Propositions and Proofs


 Amy Watts
 1 years ago
 Views:
Transcription
1 Propositions and Proofs Gert Smolka, Saarland University April 25, 2018 Proposition are logical statements whose truth or falsity can be established with proofs. Coq s type theory provides us with a language for writing propositions and proofs. Propositions are accommodated as types and proofs are accommodated as elements of types. This way proof checking reduces to type checking. Implications are accommodated as function types, and universal quantifications are accommodated as dependent function types. The remaining propositional forms such as conjunction, disjunction, and existential quantification are accommodated with inductive types. This setup provides for a basic form of logical reasoning known as intuitionistic reasoning. In contrast to classical reasoning, intuitionistic reasoning does not built in the law of excluded middle. 1 BHK Proofs Propositions are build from basic propositions with connectives and quantifiers. Here are prominent forms of propositions you will have encountered before. Name Notation Reading equality s = t s equals t truth true falsity false conjunction P Q P and Q disjunction P Q P or Q implication P Q if P then Q negation P not P equivalence P Q P if and only if Q universal quantification x : X. px for all x in X, px existential quantification x : X. px for some x in X, px Truth and falsity of propositions is established by proofs. We say that a proposition P is true if we have a proof of P, and we say that a proposition P is false if we 1
2 have a proof of P. Following a design known as BHK interpretation, 1 proofs are modelled as computational values: A proof of a conjunction P Q is a pair consisting of a proof of P and a proof of Q. A proof of a disjunction P Q is a tagged value containing either a proof of P or a proof of Q. The tag says whether we have a proof of the left proposition or the right proposition. A proof of an implication P Q is a function that for every proof of P yields a proof of Q. A proof of a universal quantification x : X. px is a function that for every term s of type X yields a proof of the proposition ps. A proof of an existential quantification x : X. px is a pair consisting of a term s of type X and a proof of the proposition ps. The term S is called the witness of the proof. has a unique proof I. has no proof. We have now said what we admit as proofs of conjunctions, disjunctions, implications, universal and existential quantifications, and. This gives us a set of proof rules modelling basic intuitions about logical reasoning. The above listing of proof forms specifies all primitive proofs of the propositional forms mentioned. Hence, every proof of an implication P Q, say, is a function from proofs of P to proofs of Q. Thus, if f is a proof of P Q and a is a proof of P, the application of f to a yields a proof of Q. We now define negation with implication and : P := P Thus a proof of P is a function that given a proof of P yields a proof of. Since has no proof, the existence of such a function guarantees that P has no proof. We define equivalence of propositions as one would expect: P Q := (P Q) (Q P) Thus a proof of an equivalence is a pair of two functions translating proofs from left to right and from right to left. From what we have said it is clear that λx : P.x is a proof P P, and that (λ_.λx.x, λ_.i) is a proof of ( ). It remains to say what counts as a proof of an equation s = t. We postpone this question until we have worked out the presented design in more detail. 1 The BHK view of proofs originated in the 1930 s in the work of the mathematicians Luitzen Brouwer, Arend Heyting, and Andrey Kolmogorov. 2
3 2 Propositions as Types Given a type theory with inductive types and dependent function types, it is natural to accommodate propositions as types and proofs as elements of propositions such that the elements of a proposition P serve as the proofs of P. This design is known as propositions as types principle and yields a natural realization of BHK proofs. We will explain the propositions as type principle as it is realized in Coq s type theory. Implications and universal quantifications are accommodated as function types P Q and dependent function types x : X. px, which establishes their proofs exactly as specified by the BHK interpretation. The remaining propositions are represented with inductive types. Coq s type theory is designed such that every proposition is a type but not every type is a proposition. To this end, Coq has a special type P (read prop ) serving as type of all propositions. We have P T, which ensures that every proposition is a type. Types in T that are not in P (e.g., N and B) are called proper types. When we define an inductive type, we can decide whether it is a proposition or a proper type. The types P and T are known as universes. Having a separate universe for propositions (e.g., P) makes is possible to impose assumptions on propositions without affecting proper types. Here are the inductive definitions we will use for truth, falsity, conjunctions, and disjunctions. 2 : P := [I : ] : P := [] (X : P)(Y : P) : P := (X : P)(Y : P) : P := [ C : X Y X Y ] [ L : X X Y R : Y X Y ] We call the value constructors of inductive propositions proof constructors. Following the BHK view of proofs, and conjunctions come with one proof constructor each, and disjunctions comes with two proof constructors. The requirement that has no proof is realized by defining as an inductive proposition that has no proof constructor. Here are the types of the constructors introduced by the inductive definitions of 2 We have named the proof constructors for conjunctions and disjunctions different from Coq. 3
4 truth, falsity, conjunctions, and disjunctions: : P : P I : : P P P C : X : P Y : P. X Y X Y : P P P L : X : P Y : P. X X Y R : X : P Y : P. Y X Y We will treat the arguments X and Y of C, L, and R as implicit arguments. For L and R this means that information from the surrounding context is needed to infer Y and X, respectively. A predicate is a function that eventually yields a proposition. Note that and are inductive predicates. The universe of propositions is closed under forming implications and universal quantifications. That is, if P and Q are propositions, then the function type P Q is a proposition. Moreover, if X is a type and p is a predicate, then the dependent function type x : X. px is a proposition. An important consequence of the propositions as types principle is the fact that proof checking reduces to type checking. Thus Coq doesn t have a specialpurpose proof checker but just a generalpurpose type checker. In short, proof checking in Coq is type checking. 3 Proof Terms We can now write proofs of propositions as terms. Here are a few straightforward examples. We assume that X, Y, and Z are propositions. Proposition Proof X Y x λxy.x X (X Y ) Y λxf.f x (X Y ) (Y Z) X Y λf gx.g(f x) ( Z : P. Z) X λf.f X In the proof terms on the right we have omitted the types of the argument variables since they can be inferred from the propositions on the left. A logical principle known as exfalso says that from falsity one can derive everything. We can prove the principle with the following proof term: x λh. match H [] The function takes a proof H of as argument and returns a proof of X. To do so, the function matches on H. Every rule of the match must yield a proof of X. Since 4
5 has no constructor, the match has no rule, and hence it is vacuously true that every rule yields a proof of X. Here are a few proof terms for propositions involving negation. Recall that negation is defined as s := s. The trick is that the definition of negation is automatically unfolded whenever this is necessary (that is, the term s is automatically replaced with the implication s ). X X λxf.f x X X Y λxf. match f x [] (X Y ) Y X λf gx. g(f x) X X λxf. f (λg.gx) X (X X) λf g. f (λx.gxx) (X X) ( X X) λf g. let x = g(λx.f xx) in f xx Here are a few proof terms for propositions involving conjunctions and disjunctions. X Y X λx. Rx X Y Y X λh. match H [ Cxy Cyx ] X Y Y X λh. match H [ L x Rx Ry Ly ] The proposition X (Y Z) X Y X Z requires a proof term with two matches: λh. match H [ CxH 1 match H 1 [ Ly L(Cxy) Rz R(Cxz) ] ] This is also the case for the proposition ( X Y ) (X Y ): λh 1 H 2. match H 2 [ Cxy match H 1 [ L f f x R g gy ] ] Russell s law is the proposition X : P. (X X). following proof term: It can be shown with the λxh. match H [Cf g let x = g(λx.f xx) in f xx] 4 Proof Diagrams The construction of proof terms requires a certain information structure. One needs to keep track of the types one has to construct proof terms for, and also of the variables and their types that have been introduced by lambdas and rules of matches. Proof term construction can be assisted with a proof diagram displaying the necessary information. Here is an example: 5
6 X : P f : X X g : X X X : P. (X X) assert X apply g X X : X apply f xx x : X apply f xx The diagram is written topdown beginning with the initial claim and records the construction of the proof term λxh. match H [ Cf g let x = g(λx.f xx) in f xx ] for the proposition X : P. (X X). Proof diagrams are havewant diagrams in that they record on the left what we have and on the right want we want. When we start, the proof diagram is partial and just consists of the first line. As the proof term construction proceeds, we add further lines and further proof goals (separated through horizontal lines) until we arrive at a complete proof diagram. With Coq one can construct proof terms interactively using commands called tactics. Coq then shows information similar to what we see in a proof diagram. There may be several proof goals open at a point in time, where each proof goal consists of a list of assumptions called context and a claim. The assumptions are typed variables as shown on the left of a proof diagram, and a claim is a type as shown on the right of a proof diagram. There may be more than one proof goal open at a point in time and one may navigate between open goals. Interactive proof term construction with Coq is convenient since the bookkeeping and verification is done by Coq and the proof goals with their assumptions and claims are displayed nicely. We show two further examples of complete proof diagrams. X (X X) f : X g : X X apply f X x : X apply gxx the proof term constructed is λf g.f (λx.gxx). 6
7 x : X X (Y Z) (X Y ) (X Z) 1. y : Y (X Y ) (X Z) apply L xy 2. z : Z (X Y ) (X Z) apply R xz the proof term constructed is λh. match H [ C xh 1 match H 1 [ L y L(C xy) R z R(C xz) ] ] Exercise 1 The following propositions formulate wellknown properties of conjunction and disjunction: X Y Y X X Y Y X commutativity X (Y Z) (X Y ) Z X (Y Z) (X Y ) Z associativity X (Y Z) X Y X Z X (Y Z) (X Y ) (X Z) distributivity X (X Y ) X X (X Y ) X absorption Make sure you can construct proof terms for all propositions using proof diagrams. Exercise 2 Prove the following propositions. a) (X Y ) X Y. b) X Y (X Y ). 5 A Difficult Proof Not every proof is easy to find. The proposition ( p. px py) p. py px where p : X P has a short proof, but takes insight to find it. The trick consists in instantiating the predicate p of the premise with λz. pz px where p is the predicate from the conclusion. Here is a proof diagram. f : ( p. px py) p : X P apply f (λz. pz px) apply λh.h ( p. px py) p. py px py px px px The proof term constructed is λf p. f (λz. pz px)(λh.h). 7
8 6 Existential Quantification We will represent an existential quantification x.s as an application ex(λx.s) of a suitable defined inductive predicate ex. The use of the abstraction λx.s ensures that x is a local variable whose scope is the term s. Let X be a type and p : X P be a predicate on X. Following the BHK design, we postulate that a proof of an existential quantification x.px is a pair consisting of a witness t and a proof of the instance pt. We realize this design with an inductive definition: ex(x : T)(p : X P) : P := [ E : x : X. px ex p ] As an example, we prove the de Morgan law for existential quantification: ( x.px) x. px. We give a prove diagram for each direction. f : ( x.px) x : X H : px apply f (E xh) ( x.px) x. px The proof term constructed is λf xh. f (E xh). f : x. px x : X H : px apply f xh ( x. px) ( x.px) The proof term constructed is λf H 1. match H 1 [ E xh f xh ]. Barber Paradox In a village, there cannot be a barber who shaves everyone who doesn t shave himself. We prove a generalisation of this statement. Fact 3 Let X be a type and p be a binary predicate on X. x y. pxy pyy is provable. Then the proposition Proof Suppose there is an x such that pxy pyy for all y. Then pxx pxx, which is contradictory by Russell s law. The barber paradox explains why there cannot be a set that contains all sets that don t contain themselves. To see this, let the type X in Fact 3 be the type of sets and 8
9 let p be the inverse membership predicate for sets (i.e., pxy := y x). The barber paradox also explains why there cannot be a Turing machine that halts on the code of a Turing machine if and only if this machine doesn t halt on its own code. For this, let X be the type of all Turing machine codes and pxy say that the machine with code x holds on code y if and only if the machine with code y doesn t halt on y. Exercise 4 Prove the following propositions with proof diagrams and give the resulting proof terms. a) ( x y. pxy) y x. pxy. b) ( x. px qx) ( x.px) ( x.qx). c) ( x.px) x. px. d) (( x.px) Z) x. px Z. 7 Impredicative Characterizations The following equivalences, known as impredicative characterisations of conjunction, disjunction, existential quantification, falsity, and truth, are easy to prove: X Y Z. (X Y Z) Z X Y Z. (X Z) (Y Z) Z x.px Z. ( x. px Z) Z Z. Z Z. Z Z The equivalences tell us that the inductively defined propositions on the left can be characterised by propositions only using universal quantification and implication. The righthand sides of the equivalences are in fact propositions expressing the proof rules realized by the matches for the inductive predicates on the left. This important insight becomes apparent in the proof terms for the equivalences. Here are the proof terms for the two directions of the equivalence for disjunctions: λhzf g. match H [ L x f x R y gy ]. λh. H(X Y ) L R The direction from left to right just applies the match for disjunctions. The other direction (right to left) instantiate the variable Z on the righthand side with the inductive proposition on the lefthand side and the two proof constructors. The proof terms for the other logical constants have the same structure. 9
10 The impredicative characterisations tell us that a type theory without inductive types suffices for the definition of conjunction, disjunction, existential quantification, falsity, and truth. Exercise 5 Make sure that you can derive the impredicative characterisations of the inductive propositions X Y, X Y, x.px,, and by expressing with a proposition the effect the tactic destruct has for the inductive proposition. The idea is that an application of a proof of the derived proposition will have the same effect as the tactic destruct. 8 Excluded Middle The proof rules coming with the propositions as types principle constitute a basic proof system known as intuitionistic logic or constructive logic. We say that a proposition follows intuitionistically or constructively if it can be shown with the proof rules coming with the propositions as types principle. Not every proposition considered true in Mathematics can be shown constructively. A proposition that cannot be shown constructively is excluded middle (XM for short): XM := X : P. X X XM says that for every proposition X the disjunction X X is true. XM is a basic assumption in standard Mathematics. If we want to use XM in Coq, we have to assume it explicitly. That is, to prove X using XM, we prove the implication XM X. Coq provides a command making it possible to assume XM for an entire development (similar to assumptions in sections). We say that a proposition follows classically if it can be shown with XM. The fact that constructive logic does not builtin XM is a virtue, not a defect. This way we can distinguish between proofs not using XM and proofs using XM. The philosophy here is that XM is a basic assumption in standard Mathematics but not a basic proof rule. We may say that constructive logic minimizes the builtin assumptions and thus provides finer proof checking (e.g., this proof doesn t rely on XM). It turns out that many interesting facts can be shown without assuming XM. Assuming XM, we can prove the following propositions for all propositions X 10
11 and Y. None of these propositions is provable without XM. X X ( X Y ) Y X (X Y ) X Y ( x.px) x. px ((X Y ) X) X (X Y ) (Y X) (X Y ) X Y Fact 6 The following propositions are equivalent. That is, if we can prove one of them, we can prove all of them. 1. X : P. X X excluded middle 2. X : P. X X double negation 3. XY : P. ( X Y ) Y X contraposition 4. XY : P. ((X Y ) X) X Peirce s law Proof It suffices to prove the implications 1 2, 2 3, 3 4, and 4 1. In each case, one applies the assumption after doing all possible intros. After this the proof is routine, the assumption is not needed further. For 4 1, one instantiates Y in 4 with. Nice exercises. Exercise 7 Consider the propositions X X ( X Y ) Y X (X Y ) X Y ((X Y ) X) X (X Y ) (Y X) (X Y ) X Y a) Prove each of the above propositions using X X. b) The double negation of a proposition s is s. Prove the double negation of each of the above propositions. Exercise 8 Prove the following propositions using XM. a) ( x.px) x. px. b) ( x.px) x. px. 11
12 Exercise 9 (Prominent Propositions) Here is a list of prominent propositions you should know: X (X X) (X Y ) X Y ( x.px) x. px (X Y ) X Y ( x.px) x. px X X X X ( X Y ) (Y X) exfalso Russell de Morgan de Morgan de Morgan de Morgan excluded middle double negation contraposition (X Y ) X Y classical implication ( x.px) x. px a) Prove the first four propositions. b) Prove the directions of all equivalences. counterexample c) Prove all quantifierfree propositions assuming X X. d) Prove the remaining propositions with excluded middle. Drinker Paradox Consider a bar populated by at least one person. Using excluded middle, one can prove that one can pick some person in the bar such that everyone in the bar drinks Whiskey if this person drinks Whiskey. The proof is easy. Either everyone in the bar drinks Whiskey or not everyone in the bar drinks Whiskey. If everyone in the bar drinks Whiskey, we pick some person in the bar and are done (we assumed that the bar is populated). If not everyone in the bar drinks Whiskey, there is some person x who doesn t drink Whiskey. Hence everyone drinks Whiskey if x drinks Whiskey (the trick is in the proof rule for implication). The formalisation and proof the statement are somewhat tricky. We will assume a type representing the persons in the bar. A type is called inhabited if it has at least one element. Fact 10 Let X be an inhabited type and p be a predicate on X. Then the proposition x. px x.px is provable assuming XM. Proof By XM we have either x. px or x. px. In the first case the proof is straightforward. We consider the second case. Let x. px. Using XM we can show x.px. Since X is inhabited, it suffices to prove px x.px for some x, which is now trivial. Makes a nice Coq exercise. 12
13 9 Notational Issues Following Coq, we use the precedence order for the notations for the logical constants. Thus we may omit parentheses as in the following example: X Y Z Z Y ((( ( X) Y ) Z) Z) Y The notations,, and are in addition right associative. As it comes to quantifiers, we use notational conveniences we also use for λabstractions. For instance, we may write xy z. s or x y z. s for x. y. z. s. 13
Propositional Logic Not Enough
Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks
More informationFirst order Logic ( Predicate Logic) and Methods of Proof
First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating
More informationCSCE 222 Discrete Structures for Computing. Predicate Logic. Dr. Hyunyoung Lee. !!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Predicate Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Predicates A function P from a set D to the set Prop of propositions is called a predicate.
More informationCS2742 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
More informationChapter 1. Logic and Proof
Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known
More information1.3 Predicates and Quantifiers
1.3 Predicates and Quantifiers INTRODUCTION Statements x>3, x=y+3 and x + y=z are not propositions, if the variables are not specified. In this section we discuss the ways of producing propositions from
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #4: Predicates and Quantifiers Based on materials developed by Dr. Adam Lee Topics n Predicates n
More informationThe predicate calculus is complete
The predicate calculus is complete Hans Halvorson The first thing we need to do is to precisify the inference rules UI and EE. To this end, we will use A(c) to denote a sentence containing the name c,
More informationLogic. 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
More information1. Propositions: Contrapositives and Converses
Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement
More informationFormal Logic: Quantifiers, Predicates, and Validity. CS 130 Discrete Structures
Formal Logic: Quantifiers, Predicates, and Validity CS 130 Discrete Structures Variables and Statements Variables: A variable is a symbol that stands for an individual in a collection or set. For example,
More informationLogical Operators. Conjunction Disjunction Negation Exclusive Or Implication Biconditional
Logical Operators Conjunction Disjunction Negation Exclusive Or Implication Biconditional 1 Statement meaning p q p implies q if p, then q if p, q when p, q whenever p, q q if p q when p q whenever p p
More informationTHE LOGIC OF QUANTIFIED STATEMENTS. Predicates and Quantified Statements I. Predicates and Quantified Statements I CHAPTER 3 SECTION 3.
CHAPTER 3 THE LOGIC OF QUANTIFIED STATEMENTS SECTION 3.1 Predicates and Quantified Statements I Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Predicates
More informationMAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
More informationIntelligent Agents. First Order Logic. Ute Schmid. Cognitive Systems, Applied Computer Science, Bamberg University. last change: 19.
Intelligent Agents First Order Logic Ute Schmid Cognitive Systems, Applied Computer Science, Bamberg University last change: 19. Mai 2015 U. Schmid (CogSys) Intelligent Agents last change: 19. Mai 2015
More informationProofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction
Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when
More information! Predicates! Variables! Quantifiers. ! Universal Quantifier! Existential Quantifier. ! Negating Quantifiers. ! De Morgan s Laws for Quantifiers
Sec$on Summary (K. Rosen notes for Ch. 1.4, 1.5 corrected and extended by A.Borgida)! Predicates! Variables! Quantifiers! Universal Quantifier! Existential Quantifier! Negating Quantifiers! De Morgan s
More information3 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
More informationMathematical Preliminaries. Sipser pages 128
Mathematical Preliminaries Sipser pages 128 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation
More informationIntroduction to Predicate Logic Part 1. Professor Anita Wasilewska Lecture Notes (1)
Introduction to Predicate Logic Part 1 Professor Anita Wasilewska Lecture Notes (1) Introduction Lecture Notes (1) and (2) provide an OVERVIEW of a standard intuitive formalization and introduction to
More informationLING 501, Fall 2004: Quantification
LING 501, Fall 2004: Quantification The universal quantifier Ax is conjunctive and the existential quantifier Ex is disjunctive Suppose the domain of quantification (DQ) is {a, b}. Then: (1) Ax Px Pa &
More informationCITS2211 Discrete Structures Proofs
CITS2211 Discrete Structures Proofs Unit coordinator: Rachel CardellOliver August 13, 2017 Highlights 1 Arguments vs Proofs. 2 Proof strategies 3 Famous proofs Reading Chapter 1: What is a proof? Mathematics
More informationStructuring Logic with Sequent Calculus
Structuring Logic with Sequent Calculus Alexis Saurin ENS Paris & École Polytechnique & CMI Seminar at IIT Delhi 17th September 2004 Outline of the talk Proofs via Natural Deduction LK Sequent Calculus
More informationArgument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday.
Logic and Proof Argument An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid
More informationTutorial on Axiomatic Set Theory. Javier R. Movellan
Tutorial on Axiomatic Set Theory Javier R. Movellan Intuitively we think of sets as collections of elements. The crucial part of this intuitive concept is that we are willing to treat sets as entities
More informationComputational Logic. Recall of FirstOrder Logic. Damiano Zanardini
Computational Logic Recall of FirstOrder Logic Damiano Zanardini UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid damiano@fi.upm.es Academic
More informationINTRODUCTION TO LOGIC 8 Identity and Definite Descriptions
8.1 Qualitative and Numerical Identity INTRODUCTION TO LOGIC 8 Identity and Definite Descriptions Volker Halbach Keith and Volker have the same car. Keith and Volker have identical cars. Keith and Volker
More informationHOW TO CREATE A PROOF. Writing proofs is typically not a straightforward, algorithmic process such as calculating
HOW TO CREATE A PROOF ALLAN YASHINSKI Abstract We discuss how to structure a proof based on the statement being proved Writing proofs is typically not a straightforward, algorithmic process such as calculating
More informationTECHNISCHE UNIVERSITEIT EINDHOVEN Faculteit Wiskunde en Informatica. Final exam Logic & Set Theory (2IT61) (correction model)
TECHNISCHE UNIVERSITEIT EINDHOVEN Faculteit Wiskunde en Informatica Final exam Logic & Set Theory (2IT61) (correction model) Thursday November 4, 2016, 9:00 12:00 hrs. (2) 1. Determine whether the abstract
More informationA 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
More informationProposi'onal Logic Not Enough
Section 1.4 Proposi'onal Logic Not Enough If we have: All men are mortal. Socrates is a man. Socrates is mortal Compare to: If it is snowing, then I will study discrete math. It is snowing. I will study
More informationDiscrete Mathematics
Department of Mathematics National Cheng Kung University 2008 2.4: The use of Quantifiers Definition (2.5) A declarative sentence is an open statement if 1) it contains one or more variables, and 1 ) quantifier:
More informationCPSC 121: Models of Computation
CPSC 121: Models of Computation Unit 6 Rewriting Predicate Logic Statements Based on slides by Patrice Belleville and Steve Wolfman Coming Up Preclass quiz #7 is due Wednesday October 25th at 9:00 pm.
More informationTo every formula scheme there corresponds a property of R. This relationship helps one to understand the logic being studied.
Modal Logic (2) There appeared to be a correspondence between the validity of Φ Φ and the property that the accessibility relation R is reflexive. The connection between them is that both relied on the
More informationIntroduction 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)
More informationApplied Logic for Computer Scientists. Answers to Some Exercises
Applied Logic for Computer Scientists Computational Deduction and Formal Proofs Springer, 2017 doi: http://link.springer.com/book/10.1007%2f9783319516530 Answers to Some Exercises Mauricio AyalaRincón
More informationPredicate Calculus. Formal Methods in Verification of Computer Systems Jeremy Johnson
Predicate Calculus Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. Motivation 1. Variables, quantifiers and predicates 2. Syntax 1. Terms and formulas 2. Quantifiers, scope
More informationPropositional 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
More informationUnit I LOGIC AND PROOFS. B. Thilaka Applied Mathematics
Unit I LOGIC AND PROOFS B. Thilaka Applied Mathematics UNIT I LOGIC AND PROOFS Propositional Logic Propositional equivalences Predicates and Quantifiers Nested Quantifiers Rules of inference Introduction
More informationPredicate Logic. CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo
Predicate Logic CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 22 Outline 1 From Proposition to Predicate
More informationPropositional and Predicate Logic
Propositional and Predicate Logic CS 53605: Science of Programming This is for Section 5 Only: See Prof. Ren for Sections 1 4 A. Why Reviewing/overviewing logic is necessary because we ll be using it
More informationFirst 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
More informationIntroduction to Isabelle/HOL
Introduction to Isabelle/HOL 1 Notes on Isabelle/HOL Notation In Isabelle/HOL: [ A 1 ;A 2 ; ;A n ]G can be read as if A 1 and A 2 and and A n then G 3 Note: Px (P x) stands for P (x) (P(x)) P(x, y) can
More informationSemantics of intuitionistic propositional logic
Semantics of intuitionistic propositional logic Erik Palmgren Department of Mathematics, Uppsala University Lecture Notes for Applied Logic, Fall 2009 1 Introduction Intuitionistic logic is a weakening
More informationIncomplete version for students of easllc2012 only. 6.6 The Model Existence Game 99
98 FirstOrder Logic 6.6 The Model Existence Game In this section we learn a new game associated with trying to construct a model for a sentence or a set of sentences. This is of fundamental importance
More informationChapter 1: The Logic of Compound Statements. January 7, 2008
Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive
More informationPeano Arithmetic. CSC 438F/2404F Notes (S. Cook) Fall, Goals Now
CSC 438F/2404F Notes (S. Cook) Fall, 2008 Peano Arithmetic Goals Now 1) We will introduce a standard set of axioms for the language L A. The theory generated by these axioms is denoted PA and called Peano
More informationECOM Discrete Mathematics
ECOM 2311 Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311 Discrete Mathematics  Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional
More informationReadings: Conjecture. Theorem. Rosen Section 1.5
Readings: Conjecture Theorem Lemma Lemma Step 1 Step 2 Step 3 : Step n1 Step n a rule of inference an axiom a rule of inference Rosen Section 1.5 Provide justification of the steps used to show that a
More informationLogic and Modelling. Introduction to Predicate Logic. Jörg Endrullis. VU University Amsterdam
Logic and Modelling Introduction to Predicate Logic Jörg Endrullis VU University Amsterdam Predicate Logic In propositional logic there are: propositional variables p, q, r,... that can be T or F In predicate
More informationThis section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.
1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as
More informationLogic and Set Notation
Logic and Set Notation Logic Notation p, q, r: statements,,,, : logical operators p: not p p q: p and q p q: p or q p q: p implies q p q:p if and only if q We can build compound sentences using the above
More informationDefinite Logic Programs
Chapter 2 Definite Logic Programs 2.1 Definite Clauses The idea of logic programming is to use a computer for drawing conclusions from declarative descriptions. Such descriptions called logic programs
More informationThe Arithmetical Hierarchy
Chapter 11 The Arithmetical Hierarchy Think of K as posing the problem of induction for computational devices, for it is impossible to tell for sure whether a given computation will never halt. Thus, K
More informationLogic Part I: Classical Logic and Its Semantics
Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model
More informationEtype interpretation without Etype pronoun: How Peirce s Graphs. capture the uniqueness implication of donkey sentences
Etype interpretation without Etype pronoun: How Peirce s Graphs capture the uniqueness implication of donkey sentences Author: He Chuansheng (PhD student of linguistics) The Hong Kong Polytechnic University
More informationCHAPTER 10. Gentzen Style Proof Systems for Classical Logic
CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationNotes on Inference and Deduction
Notes on Inference and Deduction Consider the following argument 1 Assumptions: If the races are fixed or the gambling houses are crooked, then the tourist trade will decline. If the tourist trade declines
More informationCOMP4418: Knowledge Representation and Reasoning FirstOrder Logic
COMP4418: Knowledge Representation and Reasoning FirstOrder Logic Maurice Pagnucco School of Computer Science and Engineering University of New South Wales NSW 2052, AUSTRALIA morri@cse.unsw.edu.au COMP4418
More informationSection 2.1: Introduction to the Logic of Quantified Statements
Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional
More informationExample ( x.(p(x) Q(x))) ( x.p(x) x.q(x)) premise. 2. ( x.(p(x) Q(x))) elim, 1 3. ( x.p(x) x.q(x)) elim, x. P(x) x.
Announcements CS311H: Discrete Mathematics More Logic Intro to Proof Techniques Homework due next lecture Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mathematics More Logic Intro
More informationMathematical Logic Part One
Mathematical Logic Part One Question: How do we formalize the defnitions and reasoning we use in our proofs? Where We're Going Propositional Logic (Today) Basic logical connectives. Truth tables. Logical
More informationPropositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
More informationPropositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
More informationLogic: 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
More informationThe Importance of Being Formal. Martin Henz. February 5, Propositional Logic
The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets
More informationMathematical Logic. (Based on lecture slides by Stan Burris) George Voutsadakis 1. Lake Superior State University. LSSU Math 300
Mathematical Logic (Based on lecture slides by Stan Burris) George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 300 George Voutsadakis (LSSU) Logic January
More informationThe Natural Deduction Pack
The Natural Deduction Pack Alastair Carr March 2018 Contents 1 Using this pack 2 2 Summary of rules 3 3 Worked examples 5 31 Implication 5 32 Universal quantifier 6 33 Existential quantifier 8 4 Practice
More informationNatural deduction for truthfunctional logic
Natural deduction for truthfunctional logic Phil 160  Boston University Why natural deduction? After all, we just found this nice method of truthtables, which can be used to determine the validity or
More informationThe Predicate Calculus
Math 3040 Spring 2011 The Predicate Calculus Contents 1. Introduction 1 2. Some examples 1 3. General elements of sets. 2 4. Variables and constants 2 5. Expressions 3 6. Predicates 4 7. Logical operations
More informationCHAPTER 11. Introduction to Intuitionistic Logic
CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated
More informationSymbolic Logic 3. For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true.
Symbolic Logic 3 Testing deductive validity with truth tables For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true. So, given that truth tables
More informationCS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)
CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bioconditional Converse Inverse Contrapositive Laws of
More informationDISCRETE MATHEMATICS BA202
TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION
More informationSec$on Summary. Definition of sets Describing Sets
Section 2.1 Sec$on Summary Definition of sets Describing Sets Roster Method SetBuilder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets
More informationTHE LOGIC OF COMPOUND STATEMENTS
CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.1 Logical Form and Logical Equivalence Copyright Cengage Learning. All rights reserved. Logical Form
More informationLogic and Discrete Mathematics. Section 3.5 Propositional logical equivalence Negation of propositional formulae
Logic and Discrete Mathematics Section 3.5 Propositional logical equivalence Negation of propositional formulae Slides version: January 2015 Logical equivalence of propositional formulae Propositional
More information2 Truth Tables, Equivalences and the Contrapositive
2 Truth Tables, Equivalences and the Contrapositive 12 2 Truth Tables, Equivalences and the Contrapositive 2.1 Truth Tables In a mathematical system, true and false statements are the statements of the
More informationCMPSCI 601: Tarski s Truth Definition Lecture 15. where
@ CMPSCI 601: Tarski s Truth Definition Lecture 15! "$#&%(') *+,!".#/%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 @BHI LJKK MKK )HG if H ; C if H @ 1 > > > Fitch
More informationThe Limit of Humanly Knowable Mathematical Truth
The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015 Another title for this talk could be... An Argument
More informationMA103 STATEMENTS, PROOF, LOGIC
MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking
More informationNonclassical logics (Nichtklassische Logiken)
Nonclassical logics (Nichtklassische Logiken) VU 185.249 (lecture + exercises) http://www.logic.at/lvas/ncl/ Chris Fermüller Technische Universität Wien www.logic.at/people/chrisf/ chrisf@logic.at Winter
More informationCITS2211: Test One. Student Number: 1. Use a truth table to prove or disprove the following statement.
CITS2211: Test One Name: Student Number: 1. Use a truth table to prove or disprove the following statement. ((P _ Q) ^ R)) is logically equivalent to ( P ) ^ ( Q ^ R) P Q R P _ Q (P _ Q) ^ R LHS P Q R
More informationHANDOUT AND SET THEORY. Ariyadi Wijaya
HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 11 and 41 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationA. Propositional Logic
CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals
More informationResolution: Motivation
Resolution: Motivation Steps in inferencing (e.g., forwardchaining) 1. Define a set of inference rules 2. Define a set of axioms 3. Repeatedly choose one inference rule & one or more axioms (or premices)
More informationComputational Logic. Davide Martinenghi. Spring Free University of BozenBolzano. Computational Logic Davide Martinenghi (1/26)
Computational Logic Davide Martinenghi Free University of BozenBolzano Spring 2010 Computational Logic Davide Martinenghi (1/26) Propositional Logic  algorithms Complete calculi for deciding logical
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More informationDisplay calculi in nonclassical logics
Display calculi in nonclassical logics Revantha Ramanayake Vienna University of Technology (TU Wien) Prague seminar of substructural logics March 28 29, 2014 Revantha Ramanayake (TU Wien) Display calculi
More informationPropositional Logic. CS 3234: Logic and Formal Systems. Martin Henz and Aquinas Hobor. August 26, Generated on Tuesday 31 August, 2010, 16:54
Propositional Logic CS 3234: Logic and Formal Systems Martin Henz and Aquinas Hobor August 26, 2010 Generated on Tuesday 31 August, 2010, 16:54 1 Motivation In traditional logic, terms represent sets,
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationProseminar 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,
More informationIntroduction to Metalogic 1
Philosophy 135 Spring 2012 Tony Martin Introduction to Metalogic 1 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: (i) sentence letters p 0, p 1, p 2,... (ii) connectives,
More informationLogic of Sentences (Propositional Logic) is interested only in true or false statements; does not go inside.
You are a mathematician if 1.1 Overview you say to a car dealer, I ll take the red car or the blue one, but then you feel the need to add, but not both.  1. Logic and Mathematical Notation (not in the
More informationLinear Temporal Logic (LTL)
Chapter 9 Linear Temporal Logic (LTL) This chapter introduces the Linear Temporal Logic (LTL) to reason about state properties of Labelled Transition Systems defined in the previous chapter. We will first
More informationGlossary of Logical Terms
Math 304 Spring 2007 Glossary of Logical Terms The following glossary briefly describes some of the major technical logical terms used in this course. The glossary should be read through at the beginning
More informationReal Analysis: Part I. William G. Faris
Real Analysis: Part I William G. Faris February 2, 2004 ii Contents 1 Mathematical proof 1 1.1 Logical language........................... 1 1.2 Free and bound variables...................... 3 1.3 Proofs
More information1 Introduction to Predicate Resolution
1 Introduction to Predicate Resolution The resolution proof system for Predicate Logic operates, as in propositional case on sets of clauses and uses a resolution rule as the only rule of inference. The
More information