Chapter 11 1 12142021 Formal logic and reasoning

Chapter 11. 1 12/14/2021 • Formal logic and reasoning » » Deductive & inductive reasoning Syllogisms Cognitive neuroscience Conditional reasoning ◊ The Wasson selection task Study Question. • Describe the Wasson selection task. What common type of logical errors are made by people attempting this task?

Lab results Typicality Single event Conjunction Mean 4. 0 High 3. 19 3. 48 3. 34 T-test Low 1. 68 2. 78 2. 23 Mean 2. 435 2. 785 Interaction T-test 3. 0 Conjunction Main effect (typicality) Main effect (events) 2. 0 Single event 1. 0 High Typicality Low

Logical Reasoning • Deductive vs. Inductive reasoning » Deductive Reasoning: Drawing a conclusion from a list of premises by following the rules of logic. ◊ E. g. , Canada has a better hockey team than the USA The USA has a better hockey team than Sweden therefore, Canada has a better hockey team than Sweden » Inductive Reasoning: Inferring a principle based on factual information. ◊ E. g. , A store was robbed of 15 TVs John has no alibi and 15 TVs in his house therefore, John is probably involved in the robbery

Logical Reasoning • Syllogisms - A three-statement logical form, two premises followed by a conclusion. » E. g. , All sophomores are students. All students pay tuition. Therefore, All sophomores pay tuition. » Abstract/general form (content-free) All A are B All B are C Therefore, all A are C

Logical Reasoning • Syllogisms » Try this: All whales are fish All fish are insects Therefore, all whales are insects? ? » Validity: An argument is valid if the conclusion logically follows from the premises. » Truth: An argument’s validity is not effected by the truth of the premises.

Logical Reasoning • Syllogisms » Try this: All whales are ocean dwellers Some ocean dwellers are orcas Therefore, some whales are orcas » Soundness: An argument is sound if it is valid and the premise are true.

Logical Reasoning • Categorical syllogisms: Venn diagrams » All A are B All circles are red

Logical Reasoning • Set Unions

Logical Reasoning • Syllogisms » Set Unions ◊ Some A are B A B Some Squares are red

Logical Reasoning • Mutually exclusive sets » No A are B A B No circles are blue

Logical Reasoning • Categorical syllogisms using Venn diagrams All A are B All B are C Therefore, All A are C (valid conclusion) C B A

Logical Reasoning • Categorical syllogisms using Venn diagrams (All whales are ocean dwellers) All A are B Some B are C (Some ocean dwellers are orcas) Therefore, Some A are C (Indeterminant) (Some whales are orcas) B A C Confirmatory B A C Contradictory All apples are fruits Some fruits are bananas Therefore, Some apples are bananas

Logical Reasoning • Categorical syllogisms using Venn diagrams No A are B No B are C Therefore, no As are Cs? A B No fruit are purple polka-dotted food No purple polka-dotted food are mangos Therefore, No fruit are mangos C A C Confirmatory B Contradictory

Logical Reasoning • Categorical syllogisms using Venn diagrams Some A are B Some B are C Therefore, Some As are Cs? Some apples are red things Some red things are fire trucks Therefore, Some apples are fire trucks A A B C Confirmatory B C Contradictory

Logical Reasoning • Categorical syllogisms using Venn diagrams Some A are B No B are C Therefore, No As are Cs? Some apples are green things No green things are red Therefore, No apples are red A A B C Confirmatory B C Contradictory

Logical Reasoning • Cognitive Neuroscience » Three cognitive theories 1. Mental model theory (Johnson-Laird) ◊ Draws on spatial processes ◊ Right hemisphere 2. Mental Logic (Rips) ◊ Language-based ◊ Left Hemisphere 3. Dual Mechanism ◊ Implicit: Unschooled and automatic ◊ Explicit: Formal, deliberate ◊ Two unspecified regions of processing Cf. Double dissociation procedures

Logical Reasoning • Cognitive Neuroscience » Goel’s neuroimaging studies Therefore, Content No content All dogs are pets All poodles are dogs All fish are scaly Control All poodles are pets Experimental All D are P All N are D All N are P

Logical Reasoning • Cognitive Neuroscience » Results ◊ Reasoning in the presence of content was related to activation in Wernicke’s area – Left hemishere ◊ Reasoning in the absence of content activated regions of the perietal lobe associated with visuo-spatial processing. – Right hemisphere

Logical Reasoning • Conditional Reasoning » Logical determination of whether the evidence supports, refutes, or is irrelevant to the stated conditional relationship » A conditional reasoning approach to John and the TVs: E. g. , If P -> Q If John is the robber, then he has 15 TVs Q John has 15 TVs therefore, P John is the robber – BTW: John is a TV repairer who works out of his home, and none of the TVs that he has are stolen. ◊ The above argument is not a valid argument – Affirming the consequence – This is one of the most common logical errors

Logical Reasoning • Conditional Reasoning Valid Arguments If P -> Q If it is an apple, it a fruit P It is an apple ~Q It is not a fruit therefore, Q It is a fruit therefore, ~P It is not an apple Modus Ponens Modus Tollens Invalid Arguments If P -> Q If it is an apple, it a fruit Q It is a fruit ~P It is not an apple therefore, P It is an apple therefore, ~ Q It is not a fruit Confirming the consequence Denying the antecedent

Logical Reasoning • Conditional Reasoning » Rips & Marcus’ (1977) results P -> Q P ~Q P -> Q ~P ~Q Always Sometimes Never 100 0 0 100 5 79 16 21 77 2

Logical Reasoning • Conditional Reasoning » Rips & Marcus’ (1977) results P -> Q Q P P -> Q Q ~P P -> Q ~Q ~P Always Sometimes Never 23 77 0 4 82 18 0 23 77 57 39 4

Logical Reasoning • The Wason selection task: another test » Each card has a letter on one side and a number on the other » What are the fewest cards you need to turn over to confirm or deny the following hypothesis: If it has a vowel on one side, there is an even number on the other side A B 1 2

Logical Reasoning • The Wason selection task: another test » Content knowledge Only patrons with a “wet” stamp are allowed to drink alcohol. DRY WET

Logical Reasoning • Why do we make errors? » Conditional vs. biconditional (form error) ◊ If and only if. – E. g. . If you don’t eat your supper, you get no ice cream ◊ We say or hear a conditional statement, but we think or mean a biconditional. » Confimation Bias ◊ We search for positive evidence ◊ Matching hypothesis » Memory load and Modus Tollens

Logical Reasoning • Hypothesis testing » Science as a process of disconfirmation » Statistical testing ◊ The null hypothesis ◊ If Null then No effect (if P -> Q) ◊ Is an effect (~Q) ◊ We reject the null (~P) » Research hypotheses