Review The Conditional Logical symbols Argument forms. Logic 5: Material Implication and Argument Forms Jan. 28, 2014

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1 Logic 5: Material Implication and Argument Forms Jan. 28, 2014

2 Overview I Review The Conditional Conditional statements Material implication Logical symbols Argument forms Disjunctive syllogism Disjunctive syllogism Disjunctive syllogism Disjunctive syllogism Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma

3 Review 3 basic operators we considered: p q p q T T T 1 Conjunction: T F F F T F F F F 2 Negation: 3 Disjunction: p T F p F T p q p q T T T T F T F T T F F F

4 Conditional statements Review The Conditional Logical symbols Argument forms Material implication Conditional Statements Most arguments contain a sentence of If A, then B form this is called a hypothetical, or an implication. A: antecedent B: consequent If Mr. Jones is my neighbor, then Mr. Jones has three children. antecedent: Mr. Jones is my neighbor consequent: Mr. Jones has three children. A few things to notice: A conditional statement tells us that in any case in which the antecedent is true, the consequent is also true. It does NOT tell us that the antecedent is actually true.

5 Conditional statements Review The Conditional Logical symbols Argument forms Material implication Material Implication I There are various kinds of conditional statements. 1 If all humans are mortal, and Socrates is a human, then Socrates is mortal. Logical implication 2 If John is a bachelor, then John is unmarried. Follows from the definition 3 If a piece of wood is placed in fire, then it will burn. Causal implication 4 If we lose this game, then I ll eat my hat. A decision Is there any common meaning in these? One way to find a common meaning is to check when the above sentences are false.

6 Conditional statements Review The Conditional Logical symbols Argument forms Material implication Material Implication II All the above sentences are false when their antecedents are true and their consequent are false. Material Implication The logical symbol for the conditional is (called the horseshoe ). A conditional p q is false if and only if p is true and q is false (it is true otherwise). p q p q T T T T F F F T T F F T

7 Conditional statements Review The Conditional Logical symbols Argument forms Material implication Material Implication III Things to keep in mind: Material implication includes conditionals where there is no real connection between the antecedent and the consequent. If Hitler was a military genius, then I m a monkey s uncle. This only wants to suggest that the antecedent is false. Material implication is true when the antecedent is false this is often counter-intuitive. If the world is flat, then the moon is made of green cheese. It can be shown that p q is true just when q p is true Contrapositive

8 Logical Symbols Summary We have considered four basic logical concepts. 1 Conjunction: a b is true if and only if a is true and b is true. 2 Negation: a is true if and only if a is false. 3 Disjunction: a b is false if and only if a is false and b is false. 4 Conditional: a b is false if and only if a is true and b is false. There are various relations between these; e.g., it can be shown very easily that a b is equivalent to a b.

9 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Argument Forms I Reminder: an argument is valid if whenever the the premises are true, the conclusion is also true. Thus, whenever you accept the premises of a valid argument, you must also accept the conclusion! We can test argument-validity by the truth-table method. Of course, different arguments can have the same form: The blind prisoner has a red hat or a white hat. The blind prisoner does not have a red hat. Therefore, the blind prisoner has a white hat. Either Shakespeare wrote the Hamlet or he was not a great writer. Shakespeare was a great writer. Therefore, Shakespeare wrote Hamlet.

10 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Argument Forms II Thus, it seems that if the first argument is valid, then the second one also has to be valid (they have the same form). Argument Form An argument form is a string of symbols, containing letters (variables) for statements. When these letters are filled in with sentences, we get an argument. If an argument form is valid, then every argument that falls under that form is also valid. The argument form exemplified by the arguments above: p q It is called disjunctive q syllogism. (We justified its p validity last time.)

11 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Argument Forms III The most important argument forms: 1 Disjunctive syllogism 2 Modus ponens 3 Modus tollens 4 Hypothetical syllogism 5 Constructive dilemma

12 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Modus Ponens If Shakespeare wrote Hamlet, then Shakespeare was a good writer. Shakespeare wrote Hamlet. Therefore, Shakespeare was a good writer. p = Shakespeare wrote Hamlet. q = Shakespeare was a good writer. 1 Formalize: p q p q 2 Truth-table: p q p q p q T T T T T T F F T F F T T F T F F T F F Thus, the argument form Modus Ponens is valid.

13 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Modus Tollens If Mozart was not a composer, then he did not write the Requiem. Mozart did write the Requiem. Therefore, Mozart was a composer. p = Mozart was not a composer. q = Mozart did not write the Requiem. 1 Formalize: p q q p 2 Truth-table: p q p q q p T T T F F T F F T F F T T F T F F T T T Thus, the argument form Modus Tollens is valid.

14 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Hypothetical Syllogism I If Walter Scott was British, he was not Italian. If Walter Scott was not Italian, he did not live in Italy. Therefore, if Walter Scott was British, he did not live in Italy. p = Walter Scott was British. q = Walter Scott was not Italian. r = Walter Scott did not live in Italy.

15 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Hypothetical Syllogism II 1 Formalize: p q q r p r 2 Truth-table: p q r p q q r p r T T T T T T T T F T F F T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T F F F T T T Thus, the argument form Hypothetical Syllogism is valid.

16 Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma Constructive Dilemma If Descartes was a philosopher, he wrote the Meditations, but if Beethoven was a composer, he wrote a symphony. Descartes was either a philosopher or Beethoven was a composer. Therefore, either Descartes wrote the Meditations or Beethoven wrote a symphony. p = Descartes was a philosopher. q = Descartes wrote the Meditations. r = Beethoven was a composer. s = Beethoven wrote a symphony. 1 Formalization: (p q) (r s) p r q s