Logic, reasoning and fallacies. Example 0: valid reasoning. Decide how to make a random choice. Valid reasoning. Random choice of X, Y, Z, n

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1 Logic, reasoning and fallacies and some puzzling Before we start Introductory Examples Karst Koymans Informatics Institute University of Amsterdam (version 16.3, 2016/11/21 12:58:26) Wednesday, November 23, 2016 Arguments and Reasoning Paradoxes and Fallacies Decide how to make a random choice Example 0: valid reasoning Valid reasoning Random choice of X, Y, Z, n X, Y, Z {A, E, I, O}, not necessarily all different n {1, 2, 3, 4} Niels is younger than Jaap. (premiss) Jaap is younger than Karst. (premiss) Hence Niels is younger than Karst. (conclusion) The validity of this reasoning depends on Logic, but also on the property of is younger than being transitive.

2 Example 1: invalid reasoning Example 2: iff Invalid reasoning A bad lecture is better than nothing. (premiss) Nothing is better than a good lecture. (premiss) Thus a bad lecture is better than a good lecture. (conclusion) Although is better than is also supposed to be transitive, this is an invalid argument. Why? Language and Logic are more than just form (syntax). Different contexts may give different meaning (semantics). Meaning of iff A iff B abbreviates A if and only if B A if B means If B then A (Notation: B A) A only if B means If A then B (Notation: A B) A iff B means A if B and A only if B (Notation: B A A B B A) Example 3: desda (Dutch iff) Example 4: Retrograde Analysis Meaning of desda A desda B abbreviates A dan en slechts dan als B A (dan) als B means Als B dan A (Notation: B A) (Note: It is A dan als B and not A dan B, which would mean Als A dan B ) A slechts (dan) als B means Als A dan B (Notation: A B) A desda B means A dan als B en A slechts dan als B (Notation: B A A B B A) What about the meaning of necessary and sufficient (in Dutch: noodzakelijk en voldoende )?

3 Example 5: Dead Reckoning puzzle Example 5: Dead Reckoning solution Chess Law 5.2b The game is drawn when a position has arisen in which neither player can checkmate the opponent s king with any series of legal moves. The game is said to end in a dead position. This immediately ends the game, provided that the move producing the position was legal. Example 6: Slitherlink puzzle Example 6: Slitherlink solution What are the rules of this game?

4 Example 7: Nurikabe puzzle Example 7: Nurikabe solution What are the rules of this game? Example 8: Masyu puzzle Example 8: Masyu solution What are the rules of this game?

5 Example 9: Link-A-Pix tutorial Example 9: Link-A-Pix example What are the rules of this game? d31337c45c1ce9e3d8dd757af93cc72b3182e00e89d2330ff4f2973b4381c154889f23e5263f107e430bc15d92f5ce4c/ play Definition of Argument Example of an argument Definition (Proposition) A proposition is a sentence with a truth value, being either true or false. Definition (Simple Argument) A simple argument is a series of propositions. One proposition (the conclusion) should be evidenced by the other propositions (premisses). Definition (Complex Argument) A complex argument is a series of arguments. The conclusion of one argument may be used as premiss in another argument. Women for OS3 [We do not have many female students at OS3] 1. Hence [we should intensify our PR] 2, because [it is good for society that women graduate at our master] 3. 1 is a premiss 2 is the conclusion 3 is a premiss

6 Deductive and Inductive Arguments Reasoning Example 0 Definition (Deductive argument) A deductive argument is an argument where the conclusion follows 100% certain from the premisses. Definition (Inductive argument) An inductive argument is an argument where the conclusion follows with (preferably a high) probability from the premisses. Can you give examples of both kind of arguments? Valid reasoning with true propositions All Dutch people are humans All humans are mortal All Dutch people are mortal Reasoning Example 1 Reasoning Example 2 Valid reasoning with false propositions All Dutch people are smart All smart people live forever Invalid reasoning with true propositions If I am God, I will live forever I am not God All Dutch people live forever I will not live forever This is an example of a fallacy: Denying the Antecedent

7 Reasoning Example 3 Valid arguments Valid reasoning with false premisses and a true conclusion All part-time OS3 students are smart All smart people are male Theorem (Soundness) A valid argument with true premisses always leads to a true conclusion All part-time OS3 students are male Categorical propositions Syllogisms Definition (Aristoteles) Given a subject term S and a predicate P we have the four classic categorical propositions: (A) All S are P (E) No S is P ( All S are not P) (I) Some S is P (O) Some S is not P AffIrmo (universal/particular positive) nego (universal/particular negative) Definition (Categorical Syllogism) A categorical syllogism is a deductive argument with two categorical propositions as premisses and a categorical conclusion. Example of form AAA All M are P All S are M All S are P M is called the middle term

8 Position and role of the middle term Validity of Categorical Syllogisms First role M P S M S P Third role M P M S S P Second role P M S M S P Fourth role P M M S S P 256 Possible schemes The possible schemes are XYZ-n where X, Y, Z {A, E, I, O} n {1, 2, 3, 4} Of these schemes only 15 are valid. Venn diagrams Famous paradoxes Liar paradox This sentence is false Drinker paradox There is a person such that if that person drinks then everybody drinks Grelling-Nelson paradox Autological describes itself correctly Heterological does not describe itself correctly Is heterological heterological? Berry paradox The smallest positive integer not definable in under eleven words Does such a number exist or not?

9 Mathematical paradox Fallacies Russell paradox V = {x x / x} Does V V hold? Zwicker s Hypergame A finite game is a game that always ends after a finite number of alternating moves from both players Hypergame is played by first choosing a finite game by the first player (A) and than playing that game starting with the second player (B) Is Hypergame a finite game or not? Fallacies of Relevance Fallacies of Presumption Logical/Formal Fallacies Fallacies of Relevance Argument from Ignorance Argument from Ignorance Argument Ad Hominem Appeal to Inappropriate Authority Appeal to Authority or Force Appeal to Emotion Drawing an Irrelevant Conclusion Argumentum ad Ignorantiam The fallacy that a proposition is true simply on the basis that it has not been proved false or that it is false simply because it has not been proved true. Example Argument from Ignorance Scientists cannot prove that a meteor will cause a catastrophe. Therefore it won t occur.

10 Argument Ad Hominem Appeal to Inappropriate Authority Argument Ad Hominem The fallacy of attacking the character or circumstances of an individual who is advancing a statement or an argument instead of trying to disprove the truth of the statement or the soundness of the argument. Example Argument Ad Hominem This cannot be true, since you belong to that radical party. Argumentum ad Verecundiam The fallacy of appealing to the testimony of an authority outside his special field. Example Appeal to Inappropriate Authority We from WC Eend advise WC Eend. Appeal to Authority or Force Appeal to Emotion Argumentum ad Baculum The fallacy committed when one appeals to (the threat of) force to bring about the acceptance of a conclusion. Example Appeal to Authority or Force You will lower your price, because otherwise I might reconsider your bonus. Argumentum ad Misericordiam The fallacy committed when pity or a related emotion such as sympathy or compassion is appealed to for the sake of getting a conclusion accepted. Example Appeal to Emotion Everybody else thinks cloud computing is a good idea. You should too.

11 Drawing an Irrelevant Conclusion Fallacies of Presumption Ignoratio Elenchi The fallacy of proving a conclusion not pertinent and quite different from that which was intended or required. Example Drawing an Irrelevant Conclusion A prisoner pleaded guilty as a thief. The judge allowed him to change this plea to not guilty. The jury on the trial accused the prisoner of lying. Judge reasons: You said you were a thief; the jury calls you a liar; hence you are discharged. Fallacy of False Cause Begging the Question Fallacy of False Cause Begging the Question Fallacy of False Cause The fallacy committed when an argument mistakenly attempts to establish a causal connection. Example Fallacy of False Cause The introduction of the death penalty has caused the highest criminal rate and number of prisoners of the civilized world. Petitio Principii The fallacy of assuming as a premiss a statement which has the same meaning as the conclusion. Example Begging the Question The Bible says God exists. The Bible is God s word, and God never lies. So what the Bible says is true. Therefore God exists.

12 Logical Fallacies Affirming the Consequent Affirming the Consequent Denying the Antecedent Affirming the Consequent The invalid reasoning: P Q and Q, therefore P. Example Affirming the Consequent If it s raining then the streets are wet. The streets are wet. Therefore, it s raining. Denying the Antecedent Denying the Antecedent The invalid reasoning: P Q and P, therefore Q. Example Denying the Antecedent If it s raining, then the streets are wet. It isn t raining. Therefore, the streets aren t wet.