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Introduction to Formal Logic Richard.Pettigrew@bris.ac.uk Lecture 2 The form of arguments Recap Recap Logic is the study of arguments. Recap Logic is the study of arguments. An argument is a set of premises along with the conclusion. Recap Logic is the study of arguments. An argument is a set of premises along with the conclusion. The premises and conclusions of an argument are propositions. Recap Logic is the study of arguments. An argument is a set of premises along with the conclusion. The premises and conclusions of an argument are propositions. Propositions are the sort of thing that may be true or false. Many sentences may express the same proposition. Two sentences express the same proposition if they have the same meaning. Recap on propositions Recap on propositions Le chien est brun Der Hund ist braun The dog is brown Recap on propositions Le chien est brun Der Hund ist braun The dog is brown Recap on propositions Le chien est brun Der Hund ist braun The dog is brown Three different sentences; all express the same single proposition. What properties of arguments will concern us? What properties of arguments will concern us? An argument is valid if the truth of its premises guarantees the truth of its conclusion: that is, if there is no possible circumstance in which the premises are true and the conclusion is false. What properties of arguments will concern us? An argument is valid if the truth of its premises guarantees the truth of its conclusion: that is, if there is no possible circumstance in which the premises are true and the conclusion is false. An argument is invalid if the truth of its premises does not guarantee the truth of its conclusion: that is, if there is a possible circumstance in which the premises are true and the conclusion is false. Examples of valid arguments… If Cameron wins the next election, he will abandon his environmental zeal. He will win the next election. Therefore, he will abandon his environmental zeal. Examples of valid arguments… If Cameron wins the next election, he will abandon his environmental zeal. He will win the next election. Therefore, he will abandon his environmental zeal. If grass is red, then John Sergeant is from East Anglia. The grass is red. Therefore, John Sergeant is from East Anglia. The moral of the story… An argument may be valid even if all its premises and its conclusion are false. Whether an argument is valid or not concerns only the relationship between the premises and the conclusion. It does not concern the truth of the premises, nor the truth of the conclusion. Examples of invalid arguments… If it is about to rain, the cows go under the trees. The cows have gone under the trees. Therefore, it is about to rain. Examples of invalid arguments… If it is about to rain, the cows go under the trees. The cows have gone under the trees. Therefore, it is about to rain. The sun has risen on the Earth every morning since at least 5000BC. Therefore, it will rise tomorrow morning. The moral of the story… An argument may be invalid even if its premises make its conclusion very likely. How can we show that an argument is valid? How can we show that an argument is valid? Strategy: How can we show that an argument is valid? Strategy: 1) Identify a general form of which the argument in question is a particular instance. How can we show that an argument is valid? Strategy: 1) Identify a general form of which the argument in question is a particular instance. 2) Show that all arguments of this form are valid. (We do this by one of two techniques: truth- tables or tree proofs.) How can we show that an argument is valid? Strategy: 1) Identify a general form of which the argument in question is a particular instance. 2) Show that all arguments of this form are valid. (We do this by one of two techniques: truth- tables or tree proofs.) 3) Conclude that, in particular, the argument in question is valid. What is the form of an argument? Melodic form in music: What is the form of an argument? Melodic form in music: A B A Rhyme form in poetry: Rhyme form in poetry: Hey, Mr. Tambourine Man, play a song for me, I'm not sleepy and there ain't no place I'm going to. Hey, Mr. Tambourine Man, play a song for me, In the jingle jangle morning I'll come following you. Mr Tambourine Man, Bob Dylan Rhyme form in poetry: Hey, Mr. Tambourine Man, play a song for me, A I'm not sleepy and there ain't no place I'm going to. B Hey, Mr. Tambourine Man, play a song for me, A In the jingle jangle morning I'll come following you. B Mr Tambourine Man, Bob Dylan Rhyme form in poetry: Hey, Mr. Tambourine Man, play a song for me, A I'm not sleepy and there ain't no place I'm going to. B Hey, Mr. Tambourine Man, play a song for me, A In the jingle jangle morning I'll come following you. B Mr Tambourine Man, Bob Dylan Two households, both alike in dignity, In fair Verona, where we lay our scene, From ancient grudge break to new mutiny, Where civil blood makes civil hands unclean. Romeo and Juliet, Shakespeare Rhyme form in poetry: Hey, Mr. Tambourine Man, play a song for me, A I'm not sleepy and there ain't no place I'm going to. B Hey, Mr. Tambourine Man, play a song for me, A In the jingle jangle morning I'll come following you. B Mr Tambourine Man, Bob Dylan Two households, both alike in dignity, A In fair Verona, where we lay our scene, B From ancient grudge break to new mutiny, A Where civil blood makes civil hands unclean. B Romeo and Juliet, Shakespeare Rhyme form in poetry: 12 Rhyme form in poetry: Had we but world enough, and time, This coyness, Lady, were no crime We would sit down and think which way To walk and pass our long love's day. To His Coy Mistress, Marvell 12 Rhyme form in poetry: Had we but world enough, and time, A This coyness, Lady, were no crime A We would sit down and think which way B To walk and pass our long love's day. B To His Coy Mistress, Marvell 12 Rhyme form in poetry: Had we but world enough, and time, A This coyness, Lady, were no crime A We would sit down and think which way B To walk and pass our long love's day. B To His Coy Mistress, Marvell So when the last and dreadful hour This crumbling pageant shall devour, The trumpet shall be heard on high, The dead shall live, the living die, And Music shall untune the sky! A Song for St Cecilia’s Day, Dryden 12 Rhyme form in poetry: Had we but world enough, and time, A This coyness, Lady, were no crime A We would sit down and think which way B To walk and pass our long love's day. B To His Coy Mistress, Marvell So when the last and dreadful hour A This crumbling pageant shall devour, A The trumpet shall be heard on high, B The dead shall live, the living die, B And Music shall untune the sky! B A Song for St Cecilia’s Day, Dryden 12 Argument form in logic: If Cameron wins the election, then he will abandon his environmental zeal. He will win the election. Therefore, he will abandon his environmental zeal. Argument form in logic: If Cameron wins the election, then he will abandon his environmental zeal. He will win the election. Therefore, he will abandon his environmental zeal. Argument form in logic: If Cameron wins the election, then he will abandon his environmental zeal. He will win the election. Therefore, he will abandon his environmental zeal. Argument form in logic: If Cameron wins the election, then he will abandon his environmental zeal. He will win the election. Therefore, he will abandon his environmental zeal. p = Cameron wins the election. q = Cameron will abandon his environmental zeal. Argument form in logic: If Cameron wins the election, then he will abandon his environmental zeal. He will win the election. Therefore, he will abandon his environmental zeal. p = Cameron wins the election. q = Cameron will abandon his environmental zeal. If p, then q. p. Therefore, q. Argument form in logic: If porridge oats are good for you, then he should eat them. Porridge oats are good for you. Therefore, he should eat them. Argument form in logic: If porridge oats are good for you, then he should eat them. Porridge oats are good for you. Therefore, he should eat them. Argument form in logic: If porridge oats are good for you, then he should eat them. Porridge oats are good for you. Therefore, he should eat them. Argument form in logic: If porridge oats are good for you, then he should eat them. Porridge oats are good for you. Therefore, he should eat them. p = Porridge oats are good for you. q = He should eat them. Argument form in logic: If porridge oats are good for you, then he should eat them. Porridge oats are good for you. Therefore, he should eat them. p = Porridge oats are good for you. q = He should eat them. If p, then q. p. Therefore, q. Argument form in logic: Spiders have eight legs. So, it is not true that spiders are insects, for if they were insects, then it would not be true that they have eight legs. Argument form in logic: Spiders have eight legs. So, it is not true that spiders are insects, for if they were insects, then it would not be true that they have eight legs. Argument form in logic: Spiders have eight legs. So, it is not true that spiders are insects, for if they were insects, then it would not be true that they have eight legs. Argument form in logic: Spiders have eight legs. So, it is not true that spiders are insects, for if they were insects, then it would not be true that they have eight legs. p = Spiders have eight legs. q = Spiders are insects. Argument form in logic: Spiders have eight legs. So, it is not true that spiders are insects, for if they were insects, then it would not be true that they have eight legs. p = Spiders have eight legs. q = Spiders are insects. If q, then p is not true. p. Therefore, q is not true. Argument form in logic: If the grass is green, then it is not true that John Sergeant lives in East Anglia. John Sergeant does live in East Anglia. Therefore, it is not true that the grass is green. Argument form in logic: If the grass is green, then it is not true that John Sergeant lives in East Anglia. John Sergeant does live in East Anglia. Therefore, it is not true that the grass is green. Argument form in logic: If the grass is green, then it is not true that John Sergeant lives in East Anglia. John Sergeant does live in East Anglia. Therefore, it is not true that the grass is green. Argument form in logic: If the grass is green, then it is not true that John Sergeant lives in East Anglia. John Sergeant does live in East Anglia. Therefore, it is not true that the grass is green. p = The grass is green. q = John Sergeant lives in East Anglia. Argument form in logic: If the grass is green, then it is not true that John Sergeant lives in East Anglia. John Sergeant does live in East Anglia. Therefore, it is not true that the grass is green. p = The grass is green. q = John Sergeant lives in East Anglia. If p, then q is not true. q. Therefore, p is not true. Argument form in logic: Mrs. White must have done it, because either she did it or Revd. Green did, and if he’d done it, it would have been in the Kitchen, and it was not. Argument form in logic: Mrs. White must have done it, because either she did it or Revd. Green did, and if he’d done it, it would have been in the Kitchen, and it was not. Argument form in logic: Mrs. White must have done it, because either she did it or Revd. Green did, and if he’d done it, it would have been in the Kitchen, and it was not. Argument form in logic: Mrs. White must have done it, because either she did it or Revd. Green did, and if he’d done it, it would have been in the Kitchen, and it was not. Argument form in logic: Mrs. White must have done it, because either she did it or Revd. Green did, and if he’d done it, it would have been in the Kitchen, and it was not. p = Mrs. White committed the murder. q = Revd. Green committed the murder. r = The murder was committed in the Kitchen. Argument form in logic: Mrs. White must have done it, because either she did it or Revd. Green did, and if he’d done it, it would have been in the Kitchen, and it was not. p = Mrs. White committed the murder. q = Revd. Green committed the murder. r = The murder was committed in the Kitchen. p or q. If q, then r. r is not true. Therefore, p. Argument form in logic: Matt or Tarisai won. Matt didn’t win. Therefore, Tarisai won. Argument form in logic: Matt or Tarisai won. Matt didn’t win. Therefore, Tarisai won. Argument form in logic: Matt or Tarisai won. Matt didn’t win. Therefore, Tarisai won. Argument form in logic: Matt or Tarisai won. Matt didn’t win. Therefore, Tarisai won. p = Matt won. q = Tarisai won. Argument form in logic: Matt or Tarisai won. Matt didn’t win. Therefore, Tarisai won. p = Matt won. q = Tarisai won. p or q. p is not true. Therefore, q. Summary Summary Logic is the study of arguments. Summary Logic is the study of arguments. An argument is a set of propositions, called premises, along with a single proposition, called the conclusion. Summary Logic is the study of arguments. An argument is a set of propositions, called premises, along with a single proposition, called the conclusion. The logical form of an argument is obtained by letting letters stand for the various propositions that occur in the argument: we use the same letter when a proposition occurs a further time in an argument.