PHIL12A Section answers, 14 February 2011
|
|
- Claud Terence Gregory
- 5 years ago
- Views:
Transcription
1 PHIL12A Section answers, 14 February 2011 Julian Jonker 1 How much do you know? 1. You should understand why a truth table is constructed the way it is: why are the truth values listed in the order they are? In principle, it doesn t matter how they are listed, but there is a natural way to list them systematically, which we use. (a) My favourite slot machine gives me money if I line up four cherries in a row. When I pull the lever, each of the four displayed items can come up as either a cherry or a banana. How many different combinations can the machine display? List them. This is exactly like listing all possible truth values for four atomic sentences, except with say, cherry for True and banana for False. There are = 2 4 = 16 possible combinations. Think of it in the following way: Suppose you were betting on just one item. It could come up as a cherry or banana, so there are two possible combinations. Now suppose you were betting on two items. The first item can be a cherry, but there are two possible ways for it to be a cherry if we include the possible values of the second item: cherry cherry cherry banana But this is true also if it comes up as a banana. banana cherry banana banana So there are four possible combinations. More particularly, we have 2 2 = 2 2 = 4 possibilities. If we add a third item, then the first item can come up in four different ways: if it comes up as a cherry, then there are two different ways the second item can come up (cherry or banana), and for each way the second item comes up there are two different ways the third item can come up. So all in all there are = 2 3 = 8 possibilities. The same holds for a truth table. If you have n atomic sentences, you will have 2 n rows in your truth table. In other words, the number of rows grows exponentially, which is why you will never be asked to do a truth table for more than a handful of atomic sentences! 1
2 (b) Suppose the machine displays three different items at a time my goal is to line up three cherries. But each item could be either a cherry, a banana, a palm tree or a robot. How many combinations are possible? It would take a long time to list them all, but convince yourself that there is a way to list them systematically so that you are sure you don t leave any out. What would the 17th item on your list be? There are = 4 3 = 64 possible combinations. Think of the list as a sort of tree. For the first item, draw four branches for the different possibilities: cherry, banana, palm tree, robot. At the end of the cherry branch, draw another four branches for the possible values of the second item (cherry, banana, palm tree, robot), and then do this three more times at the end of each of the other branches. You should now have sixteen end points. At the end of each end point, draw four branches for the possible values of the third item. All in all you should have 64 end points now, which you could number if you wish. If you drew the branches in the order I mentioned them here (you didn t have to, but you should have stuck with some kind of order), your 17th possibility would have been: banana cherry cherry 2. Draw up truth tables for the following sentences, and say whether they are tautologies or not. (a) (A B) ( A B) A B (A B) ( A B) T T T T F T T F T T T T F T T T F T F T T F F T F F F T T T F T T F F F T T F T T F T F (b) (Ex 4.4) (B C B) B C ( B C B) T T T T F F T F F T T F T T T T F F F T F T T F F F T F T F F F T F F T F F T F 2
3 (c) ( A ( B C)) ( (A B) C) A B C ( A ( B C)) ( (A B) C) T T T F T T F T T T T F T T T T T T T F F T F F T F F F F T T T F F T F T F T T T F T T T T T F F T T T F F F T T T F T F T T T F F T F F T T T F T F T T T T T F F T T T F T F T F T F T F F T T F F T T F F F T T F T T F T T T T F F F T T F F F T F T T F T F T T F F F T F (d) (Ex 4.6) [ A (B C) (A B)] A B C [ A (B C) (A B)] T T T F F T F F T T T T T T T T T F F F T T T T F F T T T T T F T F F T T T F F T T T F F T F F F F T T T F F F T T F F F T T F T F T F T T T T F F T F T F F T F T T T F F T F F T F F T F T F T T F F T T F F F F F F F T F T T F F F T F F F 2 Something slightly harder, if there s time. A sentence S is tautological if and only if every row of its truth table assigns True to S. S is logically necessary if and only if it is true in every logically possible circumstance. S is TW-necessary if and only it is true in every world that could possibly be constructed in Tarski s World. 1. Is the following sentence a tautology? It is not the case that both all men are mortal and Socrates is a man, otherwise Socrates is mortal. Is it logically necessary? The sentence is logically necessary, since there is no possible world in which it is false. There may be some possible worlds in which Socrates is mortal, and in that case the sentence is true. In the other worlds, at least one of the following must be true: not all men are mortal (since Socrates isn t), or Socrates isn t a man and isn t mortal like them. So in these cases the sentence is true. Now consider whether the sentence is a tautology or not. The atomic sentences are: all men are mortal, Socrates is a man, and Socrates is mortal, and they are combined by the connectives, and. So we 3
4 could abbreviate the sentence in the following way: (P Q) R, where P stands for all men are mortal, Q stands for Socrates is a man, and R stands for Socrates is mortal. This sentence is not a tautology, as drawing up a truth table will show. The moral of the story is that our truth table method is not fine-grained enough to capture the meanings of predicates and names. There is logical structure embedded within the atomic sentences that cannot be captured using truth tables. (In fact, many logic textbooks teach logic in two stages: propositional logic, dealing only with complex sentences made up of atomic sentences connected by truth-functional connectives; and predicate logic, which allows you to get into the structure of atomic sentences by working with predicates, names and quantifiers. Truth tables are taught in propositional logic, but drop out of the picture with predicate logic.) 2. Name at least two sentences that are logically necessary but not tautological. For example: (a) All bachelors are unmarried men. (b) There are infinitely many prime numbers. 3. Name at least two sentences that are TW-necessary but not logically necessary. For example: (a) Cube(a) Tet(a) Dodec(a) (b) (Large(a) Large(b)) Adjoins(a,b) 4. For each of the following, determine whether the sentence is a tautology, a logical necessity, or a TW necessity. (a) a=b b=c c=c This is not a necessity of any sort, though it is logically possible. (b) BackOf(a,b) BackOf(a,b) This is a tautology, and therefore also a logical necessity and a TW necessity. (c) (Cube(b) Cube(e)) Cube(b) A truth table shows that this is a tautology, and therefore also a logical necessity and a TW necessity. (d) (Cube(b) Cube(e)) SameShape(b,e) This is a TW necessity, since in Tarski s World there are one of two possibilities: either b and e are the same kind of block, for example they are both tetrahedrons or they are both cubes, in which case the sentence is true; or they are different shapes. But if they are different shapes, then they are certainly not both cubes, which means that the sentence is true. The sentence is not a tautology, as a truth table will show; however it is a logical necessity, as far as I can tell. 4
5 (e) SameRow(a,b) (FrontOf(a,b) BackOf(a,b)) This is a TW necessity. Either a and b are in the same row; or it is not the case that either a is in front of b or b is in front of a. I believe this is a logical necessity; it is not a tautology. (f) SameCol(a,b) SameRow(a,b) a=b This is not a necessity of any kind, since it is false if a and b are not in the same place. The following would be a TW necessity: SameCol(a,b) SameRow(a,b) (a=b), but it would not be a logical necessity. (Why?) 5. (Based on Ex 4.10) A sentence S is a logically possibility if it is true in some logically possible circumstances. S is a TW-possibility if there is at least one world that can be constructed in Tarski s World and in which S is true. S is a TT-possibility if at least one row of its truth table assigns True to S. Draw a set of nested circle indicating the relationship between logical necessities, logical possibilities, TW-possibilities, TT-possibilities, and sentences which do not belong to any of these kinds. Give an example of each kind of sentence. You should have drawn three nested circles, with the innermost being the TW possibilities, the middle circle comprising the logical possibilities, and the outer circle comprising the TT possibilities. Beyond this are those sentences which are not TT possible, and thus not logically possible and not TW possible (sentences such as a a). Here s a TW possibility (and therefore also a logical possibility and a TT possibility): SameRow(a,b) SameCol(a,b) a=b. Here s a logical possibility (and therefore also a TT possibility) that is not a TW possibility: SameRow(a,b) SameCol(a,b) (a=b). Here s a TT possibility that is not a logical possibility: SameRow(a,b) SameCol(a,b) a=b. You might want to think about these relationships along the following lines: truth tables don t look at the logical structure embedded within atomic sentences, and so they simply consider what is possible in terms of the structure of the logical connectives. The meanings of names and predicates add additional constraints: think of these as the constraints placed upon logical truths by our language. The constraints narrow the sphere of TT possibilities to the logical possibilities.the structure of the world itself then adds extra constraints on the true things we can say about the world. These constraints narrow the sphere of the logical possibilities to the TW possibilities. 3 Challenge questions 1. When determining whether a sentence is a tautology, you typically draw up a truth table and assign all possible combinations of truth values to its atomic sentences. This is pretty laborious once you have more 5
6 than three atomic sentences. Can you come up with a quicker way to determine whether a sentence is a tautology? You should try your method with the following sentence: ( ( ( ( A B) C) D) E) (A E). A tautology is true in every row of its truth table. If we try to make a tautology false, we will end up with a contradiction. On the other hand, if the sentence is not a tautology, we will end up with a valid assignment of truth values to its atomic sentences. So let s try this, starting with the connective that has widest scope: ( ( ( ( A B) C) D) E) (A E) F We know that for a disjunction to be false both of its disjuncts must be false, so we now fill these in: ( ( ( ( A B) C) D) E) (A E) F F F The disjunction on the left with the widest scope is false, so we need two false disjuncts again. On the right hand side, we have another false disjunction, allowing us to find values for A and E: ( ( ( ( A B) C) D) E) (A E) F F F F F F F T You should see the problem immediately. We need to make E on the right hand side true, but on the left hand side E comes out false. Since an atomic sentence must have the same truth value wherever it appears in a complex sentence, we have discovered a contradiction. All this means not only that we have failed to make the sentence false, but that we cannot do so. So it is a tautology. 2. The textbook says that the notion of logical necessity is annoyingly vague. Why? Can you think of a sentence whose status as a logical necessity is controversial? A concept is vague if there are borderline cases which the concept does not help us to settle. Such cases may arise for various reasons. One problem with our definition of logical necessity is its circularity. We said that a sentence is logically necessary just in case there is no logical possible circumstance in which it is false. But now we have based our definition of logical necessity on the notion of a logically possible circumstance, and in particular, on the notion of logical possibility. Which circumstances are logically possible? Those which are compatible with what is logically necessary. Another problem is that there are a number of ways in which a set of circumstances can be possible or impossible. It is not possible that 2+2=5, but this kind of impossibility is not the kind of impossibility that prohibits faster than light speed travel. Potentially, then, our notion of necessity is ambiguous. There are further kinds of impossibility: sentences that can t be true because of the logical relations between their words ( There is an unmarried person who is married. ), and sentences which can t be true because of the meanings of their words ( There is a bachelor who is married. ) Quine famously thought that it was too difficult to come up with a precise identification of the latter class of sentences, and that while our sentences are made true both by our 6
7 language and the way the world is, it is not possible to sort out the contributions of language and world sentence by sentence. Consider a sentence such as: A square has four equal sides. This, you might venture, is not true of Tahrir Square. How can I convince you that I did not mean to include Tahrir Square amongst the things I referred to as squares? Only by saying that what I meant by square was shape with four equal sides. And that would be begging the question. There is another problem with logical necessity, separate from the Quine s doubts that we can get clear on what it is for a sentence to be true in virtue of its meaning. For one might forego such sentences as logically necessary, and point simply to those sentences that are true in virtue of their logical structure. We know that a a is tautological, and thus the clearest case of a logical necessity. But we can doubt that every sentence that follows this schema is logically necessary. John has a beard or John does not have a beard follows the schema, but how could we deny that there are interesting intermediate cases in which John neither has a proper beard nor lacks one entirely? Or consider the schema (a a) - again, as clear cut a case of a logical necessity as one could possibly find. But it is not the case that the particle is spin up and the particle is spin down is an instantiation of the schema, and quantum physics seems to tell us that such sentences are often false. There are deep problems here about what counts as a possibility, and about what counts as a possible circumstance. Formal languages, such as First Order Logic, help us make these ideas more precise, but they do not necessarily solve the problems for natural language. 7
PHIL12A Section answers, 16 February 2011
PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F
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 informationFinal Exam Theory Quiz Answer Page
Philosophy 120 Introduction to Logic Final Exam Theory Quiz Answer Page 1. (a) is a wff (and a sentence); its outer parentheses have been omitted, which is permissible. (b) is also a wff; the variable
More informationPHIL12A Section answers, 28 Feb 2011
PHIL12A Section answers, 28 Feb 2011 Julian Jonker 1 How much do you know? Give formal proofs for the following arguments. 1. (Ex 6.18) 1 A B 2 A B 1 A B 2 A 3 A B Elim: 2 4 B 5 B 6 Intro: 4,5 7 B Intro:
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 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 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 informationChapter 9. Modal Language, Syntax, and Semantics
Chapter 9 Modal Language, Syntax, and Semantics In chapter 6 we saw that PL is not expressive enough to represent valid arguments and semantic relationships that employ quantified expressions some and
More informationDiscrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques Colin Stirling Informatics Some slides based on ones by Myrto Arapinis Colin Stirling (Informatics) Discrete Mathematics
More informationSupplementary 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
More informationIntroducing Proof 1. hsn.uk.net. Contents
Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction
More informationChoosing Logical Connectives
Choosing Logical Connectives 1. Too Few Connectives?: We have chosen to use only 5 logical connectives in our constructed language of logic, L1 (they are:,,,, and ). But, we might ask, are these enough?
More informationPart I: Propositional Calculus
Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully
More informationcis32-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
More informationCS 124 Math Review Section January 29, 2018
CS 124 Math Review Section CS 124 is more math intensive than most of the introductory courses in the department. You re going to need to be able to do two things: 1. Perform some clever calculations to
More informationDirect Proof and Counterexample I:Introduction
Direct Proof and Counterexample I:Introduction Copyright Cengage Learning. All rights reserved. Goal Importance of proof Building up logic thinking and reasoning reading/using definition interpreting :
More informationLogik für Informatiker Proofs in propositional logic
Logik für Informatiker Proofs in propositional logic WiSe 009/10 al consequence Q is a logical consequence of P 1,, P n, if all worlds that make P 1,, P n true also make Q true Q is a tautological consequence
More informationDirect Proof and Counterexample I:Introduction. Copyright Cengage Learning. All rights reserved.
Direct Proof and Counterexample I:Introduction Copyright Cengage Learning. All rights reserved. Goal Importance of proof Building up logic thinking and reasoning reading/using definition interpreting statement:
More informationCHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS
CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1 Language There are several propositional languages that are routinely called classical propositional logic languages. It is due to the functional dependency
More informationLogik für Informatiker Logic for computer scientists
Logik für Informatiker Logic for computer scientists Till Mossakowski WiSe 2013/14 Till Mossakowski Logic 1/ 24 Till Mossakowski Logic 2/ 24 Logical consequence 1 Q is a logical consequence of P 1,, P
More informationPropositional 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 information1.1 Statements and Compound Statements
Chapter 1 Propositional Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something
More informationChapter 4: Classical Propositional Semantics
Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical
More informationINTRODUCTION TO LOGIC
INTRODUCTION TO LOGIC L. MARIZZA A. BAILEY 1. The beginning of Modern Mathematics Before Euclid, there were many mathematicians that made great progress in the knowledge of numbers, algebra and geometry.
More informationReview. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional.
Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse,
More informationPropositional Logic. Argument Forms. Ioan Despi. University of New England. July 19, 2013
Propositional Logic Argument Forms Ioan Despi despi@turing.une.edu.au University of New England July 19, 2013 Outline Ioan Despi Discrete Mathematics 2 of 1 Order of Precedence Ioan Despi Discrete Mathematics
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 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 informationDeep Metaphysical Indeterminacy
Deep Metaphysical Indeterminacy Bradford Skow Abstract A recent theory of metaphysical indeterminacy says that metaphysical indeterminacy is multiple actuality. That is, we have a case of metaphysical
More informationPropositional 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
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 informationAnnouncements CompSci 102 Discrete Math for Computer Science
Announcements CompSci 102 Discrete Math for Computer Science Read for next time Chap. 1.4-1.6 Recitation 1 is tomorrow Homework will be posted by Friday January 19, 2012 Today more logic Prof. Rodger Most
More informationFoundation of proofs. Jim Hefferon.
Foundation of proofs Jim Hefferon http://joshua.smcvt.edu/proofs The need to prove In Mathematics we prove things To a person with a mathematical turn of mind, the base angles of an isoceles triangle are
More informationWhat is proof? Lesson 1
What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might
More informationLogik für Informatiker Logic for computer scientists
Logik für Informatiker Logic for computer scientists Till Mossakowski WiSe 2013/14 Till Mossakowski Logic 1/ 29 The language of PL1 Till Mossakowski Logic 2/ 29 The language of PL1: individual constants
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 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 informationPropositional Logic George Belic
1 Symbols and Translation (Ch 6.1) Our aim for the rest of the course is to study valid (deductive) arguments in symbolic systems. In this section, (Ch. 6-7) we look at the symbolic system of propositional
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
More informationLogic. Quantifiers. (real numbers understood). x [x is rotten in Denmark]. x<x+x 2 +1
Logic One reason for studying logic is that we need a better notation than ordinary English for expressing relationships among various assertions or hypothetical states of affairs. A solid grounding in
More informationAnnouncements For Methods of Proof for Boolean Logic Proof by Contradiction. Outline. The Big Picture Where is Today? William Starr
Announcements For 09.22 Methods of for Boolean Logic William Starr 1 HW1 grades will be on Bb by end of week 2 HW4 is due on Tuesday This one is mostly written Feel free to type it out! 3 If you have problems
More informationMathematics 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)
More informationSyllogistic Logic and its Extensions
1/31 Syllogistic Logic and its Extensions Larry Moss, Indiana University NASSLLI 2014 2/31 Logic and Language: Traditional Syllogisms All men are mortal. Socrates is a man. Socrates is mortal. Some men
More informationTheorem. 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 +
More informationProofs. Introduction II. Notes. Notes. Notes. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007
Proofs Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.5, 1.6, and 1.7 of Rosen cse235@cse.unl.edu
More informationINTRODUCTION TO LOGIC 3 Formalisation in Propositional Logic
Introduction INRODUCION O LOGIC 3 Formalisation in Propositional Logic Volker Halbach If I could choose between principle and logic, I d take principle every time. Maggie Smith as Violet Crawley in Downton
More informationHOW TO WRITE PROOFS. Dr. Min Ru, University of Houston
HOW TO WRITE PROOFS Dr. Min Ru, University of Houston One of the most difficult things you will attempt in this course is to write proofs. A proof is to give a legal (logical) argument or justification
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationPhil 110, Spring 2007 / April 23, 2007 Practice Questions for the Final Exam on May 7, 2007 (9:00am 12:00noon)
Phil 110, Spring 2007 / April 23, 2007 Practice Questions for the Final Exam on May 7, 2007 (9:00am 12:00noon) (1) Which of the following are well-formed sentences of FOL? (Draw a circle around the numbers
More informationTruth-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
More informationPHIL 422 Advanced Logic Inductive Proof
PHIL 422 Advanced Logic Inductive Proof 1. Preamble: One of the most powerful tools in your meta-logical toolkit will be proof by induction. Just about every significant meta-logical result relies upon
More informationCM10196 Topic 2: Sets, Predicates, Boolean algebras
CM10196 Topic 2: Sets, Predicates, oolean algebras Guy McCusker 1W2.1 Sets Most of the things mathematicians talk about are built out of sets. The idea of a set is a simple one: a set is just a collection
More informationFor 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
More informationProof 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
More informationLecture 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
More informationINTRODUCTION TO LOGIC 1 Sets, Relations, and Arguments. Why logic? Arguments
The Logic Manual INTRODUCTION TO LOGIC 1 Sets, Relations, and Arguments Volker Halbach Pure logic is the ruin of the spirit. Antoine de Saint-Exupéry The Logic Manual web page for the book: http://logicmanual.philosophy.ox.ac.uk/
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 informationThe 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
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 informationExamples: 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
More informationAxiomatic 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
More informationMathematical induction
Mathematical induction Notes and Examples These notes contain subsections on Proof Proof by induction Types of proof by induction Proof You have probably already met the idea of proof in your study of
More informationIntroduction to Logic
Introduction to Logic L. Marizza A. Bailey June 21, 2014 The beginning of Modern Mathematics Before Euclid, there were many mathematicians that made great progress in the knowledge of numbers, algebra
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 informationChapter 2. Mathematical Reasoning. 2.1 Mathematical Models
Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................
More informationIntroduction 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
More informationLogik für Informatiker Logic for computer scientists
Logik für Informatiker for computer scientists WiSe 2009/10 Rooms Monday 12:00-14:00 MZH 1400 Thursday 14:00-16:00 MZH 5210 Exercises (bring your Laptops with you!) Wednesday 8:00-10:00 Sportturm C 5130
More informationA Primer on Logic Part 2: Statements, Truth Functions, and Argument Schemas Jason Zarri. 1. Introduction
A Primer on Logic Part 2: Statements, Truth Functions, and Argument Schemas Jason Zarri 1. Introduction In this Part our goal is to learn how to determine the validity of argument schemas, abstract forms
More informationRussell s logicism. Jeff Speaks. September 26, 2007
Russell s logicism Jeff Speaks September 26, 2007 1 Russell s definition of number............................ 2 2 The idea of reducing one theory to another.................... 4 2.1 Axioms and theories.............................
More informationLogic - recap. So far, we have seen that: Logic is a language which can be used to describe:
Logic - recap So far, we have seen that: Logic is a language which can be used to describe: Statements about the real world The simplest pieces of data in an automatic processing system such as a computer
More informationDEEP METAPHYSICAL INDETERMINACY
The Philosophical Quarterly June 2010 doi: 10.1111/j.1467-9213.2010.672.x The Scots Philosophical Association and the University of St Andrews DEEP METAPHYSICAL INDETERMINACY BY BRADFORD SKOW A recent
More information3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations
Logic Propositions and logical operations Main concepts: propositions truth values propositional variables logical operations 1 Propositions and logical operations A proposition is the most basic element
More informationLecture 3 : Predicates and Sets DRAFT
CS/Math 240: Introduction to Discrete Mathematics 1/25/2010 Lecture 3 : Predicates and Sets Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last time we discussed propositions, which are
More informationPhilosophy 220. Truth-Functional Equivalence and Consistency
Philosophy 220 Truth-Functional Equivalence and Consistency Review Logical equivalency: The members of a pair of sentences [of natural language] are logically equivalent if and only if it is not [logically]
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 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 informationINTENSIONS MARCUS KRACHT
INTENSIONS MARCUS KRACHT 1. The Way Things Are This note accompanies the introduction of Chapter 4 of the lecture notes. I shall provide some formal background and technology. Let a language L be given
More informationINTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims
Introduction INTRODUCTION TO LOGIC 2 Syntax and Semantics of Propositional Logic Volker Halbach In what follows I look at some formal languages that are much simpler than English and define validity of
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities
More information4.1 Real-valued functions of a real variable
Chapter 4 Functions When introducing relations from a set A to a set B we drew an analogy with co-ordinates in the x-y plane. Instead of coming from R, the first component of an ordered pair comes from
More informationDeduction by Daniel Bonevac. Chapter 3 Truth Trees
Deduction by Daniel Bonevac Chapter 3 Truth Trees Truth trees Truth trees provide an alternate decision procedure for assessing validity, logical equivalence, satisfiability and other logical properties
More informationRevisit summer... go to the Fitzwilliam Museum!
Revisit summer... go to the Fitzwilliam Museum! Faculty of Philosophy Formal Logic Lecture 5 Peter Smith Peter Smith: Formal Logic, Lecture 5 2 Outline Propositional connectives, and the assumption of
More informationPropositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
More informationPUZZLE. You meet A, B, and C in the land of knights and knaves. A says Either B and I are both knights or we are both knaves.
PUZZLE You meet A, B, and C in the land of knights and knaves. A says Either B and I are both knights or we are both knaves. B says C and I are the same type. C says Either A is a knave or B is a knave.
More informationTautological equivalence. entence equivalence. Tautological vs logical equivalence
entence equivalence Recall two definitions from last class: 1. A sentence is an X possible sentence if it is true in some X possible world. Cube(a) is TW possible sentence. 2. A sentence is an X necessity
More informationPhilosophy of Mathematics Structuralism
Philosophy of Mathematics Structuralism Owen Griffiths oeg21@cam.ac.uk St John s College, Cambridge 17/11/15 Neo-Fregeanism Last week, we considered recent attempts to revive Fregean logicism. Analytic
More informationUnary negation: T F F T
Unary negation: ϕ 1 ϕ 1 T F F T Binary (inclusive) or: ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ) T T T T F T F T T F F F Binary (exclusive) or: ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ) T T F T F T F T T F F F Classical (material) conditional: ϕ 1
More informationGödel 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
More informationFirst 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
More informationArguments and Proofs. 1. A set of sentences (the premises) 2. A sentence (the conclusion)
Arguments and Proofs For the next section of this course, we will study PROOFS. A proof can be thought of as the formal representation of a process of reasoning. Proofs are comparable to arguments, since
More informationIntroduction to fuzzy logic
Introduction to fuzzy logic Andrea Bonarini Artificial Intelligence and Robotics Lab Department of Electronics and Information Politecnico di Milano E-mail: bonarini@elet.polimi.it URL:http://www.dei.polimi.it/people/bonarini
More informationDiscrete Mathematics for CS Fall 2003 Wagner Lecture 3. Strong induction
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 3 This lecture covers further variants of induction, including strong induction and the closely related wellordering axiom. We then apply these
More informationIntermediate 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
More informationMath 13, Spring 2013, Lecture B: Midterm
Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # E-mail address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.
More informationLogic for Computer Science - Week 2 The Syntax of Propositional Logic
Logic for Computer Science - Week 2 The Syntax of Propositional Logic Ștefan Ciobâcă November 30, 2017 1 An Introduction to Logical Formulae In the previous lecture, we have seen what makes an argument
More information1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, 10 Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual
More informationNon-normal Worlds. Daniel Bonevac. February 5, 2012
Non-normal Worlds Daniel Bonevac February 5, 2012 Lewis and Langford (1932) devised five basic systems of modal logic, S1 - S5. S4 and S5, as we have seen, are normal systems, equivalent to K ρτ and K
More informationcse541 LOGIC FOR COMPUTER SCIENCE
cse541 LOGIC FOR COMPUTER SCIENCE Professor Anita Wasilewska Spring 2015 LECTURE 2 Chapter 2 Introduction to Classical Propositional Logic PART 1: Classical Propositional Model Assumptions PART 2: Syntax
More informationWilliamson s Modal Logic as Metaphysics
Williamson s Modal Logic as Metaphysics Ted Sider Modality seminar 1. Methodology The title of this book may sound to some readers like Good as Evil, or perhaps Cabbages as Kings. If logic and metaphysics
More information