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Deductive Reasoning Deductive Reasoning and Inductive Deductive Reasoning and Inductive Reasoning “There has been a murder done, and the murderer was a man. He was more than 6 feet high, was in the Deductive Reasoning prime of life, had small feet for his height, wore coarse, square-toed boots and smoked a Trichinopoly cigar.” –Doyle “Orr would be crazy to fly more missions and sane if he didn’t, but if he was sane he had to fly them. If he flew then he was crazy and didn’t have to; but if he didn’t want to he was sane and had to.” – Heller Deductive vs. Inductive Reasoning Deductive Reasoning Inductive Reasoning • Concerned with beliefs • Concerned with beliefs licensing or being supporting or being logically required by other supported by other beliefs beliefs • Considers all possible • Considers most relevant states of affairs states of affairs • Leads to conclusions that • Leads to conclusions that are necessary are probable • Infallible Conclusions • Fallible Conclusions (when premises true) (even when premises true) Logic & Reasoning Formalization • Arguments often evaluated not only in terms of whether • Some reasoning problems occur due to lack of clarity in how to map human they are valid, but also whether they are empirically true understanding onto abstract symbols • David Lewis • Truth versus Validity – If J. Edgar Hoover had been born in Russia, then he would have been a Premise 1: All doctors are professional people. communist. – If J. Edgar Hoover had been a communist, then he would have been a traitor. Premise 2: Some professional people are rich. – Therefore, if J. Edgar Hoover had been born in Russia, then he would have been Conclusion: Some doctors are rich. a traitor. • Content Effects • Formalization – (Premise 1) If A, then B. A Æ B – Finding that people judge the same logical argument differently – (Premise 2) If B, then C. B Æ C depending on what the topic is – (Conclusion) If A, then C. A Æ C – But, laws of logic tell us which beliefs follow from other beliefs •But based on their form, not their content! – Presumes that in Premise 2, J. Edgar Hoover had been living in the US (L), and • Content effects may stem from fact that reasoning was head of the FBI (F), …and that he was a traitor to the US (C), not Russia. –A Æ B typically embedded in a context where truth and validity – B & (L & F) Æ C are important 1 Relevance Conditional Reasoning • Relevance Modus Ponens Modus Tollens Premise 1: If it is raining, the picnic will not be (1) P Æ Q (1) P Æ Q held. (2) P (2) ~Q Premise 2: It is raining. (3) Therefore: Q (3) Therefore: ~P Conclusion: Either the picnic will not be held or cats have 6 legs. • Researchers rarely study which P: John gets B or better on final exam conclusions people find intuitively natural Q: John passes the course Can people do conditional Invalid Inferences reasoning? Denying the Affirming the Antecedant Consequent (1) P Æ Q (1) P Æ Q Modus Ponens (2) ~P (2) Q (3) Therefore: ~Q (3) Therefore: P Denying the Antecedant Affirming the P: The object is square Consequent Q: The object is blue. Modus Tollens Conditional vs. Bi-conditional Conditional vs. Bi-conditional P Q PÆQ PÅÆQ Affirming the TTT T P Q PÆQ PÅÆQ Consequent TFF F T TT T Æ FTT F TFF F(1) P Q FFT T F TT F(2) Q FFT T (3) Therefore: P If you pick up your toys, I’ll read you a story. If our quarterback is injured, then our team will lose. 2 Conditional vs. Bi-conditional Conditional vs. Bi-conditional ‘Affirming the Denying the P Q PÆQ PÅÆQ P Q PÆQ PÅÆQ Consequent’ Antecedant T TT T TTT T TFF F(1) P ÅÆ Q TFF F(1) P Æ Q FTT F(2) Q F T TF(2) ~P FFT T F F TT (3) Therefore: P (3) Therefore: ~Q On the biconditional reading of “if”, ‘Affirming the Consequent’ is a valid inference schema! Modus Ponens vs Modus Conditional vs. Bi-conditional Tollens ‘Denying the • Modus Ponens is P Q PÆQ PÅÆQ easy Antecedant’ TTT T • Modus Tollens is hard TFF F(1) P ÅÆ Q Use of causal schemas FTT F(2) ~P F F TT (3) Therefore: ~Q On the biconditional reading of “if” ‘Denying the Antecedant’ is a valid inference schema. “If” Interpretation “If” Interpretation • Depends on causal schemas associated with • Depends on existence of alternative content of argument explanations for Q • Example that biases ÅÆ (1) If the horses had been to the waterhole, then the (1) If the horses had been to the waterhole, we would food we left out would be gone. see their tracks. (2) The food we left out is not gone. (2) We see no tracks. (3) Therefore: The horses have not been to the (3) Therefore: The horses have not been to the waterhole. waterhole. (2a) We see their tracks. (2a) The food we left out is gone. (3a) Therefore: The horses have been to the waterhole. (3a) Therefore: The horses have been to the waterhole. (?) 3 Causal Schemas & Conditional Conditional Reasoning in Reasoning Hypothesis Testing • If-then argument form not • Difficulty w/modus tollens inferences seen equivalent to cause-effect relations in performance on hypothesis testing tasks • Cause-effect relations • Confirmation Bias – tendency to look for affect “if” interpretation and ease of modus evidence that confirms hypothesis rather tollens than falsifying evidence • Ready alternatives for Q “If” P then Q = PÆQ • No ready alternatives for Q “If” P then Q = PÅÆQ Wason Selection Task Wason Selection Task E 4 7 F P Q ~Q ~P If a card has a vowel on one side, it has an even P (correct) number on the other. 50% E Q affirming the consequent 46% E & 4 ~Q (correct) 4% E&7 ~P denying the antecedant 4.
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