Logic: Sets, Relations, and Arguments - An Introduction, Summaries of Logic

Download Logic: Sets, Relations, and Arguments - An Introduction and more Summaries Logic in PDF only on Docsity! INTRODUCTION TO LOGIC  Sets, Relations, and Arguments Volker Halbach Pure logic is the ruin of the spirit. Antoine de Saint-Exupéry The Logic Manual e Logic Manual web page for the book: http://logicmanual.philosophy.ox.ac.uk/ Exercises Booklet More Exercises by Peter Fritz slides of the lectures worked examples past examination papers with solutions Mark Sainsbury: Logical Forms: An Introduction to Philosophical Logic, Blackwell, second edition, 2001 The Logic Manual e Logic Manual web page for the book: http://logicmanual.philosophy.ox.ac.uk/ Exercises Booklet More Exercises by Peter Fritz slides of the lectures worked examples past examination papers with solutions Mark Sainsbury: Logical Forms: An Introduction to Philosophical Logic, Blackwell, second edition, 2001 The Logic Manual e Logic Manual web page for the book: http://logicmanual.philosophy.ox.ac.uk/ Exercises Booklet More Exercises by Peter Fritz slides of the lectures worked examples past examination papers with solutions Mark Sainsbury: Logical Forms: An Introduction to Philosophical Logic, Blackwell, second edition, 2001 The Logic Manual e Logic Manual web page for the book: http://logicmanual.philosophy.ox.ac.uk/ Exercises Booklet More Exercises by Peter Fritz slides of the lectures worked examples past examination papers with solutions Mark Sainsbury: Logical Forms: An Introduction to Philosophical Logic, Blackwell, second edition, 2001 Why logic? Logic is the scientic study of valid argument. Philosophy is all about arguments and reasoning. Logic allows us to test validity rigorously. Modern philosophy assumes familiarity with logic. Used in linguistics, mathematics, computer science,. . . Helps us make ne-grained conceptual distinctions. Logic is compulsory. Why logic? Logic is the scientic study of valid argument. Philosophy is all about arguments and reasoning. Logic allows us to test validity rigorously. Modern philosophy assumes familiarity with logic. Used in linguistics, mathematics, computer science,. . . Helps us make ne-grained conceptual distinctions. Logic is compulsory. Why logic? Logic is the scientic study of valid argument. Philosophy is all about arguments and reasoning. Logic allows us to test validity rigorously. Modern philosophy assumes familiarity with logic. Used in linguistics, mathematics, computer science,. . . Helps us make ne-grained conceptual distinctions. Logic is compulsory. Why logic? Logic is the scientic study of valid argument. Philosophy is all about arguments and reasoning. Logic allows us to test validity rigorously. Modern philosophy assumes familiarity with logic. Used in linguistics, mathematics, computer science,. . . Helps us make ne-grained conceptual distinctions. Logic is compulsory. Why logic? Logic is the scientic study of valid argument. Philosophy is all about arguments and reasoning. Logic allows us to test validity rigorously. Modern philosophy assumes familiarity with logic. Used in linguistics, mathematics, computer science,. . . Helps us make ne-grained conceptual distinctions. Logic is compulsory. 1.5 Arguments, Validity, and Contradiction Arguments Denition Sentences that are true or false are called declarative sentences. In what follows I will focus exclusively on declarative sentences. Denition An argument consists of a set of declarative sentences (the premisses) and a declarative sentence (the conclusion) somehow marked as the concluded sentence. 1.5 Arguments, Validity, and Contradiction Example I’m not dreaming if I can see the computer in front of me. I can see the computer in front of me. erefore I’m not dreaming. ‘I’m not dreaming if I can see the computer in front of me’ is a premiss. ‘I can see the computer in front of me’ is a premiss. ‘I’m not dreaming’ is the conclusion, which is marked by ‘therefore’. 40 1.5 Arguments, Validity, and Contradiction Example I’m not dreaming if I can see the computer in front of me. I can see the computer in front of me. erefore I’m not dreaming. ‘I’m not dreaming if I can see the computer in front of me’ is a premiss. ‘I can see the computer in front of me’ is a premiss. ‘I’m not dreaming’ is the conclusion, which is marked by ‘therefore’. 40 1.5 Arguments, Validity, and Contradiction Example I’m not dreaming if I can see the computer in front of me. I can see the computer in front of me. erefore I’m not dreaming. ‘I’m not dreaming if I can see the computer in front of me’ is a premiss. ‘I can see the computer in front of me’ is a premiss. ‘I’m not dreaming’ is the conclusion, which is marked by ‘therefore’. 40 1.5 Arguments, Validity, and Contradiction Occasionally the conclusion precedes the premisses or is found between premisses. e conclusion needn’t be marked as such by ‘therefore’ or a similar phrase. Alternative ways to express the argument: Example I’m not dreaming. For if I can see the computer in front of me I’m not dreaming, and I can see the computer in front of me. Example I’m not dreaming, if I can see the computer in front of me. us, I’m not dreaming. is is because I can see the computer in front of me. 1.5 Arguments, Validity, and Contradiction Occasionally the conclusion precedes the premisses or is found between premisses. e conclusion needn’t be marked as such by ‘therefore’ or a similar phrase. Alternative ways to express the argument: Example I’m not dreaming. For if I can see the computer in front of me I’m not dreaming, and I can see the computer in front of me. Example I’m not dreaming, if I can see the computer in front of me. us, I’m not dreaming. is is because I can see the computer in front of me. 1.5 Arguments, Validity, and Contradiction e point of ‘good’ arguments is that the truth of the premisses guarantees the truth of the conclusion. Many arguments with this property exhibit certain patterns. Example I’m not dreaming if I can see the computer in front of me. I can see the computer in front of me. erefore I’m not dreaming. Example Fiona can open the dvi-le if yap is installed. yap is installed. erefore Fiona can open the dvi-le. general form of both arguments A if B. B. erefore A. Logicians are interested in the patterns of ‘good’ arguments that cannot take one from true premisses to a false conclusion. 1.5 Arguments, Validity, and Contradiction Characterisation An argument is logically (or formally) valid if and only if there is no interpretation under which the premisses are all true and the conclusion is false. Example Zeno is a tortoise. All tortoises are toothless. erefore Zeno is toothless. Example Socrates is a man. All men are mortal. erefore Socrates is mortal. 1.5 Arguments, Validity, and Contradiction Characterisation An argument is logically (or formally) valid if and only if there is no interpretation under which the premisses are all true and the conclusion is false. Example Zeno is a tortoise. All tortoises are toothless. erefore Zeno is toothless. Example Socrates is a man. All men are mortal. erefore Socrates is mortal. 1.5 Arguments, Validity, and Contradiction Characterisation An argument is logically (or formally) valid if and only if there is no interpretation under which the premisses are all true and the conclusion is false. Example Zeno is a tortoise. All tortoises are toothless. erefore Zeno is toothless. Example Socrates is a man. All men are mortal. erefore Socrates is mortal. 1.5 Arguments, Validity, and Contradiction Features of logically valid arguments: e truth of the conclusion follows from the truth of the premisses independently what the subject-specic expressions mean. Whatever tortoises are, whoever Zeno is, whatever exists: if the premisses of the argument are true the conclusion will be true. e truth of the conclusion follows from the truth of the premisses purely in virtue of the ‘form’ of the argument and independently of any subject-specic assumptions. It’s not possible that the premisses of a logically valid argument are true and its conclusion is false. In a logically valid argument the conclusion can be false (in that case at least one of its premisses is false). Validity does not depend on the meanings of subject-specic expressions. 1.5 Arguments, Validity, and Contradiction Features of logically valid arguments: e truth of the conclusion follows from the truth of the premisses independently what the subject-specic expressions mean. Whatever tortoises are, whoever Zeno is, whatever exists: if the premisses of the argument are true the conclusion will be true. e truth of the conclusion follows from the truth of the premisses purely in virtue of the ‘form’ of the argument and independently of any subject-specic assumptions. It’s not possible that the premisses of a logically valid argument are true and its conclusion is false. In a logically valid argument the conclusion can be false (in that case at least one of its premisses is false). Validity does not depend on the meanings of subject-specic expressions. 1.5 Arguments, Validity, and Contradiction Features of logically valid arguments: e truth of the conclusion follows from the truth of the premisses independently what the subject-specic expressions mean. Whatever tortoises are, whoever Zeno is, whatever exists: if the premisses of the argument are true the conclusion will be true. e truth of the conclusion follows from the truth of the premisses purely in virtue of the ‘form’ of the argument and independently of any subject-specic assumptions. It’s not possible that the premisses of a logically valid argument are true and its conclusion is false. In a logically valid argument the conclusion can be false (in that case at least one of its premisses is false). Validity does not depend on the meanings of subject-specic expressions. 1.5 Arguments, Validity, and Contradiction Characterisation (consistency) A set of sentences is consistent if and only if there is a least one interpretation under which all sentences of the set are true. Characterisation (logical truth) A sentence is logically true if and only if it is true under any interpretation. ‘All metaphysicians are metaphysicians.’ 1.5 Arguments, Validity, and Contradiction Characterisation (consistency) A set of sentences is consistent if and only if there is a least one interpretation under which all sentences of the set are true. Characterisation (logical truth) A sentence is logically true if and only if it is true under any interpretation. ‘All metaphysicians are metaphysicians.’ 1.5 Arguments, Validity, and Contradiction Characterisation (contradiction) A sentence is a contradiction if and only if it is false under any interpretation. ‘Some metaphysicians who are also ethicists aren’t metaphysicians.’ I’ll make these notions precise for the formal languages or propositional and predicate logic. 1.1 Sets Sets e following is not really logic in the strict sense but we’ll need it later and it is useful in other areas as well. Characterisation A set is a collection of objects. e objects in the set are the elements of the set. ere is a set that has exactly all books as elements. ere is a set that has Volker Halbach as its only element. 25 1.1 Sets Sets e following is not really logic in the strict sense but we’ll need it later and it is useful in other areas as well. Characterisation A set is a collection of objects. e objects in the set are the elements of the set. ere is a set that has exactly all books as elements. ere is a set that has Volker Halbach as its only element. 25 1.1 Sets Sets e following is not really logic in the strict sense but we’ll need it later and it is useful in other areas as well. Characterisation A set is a collection of objects. e objects in the set are the elements of the set. ere is a set that has exactly all books as elements. ere is a set that has Volker Halbach as its only element. 25 1.1 Sets e claim ‘a is an element of S’ can be written as ‘a ∈ S’. One also says ‘S contains a’ or ‘a is in S’. ere is exactly one set with no elements. e symbol for this set is ‘∅’. e set {Oxford, ∅, Volker Halbach} has as its elements exactly three things: Oxford, the empty set ∅, and me. Here is another way to denote sets: { d ∶ d is an animal with a heart} is the set of all animals with a heart. It has as its elements exactly all animals with a heart. 1.1 Sets e claim ‘a is an element of S’ can be written as ‘a ∈ S’. One also says ‘S contains a’ or ‘a is in S’. ere is exactly one set with no elements. e symbol for this set is ‘∅’. e set {Oxford, ∅, Volker Halbach} has as its elements exactly three things: Oxford, the empty set ∅, and me. Here is another way to denote sets: { d ∶ d is an animal with a heart} is the set of all animals with a heart. It has as its elements exactly all animals with a heart. 1.1 Sets e claim ‘a is an element of S’ can be written as ‘a ∈ S’. One also says ‘S contains a’ or ‘a is in S’. ere is exactly one set with no elements. e symbol for this set is ‘∅’. e set {Oxford, ∅, Volker Halbach} has as its elements exactly three things: Oxford, the empty set ∅, and me. Here is another way to denote sets: { d ∶ d is an animal with a heart} is the set of all animals with a heart. It has as its elements exactly all animals with a heart. 1.1 Sets Example {Oxford, ∅, Volker Halbach} = {Volker Halbach, Oxford, ∅ } Example {the capital of England, Munich} = {London, Munich, the capital of England} Example Mars ∈ {d ∶ d is a planet } Example ∅ ∈ {∅} 15 1.1 Sets Example {Oxford, ∅, Volker Halbach} = {Volker Halbach, Oxford, ∅ } Example {the capital of England, Munich} = {London, Munich, the capital of England} Example Mars ∈ {d ∶ d is a planet } Example ∅ ∈ {∅} 15 1.1 Sets Example {Oxford, ∅, Volker Halbach} = {Volker Halbach, Oxford, ∅ } Example {the capital of England, Munich} = {London, Munich, the capital of England} Example Mars ∈ {d ∶ d is a planet } Example ∅ ∈ {∅} 15 1.2 Binary relations Relations e set {London, Munich} is the same set as {Munich, London}. e ordered pair ⟨London, Munich⟩ is dišerent from the ordered pair ⟨Munich, London⟩. Ordered pairs are identical if and only if the agree in their rst and second components, or more formally: ⟨d , e⟩ = ⟨ f , g⟩ iš (d = f and e = g) e abbreviation ‘iš’ stands for ‘if and only if’. ere are also triples (3-tuples) like ⟨London, Munich, Rome⟩, quadruples, 5-tuples, 6-tuples etc. 1.2 Binary relations Relations e set {London, Munich} is the same set as {Munich, London}. e ordered pair ⟨London, Munich⟩ is dišerent from the ordered pair ⟨Munich, London⟩. Ordered pairs are identical if and only if the agree in their rst and second components, or more formally: ⟨d , e⟩ = ⟨ f , g⟩ iš (d = f and e = g) e abbreviation ‘iš’ stands for ‘if and only if’. ere are also triples (3-tuples) like ⟨London, Munich, Rome⟩, quadruples, 5-tuples, 6-tuples etc. 1.2 Binary relations Denition A set is a binary relation if and only if it contains only ordered pairs. e empty set ∅ doesn’t contain anything that’s not an ordered pair; therefore it’s a relation. Example e relation of being a bigger city than is the set {⟨London, Munich⟩, ⟨London, Birmingham⟩, ⟨Paris, Milan⟩. . .}. 1.2 Binary relations e following set is a binary relation: {⟨France, Italy⟩, ⟨Italy, Austria⟩, ⟨France, France⟩, ⟨Italy, Italy⟩, ⟨Austria, Austria⟩} Some relations can be visualised by diagrams. Every pair corresponds to an arrow: France $$  Austria  Italy :: UU 1.2 Binary relations e following set is a binary relation: {⟨France, Italy⟩, ⟨Italy, Austria⟩, ⟨France, France⟩, ⟨Italy, Italy⟩, ⟨Austria, Austria⟩} Some relations can be visualised by diagrams. Every pair corresponds to an arrow: France $$  Austria  Italy :: UU 1.2 Binary relations I’ll mention only some properties of relations. Denition A binary relation R is symmetric iš for all d , e: if ⟨d , e⟩ ∈ R then ⟨e , d⟩ ∈ R. e relation with the following diagram isn’t symmetric: France   Austria zz Italy UU \\ e pair ⟨Austria, Italy⟩ is in the relation, but the pair ⟨Italy, Austria⟩ isn’t. 1.2 Binary relations e relation with the following diagram is symmetric. France   Austria vv Italy UU \\ 66 1.2 Binary relations Denition A binary relation is transitive iš for all d , e , f : if ⟨d , e⟩ ∈ R and ⟨e , f ⟩ ∈ R, then also ⟨d , f ⟩ ∈ R In the diagram of a transitive relation there is for any two-arrow way from an point to a point a direct arrow. is is the diagram of a relation that’s not transitive: France // Austria // Italy is is the diagram of a relation that is transitive: France // --Austria // Italy 1.2 Binary relations Denition A binary relation is transitive iš for all d , e , f : if ⟨d , e⟩ ∈ R and ⟨e , f ⟩ ∈ R, then also ⟨d , f ⟩ ∈ R In the diagram of a transitive relation there is for any two-arrow way from an point to a point a direct arrow. is is the diagram of a relation that’s not transitive: France // Austria // Italy is is the diagram of a relation that is transitive: France // --Austria // Italy 1.2 Binary relations e relation with the following diagram is not režexive on {France, Austria, Italy}, but režexive on {France, Austria}: France $$  Austria  Italy :: 1.3 Functions Functions Denition A binary relation R is a function iš for all d , e , f : if ⟨d , e⟩ ∈ R and ⟨d , f ⟩ ∈ R then e = f . e relation with the following diagram is a function: France $$ Austria zz Italy UU ere is at most one arrow leaving from every point in the diagram of a function. 1.3 Functions Functions Denition A binary relation R is a function iš for all d , e , f : if ⟨d , e⟩ ∈ R and ⟨d , f ⟩ ∈ R then e = f . e relation with the following diagram is a function: France $$ Austria zz Italy UU ere is at most one arrow leaving from every point in the diagram of a function.