one more example of a universal statement always coming out true on an empty domain, since the definition of intersection involves a universal quantifier.

7 V. Borschev and B. Partee, October 17-19, 2006 p. 7 Closure operator A closure operator on a set A is a mapping J of (A) in itself with the properties: J.1. If X Y, then J(X) J(Y) J.2. X J(X) J.3. JJ(X) = J(X) Theorem: Every closure system R on a set A defines a closure operator J on A by the rule J(X) = { Y R Y X}. Conversely, every closure operator J on A defines a closure system by R = { X A J(X) = X }, and the correspondence R J between closure systems and closure operators thus defined is bijective (one to one). [Try to prove that Theorem. Make use of the definitions of closure system and closure operator.] Examples Consider a set S of all sentences of some first order language and define operators J D and J E on S such that for every S 1. J D ( ) = c. 2. J E ( ) = ** where ** ={φ φ } It is easy to see that J D and J E are closure operators on S and J D = J E. In particular, if is a set of axioms for Γ then J D ( ) = J D (Γ) = c = Γ c Galois connection Let A and B be any sets and R A B a binary relation from A to B. For any subset X of A we define a subset X* of B by the equation X* = {y B <x,y> R for all x X } And similarly, for any subset Y of B we define a subset Y* of A by Y* = {x A <x,y> R for all y Y } Thus we have the mappings (1) f: (A) (B) and g: (B) (A) such that f(x) = X*, g(y) = Y* and it is easy to see that the properties (2) (4) hold (2) If X 1 X 2, then X 1 * X 2 *, If Y 1 Y 2, then Y 1 * Y 2 * (3) X X**, Y Y** (4) X*** = X*, Y*** = Y* Conditions (2) and (3) follow immediately from the definitions. If (2) is applied to (3) we get X* X***, while (3) applied to X* gives the reverse inequality. Thus any mappings (1) which satisfy (2) and (3) also satisfy (4). A pair of mappings (1) between (A) and (B) is called a Galois connection if it satisfies (2) and (3) (and hence (4).

8 V. Borschev and B. Partee, October 17-19, 2006 p. 8 To establish the link with closure systems we observe that in any Galois connection the mapping X X** is a closure operator in A and Y Y** is a closure operator in B (by (2)-(4)). Moreover, the mapping (1) gives a bijection between thes two closure systems. Examples 1. Our main example is a Galois connection based on a relation M ϕ (ϕ is valid in M) from class Mod(L) of all models for first order language L to the set S of all sentences of L. In this way we obtain a Galois connection between (Mod(L)) and (S). By this connection, (i) to any class MOD Mod(L), there corresponds the set MOD* of all sentences which are true in each M MOD, and (ii) to any set S there corresponds the class * of all those models in which all the sentences of are true. In a notation we used before, MOD* = {ϕ M ϕ for every M MOD } and * = MOD( ). So any model from the class * = MOD( ) is called a model for. Any sentence of MOD* is called a theorem in MOD. A class of models which is of the form * is said to be an axiomatic class and a set of its axioms. ** is the set of all theorems in * = MOD( ), a theory of * = MOD( ). So the semantic notion of axioms and theory given here corresponds to syntactic one given in 2.1 above. 2. Let Obj be a set of objects, Feat a set of their features, and R a relation from Obj to Feat to have a feature ( object O has a feature F ). The Galois connection based on this relation is similar to the previous one. It is used in Formal concepts theory, rather popular now. 4. Morphisms for models; categoricity Earlier we defined isomorphism, homomorphism, and automorphism for Ω-algebras: systems with a carrier set and operations of various arities on this carrier corresponding to symbols from the signature. Similar notions (different kinds of morphisms) are also considered for models. Describing Predicate Logic (the First Order Language) in the Lecture 6 we did not use the word signature but really used the notion. For every concrete language L we fix the set Const of constants and the set Pred of predicate symbols of different arities, Pred = Pred1 Pred 2 We can define the signature Ω of this language as a the union of its constants and predicate symbols, Ω = Const Pred. Then we can call all the models of this language of the kind M = <D,I> as models over signature Ω or Ω-models. Consider two Ω-models M = <D M,I M > and N = <D N,I N >. A mapping f: D M D N is called a homomorphism from M to N (and is denoted also as f: M N ) if (1) for every constant c Const, f(i M (c)) = I N (c) (2) for every n-ary predicate symbol P Pred and every n -tuple <d 1,, d n > D n M, if <d 1,, d n > I M (P) then <f(d 1 ),, f(d n ) > I N (P) A homomorphism f: M N is called an isomorphism between M and N if the inverse relation f 1 : D N D M is a function and it is a homomorphism.

9 V. Borschev and B. Partee, October 17-19, 2006 p. 9 An isomorphism f: M M is called an automorphism. An axiom system is called categorical if all of its models are isomorphic. (See homework questions 3,6,8.) Note. The notion of homomorphism for models is weaker than homomorphism for algebras. We have from the definition that for every n-tuple <d 1,, d n > I M (P) its image <f(d 1 ),, f(d n )> belongs to I N (P). But the opposite need not be true: if some n-tuple <d 1,, d n > doesn t belong to I M (P), its image <f(d 1 ),, f(d n )> can be in I N (P). So for every relation I M (P) in the model M, its image f(i M (P)) is a subset of corresponding relation I N (P) in the model N, i.e. we have f(i M (P)) I N (P), but we don t in general have the equality f(i M (P)) = I N (P). So in the case of models, in contrast to algebras, a bijective homomorphism f: M N (a homomorphism such that the mapping f: D M D N is a bijection (one-to-one and onto)) need not be an isomorphism between the models M and N because the inverse mapping f 1 : D N D M is not obligatorily a homomorphism. Appendix: An axiomatization of statement logic. This is from PtMW, Chapter 8.6, p This axiomatization of statement logic comes from Hilbert and Ackermann, who got it by deleting one non-independent axiom from the system of Whitehead and Russell s Principia Mathematica. This logic is provably complete, and its axioms are all independent. For more details and notational conventions, see PtMW, p.218. Axioms: A1: (p p) p A2. p (p q) A3. (p q) (q p) A4. (p q) ((r p) (r q)) Rules of Inference: R1: Rule of Substitution. For a statement variable in any statement Q we may substitute a statement P, provided that P is substituted for every occurrence of that statement variable in Q. [Note: this rule of inference is valid within the system of logic itself, where the premises are all tautologies and all rules of inference lead to further tautologies. It would not be a sound rule of inference within a formal system considering arbitrary contingent premises and deriving conclusions from them.] R2. Modus ponens. From P and P Q infer Q. [Unlike R2, this is valid everywhere.] From these four axioms with these two rules, all the tautologies of statement logic can be derived as theorems. Appendix: Note on model-theoretic vs. proof-theoretic syntax To see the link between proof theory in logic and derivations with respect to formal grammars, read Sections in PtMW (pp ). Model-theoretic syntax characterizes well-formed syntactic objects, e.g. trees for sentences of language L, not via derivations from a grammar but via a set of axioms ( constraints ) that each tree in L must satisfy; the set of well-formed trees for L then constitutes a model for that set of axioms. See the Postal extract that we will copy.

### Lecture 12. Statement Logic as a word algebra on the set of atomic statements. Lindenbaum algebra.

V. Borschev and B. Partee, October 26, 2006 p. 1 Lecture 12. Statement Logic as a word algebra on the set of atomic statements. Lindenbaum algebra. 0. Preliminary notes...1 1. Freedom for algebras. Word

### Lecture 7. Logic. Section1: Statement Logic.

Ling 726: Mathematical Linguistics, Logic, Section : Statement Logic V. Borschev and B. Partee, October 5, 26 p. Lecture 7. Logic. Section: Statement Logic.. Statement Logic..... Goals..... Syntax of Statement

### Lecture 10. Statement Logic as a word algebra on the set of atomic statements. Lindenbaum algebra.

Lecture 10. Statement Logic as a word algebra on the set of atomic statements. Lindenbaum algebra. 0. Preliminary notes...1 1. Freedom for algebras. Word algebras, initial algebras....2 1.1. Word algebras

### Lecture 4. Algebra, continued Section 2: Lattices and Boolean algebras

V. Borschev and B. Partee, September 21-26, 2006 p. 1 Lecture 4. Algebra, continued Section 2: Lattices and Boolean algebras CONTENTS 1. Lattices.... 1 1.0. Why lattices?... 1 1.1. Posets... 1 1.1.1. Upper

### Lecture 4. Algebra. Section 1:. Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras.

V. Borschev and B. Partee, September 18, 2001 p. 1 Lecture 4. Algebra. Section 1:. Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras. CONTENTS 0. Why algebra?...1

185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted

### Equational Logic. Chapter Syntax Terms and Term Algebras

Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from

### Introduction to Metalogic

Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)

### Natural Deduction for Propositional Logic

Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic

### The Converse of Deducibility: C.I. Lewis and the Origin of Modern AAL/ALC Modal 2011 Logic 1 / 26

The Converse of Deducibility: C.I. Lewis and the Origin of Modern Modal Logic Edwin Mares Victoria University of Wellington AAL/ALC 2011 The Converse of Deducibility: C.I. Lewis and the Origin of Modern

### Introduction to Metalogic

Introduction to Metalogic Hans Halvorson September 21, 2016 Logical grammar Definition. A propositional signature Σ is a collection of items, which we call propositional constants. Sometimes these propositional

### Classical Propositional Logic

The Language of A Henkin-style Proof for Natural Deduction January 16, 2013 The Language of A Henkin-style Proof for Natural Deduction Logic Logic is the science of inference. Given a body of information,

### First Order Logic: Syntax and Semantics

CS1081 First Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Problems Propositional logic isn t very expressive As an example, consider p = Scotland won on Saturday

### ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes)

ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group Birmingham 2. Theories and models Categorical approach to many-sorted

### cis32-ai lecture # 18 mon-3-apr-2006

cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem

### MAGIC Set theory. lecture 1

MAGIC Set theory lecture 1 David Asperó University of East Anglia 15 October 2014 Welcome Welcome to this set theory course. This will be a 10 hour introduction to set theory. The only prerequisite is

### 1. Propositional Calculus

1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:

### Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture - 15 Propositional Calculus (PC)

Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture - 15 Propositional Calculus (PC) So, now if you look back, you can see that there are three

### Propositional Logic: Part II - Syntax & Proofs 0-0

Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems

### Systems of modal logic

499 Modal and Temporal Logic Systems of modal logic Marek Sergot Department of Computing Imperial College, London utumn 2008 Further reading: B.F. Chellas, Modal logic: an introduction. Cambridge University

### Formal Epistemology: Lecture Notes. Horacio Arló-Costa Carnegie Mellon University

Formal Epistemology: Lecture Notes Horacio Arló-Costa Carnegie Mellon University hcosta@andrew.cmu.edu Logical preliminaries Let L 0 be a language containing a complete set of Boolean connectives, including

### Meta-logic derivation rules

Meta-logic derivation rules Hans Halvorson February 19, 2013 Recall that the goal of this course is to learn how to prove things about (as opposed to by means of ) classical first-order logic. So, we will

### Mathematics 114L Spring 2018 D.A. Martin. Mathematical Logic

Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)

### 1. Propositional Calculus

1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:

### What are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos

What are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos armandobcm@yahoo.com February 5, 2014 Abstract This note is for personal use. It

### Modal and temporal logic

Modal and temporal logic N. Bezhanishvili I. Hodkinson C. Kupke Imperial College London 1 / 83 Overview Part II 1 Soundness and completeness. Canonical models. 3 lectures. 2 Finite model property. Filtrations.

### Přednáška 12. Důkazové kalkuly Kalkul Hilbertova typu. 11/29/2006 Hilbertův kalkul 1

Přednáška 12 Důkazové kalkuly Kalkul Hilbertova typu 11/29/2006 Hilbertův kalkul 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: A. a language B. a set of axioms C. a set of

### Overview of Logic and Computation: Notes

Overview of Logic and Computation: Notes John Slaney March 14, 2007 1 To begin at the beginning We study formal logic as a mathematical tool for reasoning and as a medium for knowledge representation The

### COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof

### Propositional Logic Arguments (5A) Young W. Lim 11/8/16

Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

### Basic Algebraic Logic

ELTE 2013. September Today Past 1 Universal Algebra 1 Algebra 2 Transforming Algebras... Past 1 Homomorphism 2 Subalgebras 3 Direct products 3 Varieties 1 Algebraic Model Theory 1 Term Algebras 2 Meanings

### The semantics of propositional logic

The semantics of propositional logic Readings: Sections 1.3 and 1.4 of Huth and Ryan. In this module, we will nail down the formal definition of a logical formula, and describe the semantics of propositional

### Propositional Logic Arguments (5A) Young W. Lim 10/11/16

Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

### Gödel s Completeness Theorem

A.Miller M571 Spring 2002 Gödel s Completeness Theorem We only consider countable languages L for first order logic with equality which have only predicate symbols and constant symbols. We regard the symbols

### Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson

Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness

### COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof

### 2. Introduction to commutative rings (continued)

2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of

### Axiomatic set theory. Chapter Why axiomatic set theory?

Chapter 1 Axiomatic set theory 1.1 Why axiomatic set theory? Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its

### Intermediate Logic. Natural Deduction for TFL

Intermediate Logic Lecture Two Natural Deduction for TFL Rob Trueman rob.trueman@york.ac.uk University of York The Trouble with Truth Tables Natural Deduction for TFL The Trouble with Truth Tables The

### Examples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula:

Logic: The Big Picture Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and

### 3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.

1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is

### Modal Logic XX. Yanjing Wang

Modal Logic XX Yanjing Wang Department of Philosophy, Peking University May 6th, 2016 Advanced Modal Logic (2016 Spring) 1 Completeness A traditional view of Logic A logic Λ is a collection of formulas

### 3 The Semantics of the Propositional Calculus

3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical

### CMPSCI 601: Tarski s Truth Definition Lecture 15. where

@ CMPSCI 601: Tarski s Truth Definition Lecture 15! "\$#&%(') *+,-!".#/%0'!12 43 5 6 7 8:9 4; 9 9 < = 9 = or 5 6?>A@B!9 2 D for all C @B 9 CFE where ) CGE @B-HI LJKK MKK )HG if H ; C if H @ 1 > > > Fitch

### Gödel s Incompleteness Theorems

Seminar Report Gödel s Incompleteness Theorems Ahmet Aspir Mark Nardi 28.02.2018 Supervisor: Dr. Georg Moser Abstract Gödel s incompleteness theorems are very fundamental for mathematics and computational

### All psychiatrists are doctors All doctors are college graduates All psychiatrists are college graduates

Predicate Logic In what we ve discussed thus far, we haven t addressed other kinds of valid inferences: those involving quantification and predication. For example: All philosophers are wise Socrates is

### Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)

http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product

### KRIPKE S THEORY OF TRUTH 1. INTRODUCTION

KRIPKE S THEORY OF TRUTH RICHARD G HECK, JR 1. INTRODUCTION The purpose of this note is to give a simple, easily accessible proof of the existence of the minimal fixed point, and of various maximal fixed

### First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms

First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO

### Formal (natural) deduction in propositional logic

Formal (natural) deduction in propositional logic Lila Kari University of Waterloo Formal (natural) deduction in propositional logic CS245, Logic and Computation 1 / 67 I know what you re thinking about,

### Truth-Functional Logic

Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence

### Proseminar on Semantic Theory Fall 2013 Ling 720 Propositional Logic: Syntax and Natural Deduction 1

Propositional Logic: Syntax and Natural Deduction 1 The Plot That Will Unfold I want to provide some key historical and intellectual context to the model theoretic approach to natural language semantics,

### Modal Logic: Exercises

Modal Logic: Exercises KRDB FUB stream course www.inf.unibz.it/ gennari/index.php?page=nl Lecturer: R. Gennari gennari@inf.unibz.it June 6, 2010 Ex. 36 Prove the following claim. Claim 1. Uniform substitution

### First-Degree Entailment

March 5, 2013 Relevance Logics Relevance logics are non-classical logics that try to avoid the paradoxes of material and strict implication: p (q p) p (p q) (p q) (q r) (p p) q p (q q) p (q q) Counterintuitive?

### Applied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw

Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018

### Deductive Systems. Lecture - 3

Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth

### Boolean Algebra and Propositional Logic

Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection

### Theorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)

Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +

### Lecture 2: Syntax. January 24, 2018

Lecture 2: Syntax January 24, 2018 We now review the basic definitions of first-order logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified

### Propositional Logic Review

Propositional Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane The task of describing a logical system comes in three parts: Grammar Describing what counts as a formula Semantics Defining

### Lecture 7. Logic III. Axiomatic description of properties of relations.

V. Borschev and B. Partee, October 11, 2001 p. 1 Lecture 7. Logic III. Axiomatic description of properties of relations. CONTENTS 1. Axiomatic description of properties and classes of relations... 1 1.1.

### Propositional Logic: Syntax

4 Propositional Logic: Syntax Reading: Metalogic Part II, 22-26 Contents 4.1 The System PS: Syntax....................... 49 4.1.1 Axioms and Rules of Inference................ 49 4.1.2 Definitions.................................

### GS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen Anthony Hall

GS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen www.cs.ucl.ac.uk/staff/j.bowen/gs03 Anthony Hall GS03 W1 L3 Predicate Logic 12 January 2007 1 Overview The need for extra structure

### A Little Deductive Logic

A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that

### Lecture 6: Finite Fields

CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going

### A Little Deductive Logic

A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that

### Supplementary Notes on Inductive Definitions

Supplementary Notes on Inductive Definitions 15-312: Foundations of Programming Languages Frank Pfenning Lecture 2 August 29, 2002 These supplementary notes review the notion of an inductive definition

### A Guide to Proof-Writing

A Guide to Proof-Writing 437 A Guide to Proof-Writing by Ron Morash, University of Michigan Dearborn Toward the end of Section 1.5, the text states that there is no algorithm for proving theorems.... Such

### Overview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence Lecture 19: Logic for KR Last time Expert Systems and Ontologies oday Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof theory Natural

### Boolean Algebra and Propositional Logic

Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a

### Lecture Notes on Inductive Definitions

Lecture Notes on Inductive Definitions 15-312: Foundations of Programming Languages Frank Pfenning Lecture 2 August 28, 2003 These supplementary notes review the notion of an inductive definition and give

### Advanced Topics in LP and FP

Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection

### Lecture 15 The Second Incompleteness Theorem. Michael Beeson

Lecture 15 The Second Incompleteness Theorem Michael Beeson The Second Incompleteness Theorem Let Con PA be the formula k Prf(k, 0 = 1 ) Then Con PA expresses the consistency of PA. The second incompleteness

### CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:

x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which

### For all For every For each For any There exists at least one There exists There is Some

Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following

### 02 Propositional Logic

SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or

### Logic: Propositional Logic Truth Tables

Logic: Propositional Logic Truth Tables Raffaella Bernardi bernardi@inf.unibz.it P.zza Domenicani 3, Room 2.28 Faculty of Computer Science, Free University of Bolzano-Bozen http://www.inf.unibz.it/~bernardi/courses/logic06

### Proof strategies, or, a manual of logical style

Proof strategies, or, a manual of logical style Dr Holmes September 27, 2017 This is yet another version of the manual of logical style I have been working on for many years This semester, instead of posting

### AN EXTENSION OF THE PROBABILITY LOGIC LP P 2. Tatjana Stojanović 1, Ana Kaplarević-Mališić 1 and Zoran Ognjanović 2

45 Kragujevac J. Math. 33 (2010) 45 62. AN EXTENSION OF THE PROBABILITY LOGIC LP P 2 Tatjana Stojanović 1, Ana Kaplarević-Mališić 1 and Zoran Ognjanović 2 1 University of Kragujevac, Faculty of Science,

### Lecture 13: Soundness, Completeness and Compactness

Discrete Mathematics (II) Spring 2017 Lecture 13: Soundness, Completeness and Compactness Lecturer: Yi Li 1 Overview In this lecture, we will prvoe the soundness and completeness of tableau proof system,

### Logic. Propositional Logic: Syntax

Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about

### Part II. Logic and Set Theory. Year

Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]

### Proof Techniques (Review of Math 271)

Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil

### Semantics and Pragmatics of NLP

Semantics and Pragmatics of NLP Alex Ewan School of Informatics University of Edinburgh 28 January 2008 1 2 3 Taking Stock We have: Introduced syntax and semantics for FOL plus lambdas. Represented FOL

### Logic: The Big Picture

Logic: The Big Picture A typical logic is described in terms of syntax: what are the legitimate formulas semantics: under what circumstances is a formula true proof theory/ axiomatization: rules for proving

### GÖDEL S COMPLETENESS AND INCOMPLETENESS THEOREMS. Contents 1. Introduction Gödel s Completeness Theorem

GÖDEL S COMPLETENESS AND INCOMPLETENESS THEOREMS BEN CHAIKEN Abstract. This paper will discuss the completeness and incompleteness theorems of Kurt Gödel. These theorems have a profound impact on the philosophical

### Introduction to Logic and Axiomatic Set Theory

Introduction to Logic and Axiomatic Set Theory 1 Introduction In mathematics, we seek absolute rigor in our arguments, and a solid foundation for all of the structures we consider. Here, we will see some

### Propositional logic. First order logic. Alexander Clark. Autumn 2014

Propositional logic First order logic Alexander Clark Autumn 2014 Formal Logic Logical arguments are valid because of their form. Formal languages are devised to express exactly that relevant form and

### Supplementary Logic Notes CSE 321 Winter 2009

1 Propositional Logic Supplementary Logic Notes CSE 321 Winter 2009 1.1 More efficient truth table methods The method of using truth tables to prove facts about propositional formulas can be a very tedious

### Equational Logic and Term Rewriting: Lecture I

Why so many logics? You all know classical propositional logic. Why would we want anything more? Equational Logic and Term Rewriting: Lecture I One reason is that we might want to change some basic logical

### Logic via Algebra. Sam Chong Tay. A Senior Exercise in Mathematics Kenyon College November 29, 2012

Logic via Algebra Sam Chong Tay A Senior Exercise in Mathematics Kenyon College November 29, 2012 Abstract The purpose of this paper is to gain insight to mathematical logic through an algebraic perspective.

### Propositional Logic: Syntax

Logic Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and programs) epistemic

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.

### Class 29 - November 3 Semantics for Predicate Logic

Philosophy 240: Symbolic Logic Fall 2010 Mondays, Wednesdays, Fridays: 9am - 9:50am Hamilton College Russell Marcus rmarcus1@hamilton.edu Class 29 - November 3 Semantics for Predicate Logic I. Proof Theory

### CONTENTS. Appendix C: Gothic Alphabet 109

Contents 1 Sentential Logic 1 1.1 Introduction............................ 1 1.2 Sentences of Sentential Logic................... 2 1.3 Truth Assignments........................ 7 1.4 Logical Consequence.......................

### Propositional Logics and their Algebraic Equivalents

Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic

### First Order Logic (1A) Young W. Lim 11/18/13

Copyright (c) 2013. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software