Is the basic conditional probabilistic? Geoff Goodwin University of Pennsylvania.

Is the basic conditional probabilistic? Geoff Goodwin University of Pennsylvania

Plan Study 1: Select conditionals to describe set of facts Study 1: Select conditionals to describe set of facts Study 2: Generate conditionals to describe set of facts Study 2: Generate conditionals to describe set of facts Study 3: Select conditionals to describe possibilities Study 3: Select conditionals to describe possibilities

What is the meaning of conditional statements? Model theory posits a basic conditional (J-L & Byrne, 2002)  If p then q: Model theory posits a basic conditional (J-L & Byrne, 2002)  If p then q: pqpqpqpq ¬ pq ¬ pq ¬p ¬q ¬p ¬q Implication: Basic conditional is not probabilistic Implication: Basic conditional is not probabilistic

Basic conditionals: Examples If there’s a circle on one side of this card, then there’s a triangle on the other side: If there’s a circle on one side of this card, then there’s a triangle on the other side: circletriangle not circletriangle not circlenot triangle If Arsenal lose today then they won’t make it out of the group stage : If Arsenal lose today then they won’t make it out of the group stage : Arsenal lose do not make it Arsenal do not losedo not make it Arsenal do not lose do make it

What about probabilistic conditionals? Examples: Examples: If you smoke, then you will get lung cancer If you smoke, then you will get lung cancer If you don’t do you homework, then you will get bad grades If you don’t do you homework, then you will get bad grades If Arsenal don’t get out of the group stage, then Wenger (their coach) will be sacked If Arsenal don’t get out of the group stage, then Wenger (their coach) will be sacked Relation between antecedent and consequent seems only probabilistic Relation between antecedent and consequent seems only probabilistic

Two alternatives The meaning of conditional statements is inherently probabilistic. Conditional probability views: The meaning of conditional statements is inherently probabilistic. Conditional probability views: Evans, Over, and Handley; suppositional account / defective truth tables Evans, Over, and Handley; suppositional account / defective truth tables Oaksford and Chater; Bayesian approach Oaksford and Chater; Bayesian approach Pfeifer and Kleiter; conditional probability logic Pfeifer and Kleiter; conditional probability logic Adams Adams The conditionals on the previous slide are imprecise and false The conditionals on the previous slide are imprecise and false mental model theory mental model theory formal rule theories formal rule theories

To make them true.... If you smoke, then you have a greater chance of getting lung cancer If you smoke, then you have a greater chance of getting lung cancer If you don’t do you homework, then you will most likely get bad grades If you don’t do you homework, then you will most likely get bad grades If Arsenal don’t get out of the group stage, then Wenger will probably be sacked If Arsenal don’t get out of the group stage, then Wenger will probably be sacked

Conditional probability accounts If conditionals are inherently probabilistic (conditional probability views), insertion of probabilistic terminology: if p then probably q, is redundant. If conditionals are inherently probabilistic (conditional probability views), insertion of probabilistic terminology: if p then probably q, is redundant. If p then q: conditional probability p(q/p); If p then probably q, …. If p then q: conditional probability p(q/p); If p then probably q, ….

Question: How do people generate and interpret conditionals in response to patterns of data? Strategy: Strategy: Present two kinds of data patterns: probabilistic and deterministic Present two kinds of data patterns: probabilistic and deterministic One task: ask individuals to choose between conditional statements which describe the data One task: ask individuals to choose between conditional statements which describe the data Second task: ask individuals to describe the data Second task: ask individuals to describe the data Do people prefer conditionals which explicitly use probabilistic qualification to describe probabilistic patterns of data? Do people prefer conditionals which explicitly use probabilistic qualification to describe probabilistic patterns of data?

Study 1 41 undergraduates 41 undergraduates Presented with 6 descriptions of data, and had to choose one conditional description of the data Presented with 6 descriptions of data, and had to choose one conditional description of the data Data described deterministic and probabilistic relationships. Data described deterministic and probabilistic relationships.

Example 1 Say that you are told the following facts: A particularly nasty virus has been sweeping through Africa. After some investigation it turns out that all/ close to all (approximately 95%) of the countries in which the virus has appeared have an annual per person income of less than $500. Please indicate which of the following statements best describes this set of facts. You may only select one of the statements. Please read each of them carefully. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it probably has an annual per person income of less than $500. If a country has the virus, then it probably has an annual per person income of less than $500. If a country has an annual per person income of less than $500, then it has the virus. If a country has an annual per person income of less than $500, then it has the virus. If a country has an annual per person income of less than $500, then it probably has the virus. If a country has an annual per person income of less than $500, then it probably has the virus.

Example 1 Say that you are told the following facts: A particularly nasty virus has been sweeping through Africa. After some investigation it turns out that all of the countries in which the virus has appeared have an annual per person income of less than $500. Please indicate which of the following statements best describes this set of facts. You may only select one of the statements. Please read each of them carefully. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it probably has an annual per person income of less than $500. If a country has the virus, then it probably has an annual per person income of less than $500. If a country has an annual per person income of less than $500, then it has the virus. If a country has an annual per person income of less than $500, then it has the virus. If a country has an annual per person income of less than $500, then it probably has the virus. If a country has an annual per person income of less than $500, then it probably has the virus.

Example 1 Say that you are told the following facts: A particularly nasty virus has been sweeping through Africa. After some investigation it turns out that close to all (approximately 95%) of the countries in which the virus has appeared have an annual per person income of less than $500. Please indicate which of the following statements best describes this set of facts. You may only select one of the statements. Please read each of them carefully. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it probably has an annual per person income of less than $500. If a country has the virus, then it probably has an annual per person income of less than $500. If a country has an annual per person income of less than $500, then it has the virus. If a country has an annual per person income of less than $500, then it has the virus. If a country has an annual per person income of less than $500, then it probably has the virus. If a country has an annual per person income of less than $500, then it probably has the virus.

Example 2 Each day when you go to work you park your car illegally in a spot that is closer to where you need to go. Most of the time, you do not get ticketed, however occasionally you do. You would like to see whether there is any pattern to when you get ticketed so you collect some data. It turns out that all/ a large majority (approximately 80%) of the days you get ticketed are ones when it is raining. Please indicate (by circling the letter) which of the following statements best describes this set of facts. You may only select one of the statements. Please read each of them carefully. If you get ticketed, then it is a rainy day. If you get ticketed, then it is a rainy day. If you get ticketed, then it is probably a rainy day. If you get ticketed, then it is probably a rainy day. If it is a rainy day, then you get ticketed. If it is a rainy day, then you get ticketed. If it is a rainy day, then you probably get ticketed. If it is a rainy day, then you probably get ticketed.

Example 2 Each day when you go to work you park your car illegally in a spot that is closer to where you need to go. Most of the time, you do not get ticketed, however occasionally you do. You would like to see whether there is any pattern to when you get ticketed so you collect some data. It turns out that all of the days you get ticketed are ones when it is raining. Please indicate (by circling the letter) which of the following statements best describes this set of facts. You may only select one of the statements. Please read each of them carefully. If you get ticketed, then it is a rainy day. If you get ticketed, then it is a rainy day. If you get ticketed, then it is probably a rainy day. If you get ticketed, then it is probably a rainy day. If it is a rainy day, then you get ticketed. If it is a rainy day, then you get ticketed. If it is a rainy day, then you probably get ticketed. If it is a rainy day, then you probably get ticketed.

Example 2 Each day when you go to work you park your car illegally in a spot that is closer to where you need to go. Most of the time, you do not get ticketed, however occasionally you do. You would like to see whether there is any pattern to when you get ticketed so you collect some data. It turns out that a large majority (approximately 80%) of the days you get ticketed are ones when it is raining. Please indicate (by circling the letter) which of the following statements best describes this set of facts. You may only select one of the statements. Please read each of them carefully. If you get ticketed, then it is a rainy day. If you get ticketed, then it is a rainy day. If you get ticketed, then it is probably a rainy day. If you get ticketed, then it is probably a rainy day. If it is a rainy day, then you get ticketed. If it is a rainy day, then you get ticketed. If it is a rainy day, then you probably get ticketed. If it is a rainy day, then you probably get ticketed.

Remaining examples Say that you are told the following facts: A large study has been performed on the most effective therapies for depression. The results show that all/ a very large majority (approximately 90%) of the most effective therapies for depression include some element of drug treatment. Say that you are told the following facts: A large study has been performed on the most effective therapies for depression. The results show that all/ a very large majority (approximately 90%) of the most effective therapies for depression include some element of drug treatment. You and a close friend are planning to go on vacation together. Your friend asks you to make a list of the places you would like to visit. It turns out that all/a large majority (approximately 85%) of the vacation spots you list are ones that your friend has already visited. You and a close friend are planning to go on vacation together. Your friend asks you to make a list of the places you would like to visit. It turns out that all/a large majority (approximately 85%) of the vacation spots you list are ones that your friend has already visited.

Remaining examples Imagine that you are smoker who is trying to give up. Although you try hard, there are still some occasions when you can’t help having a cigarette. You want to try to find out what might cause you to have lapses, so you start observing your behavior carefully. It turns out that all/ a majority (approximately 75%) of the times when you smoke are ones where you are anticipating giving a public talk. Imagine that you are smoker who is trying to give up. Although you try hard, there are still some occasions when you can’t help having a cigarette. You want to try to find out what might cause you to have lapses, so you start observing your behavior carefully. It turns out that all/ a majority (approximately 75%) of the times when you smoke are ones where you are anticipating giving a public talk. Imagine that you are a researcher who is trying to work out why people commit various crimes. You commission a large national study which reveals that all/ a majority of individuals in your sample (approximately 70%) who committed violent crimes did not complete high school. Imagine that you are a researcher who is trying to work out why people commit various crimes. You commission a large national study which reveals that all/ a majority of individuals in your sample (approximately 70%) who committed violent crimes did not complete high school.

Study 1: Design First three data patterns were either probabilistic or deterministic, vice versa for second three. First three data patterns were either probabilistic or deterministic, vice versa for second three. Probabilities varied,.70 -.95, but were fixed to content Probabilities varied,.70 -.95, but were fixed to content Are probabilistic conditionals preferred for probabilistic patterns of data? Are probabilistic conditionals preferred for probabilistic patterns of data?

Study 1: Results

Study 1: First three conditionals

Study 1: Probabilistic relations

‘Reverse’ conditionals 19% of the time, participants selected the wrong, ‘reverse’ conditional. 19% of the time, participants selected the wrong, ‘reverse’ conditional. I.e., given: I.e., given: 80% of the days you get ticketed are ones when it is raining  If it is raining, then you will probably get ticketed. 80% of the days you get ticketed are ones when it is raining  If it is raining, then you will probably get ticketed. Results the same when including (previous slides) or excluding this data: Results the same when including (previous slides) or excluding this data: Overall: Probabilistic: 99% vs. deterministic: 37% Overall: Probabilistic: 99% vs. deterministic: 37% First 3: Probabilistic: 98% vs. deterministic: 44% First 3: Probabilistic: 98% vs. deterministic: 44%

Study 1: Summary Probabilistic patterns of data, Ss almost always preferred explicitly probabilistic conditionals. Probabilistic patterns of data, Ss almost always preferred explicitly probabilistic conditionals. Ss much more likely to choose a probabilistic conditional when data was probabilistic than when deterministic. Ss much more likely to choose a probabilistic conditional when data was probabilistic than when deterministic. How is this explained on a view that conditional is inherently probabilistic? Should be no preference between explicitly probabilistic and implicitly probabilistic conditionals. How is this explained on a view that conditional is inherently probabilistic? Should be no preference between explicitly probabilistic and implicitly probabilistic conditionals.

Objection Cannot rule out some demand or ‘linguistic matching’ effect, since response options included comparison between probabilistic and deterministic conditionals Cannot rule out some demand or ‘linguistic matching’ effect, since response options included comparison between probabilistic and deterministic conditionals Next study: Generate conditionals from data Next study: Generate conditionals from data

Study 2: Task Say that you are told the following facts: A particularly nasty virus has been sweeping through Africa. After some investigation it turns out that all/ close to all (approximately 95%) of the countries in which the virus has appeared have an annual per person income of less than $500. How would you describe this set of facts using an IF- THEN statement:

Study 2: Task Say that you are told the following facts: A particularly nasty virus has been sweeping through Africa. After some investigation it turns out that all of the countries in which the virus has appeared have an annual per person income of less than $500. How would you describe this set of facts using an IF- THEN statement: If a country has the virus, then it has an annual per person income of less than $500.

Study 2: Task Say that you are told the following facts: A particularly nasty virus has been sweeping through Africa. After some investigation it turns out that close to all (approximately 95%) of the countries in which the virus has appeared have an annual per person income of less than $500. How would you describe this set of facts using an IF- THEN statement: If a country has the virus, then it most likely has an annual per person income of less than $500.

Study 2: Task Imagine that you are a researcher who is trying to work out why people commit various crimes. You commission a large national study which reveals that all/ a majority of individuals in your sample (approximately 70%) who committed violent crimes did not complete high school. How would you describe this set of facts using an IF-THEN statement:

Study 2: Task Imagine that you are a researcher who is trying to work out why people commit various crimes. You commission a large national study which reveals that all of individuals in your sample who committed violent crimes did not complete high school. How would you describe this set of facts using an IF- THEN statement: If an individual in your sample committed a violent crime, then they did not complete high school.

Study 2: Task Imagine that you are a researcher who is trying to work out why people commit various crimes. You commission a large national study which reveals that a majority of individuals in your sample (approximately 70%) who committed violent crimes did not complete high school. How would you describe this set of facts using an IF- THEN statement: If an individual in your sample committed a violent crime, then they most likely did not complete high school.

Study 2: Design 39 undergraduates 39 undergraduates Each generated conditional descriptions for the same 6 data sets as before. First three were probabilistic/ deterministic; second three were opposite. Each generated conditional descriptions for the same 6 data sets as before. First three were probabilistic/ deterministic; second three were opposite. Probabilities varied in the same way as before. Probabilities varied in the same way as before.

Study 2: Results Probabilistic conditionals more likely for probabilistic than deterministic data, p <.001. Probabilistic conditionals more likely than chance, p <.001.

Study 2: Results (first 3 problems) Probabilistic conditionals more likely for probabilistic than deterministic data, p <.001. Probabilistic conditionals more likely than chance, p <.01.

Study 2: Probabilistic relations

‘Reverse’ conditionals 38% of the time, participants generated the wrong, ‘reverse’ conditional. 38% of the time, participants generated the wrong, ‘reverse’ conditional. I.e., given: I.e., given: 80% of the days you get ticketed are ones when it is raining  If it is raining, then you will probably get ticketed. 80% of the days you get ticketed are ones when it is raining  If it is raining, then you will probably get ticketed. Results the same when including (previous slides) or excluding this data: Results the same when including (previous slides) or excluding this data: Overall: Probabilistic: 89% vs. deterministic: 20% Overall: Probabilistic: 89% vs. deterministic: 20% First 3: Probabilistic: 81% vs. deterministic: 6% First 3: Probabilistic: 81% vs. deterministic: 6%

Study 2: Summary People prefer to describe probabilistic patterns of data using conditionals that are qualified probabilistically. People prefer to describe probabilistic patterns of data using conditionals that are qualified probabilistically. Qualification requires extra work. Qualification requires extra work. Why would this happen if the conditional is intrinsically probabilistic? Why would this happen if the conditional is intrinsically probabilistic? Can’t be due simply to a contrast between probabilistic and deterministic materials. Can’t be due simply to a contrast between probabilistic and deterministic materials. Conclusion: The conditional is not treated as inherently probabilistic. Conclusion: The conditional is not treated as inherently probabilistic.

Study 2: Alternative explanation? What if people use probabilistic terms to convey a weakening of the strength of the relation? What if people use probabilistic terms to convey a weakening of the strength of the relation? Two problems: Two problems: Probabilistic conditionals not more likely as strength of conditional probability declined. Probabilistic conditionals not more likely as strength of conditional probability declined. Abrupt difference between deterministic relations: 41% (Experiment 1), 25% (Experiment 2) and conditional probabilities of.95 100% (Experiment 1), 78% (Experiment 2) Abrupt difference between deterministic relations: 41% (Experiment 1), 25% (Experiment 2) and conditional probabilities of.95 100% (Experiment 1), 78% (Experiment 2)

Study 3: Conditional descriptions Prior evidence for MMT’s basic conditional involves possibility listing. Prior evidence for MMT’s basic conditional involves possibility listing. Given conditional, what is possible? Given conditional, what is possible? People typically list the models consistent with basic conditional. E.g., If Arsenal lose today, then they won’t make it out of the group stage: People typically list the models consistent with basic conditional. E.g., If Arsenal lose today, then they won’t make it out of the group stage: Arsenal lose do not make it Arsenal do not losedo not make it Arsenal do not lose do make it Barrouillet and others: Children more likely to do so as they get older. Seem to add one possibility at a time. Barrouillet and others: Children more likely to do so as they get older. Seem to add one possibility at a time. Criticized by proponents of defective truth tables (only p cases are relevant). Criticized by proponents of defective truth tables (only p cases are relevant).

Novel task Aim: Examine a different task, which can elucidate interpretation of basic conditional. Aim: Examine a different task, which can elucidate interpretation of basic conditional. Given a list of three possibilities compatible with the basic conditional, how do people describe them? Given a list of three possibilities compatible with the basic conditional, how do people describe them? If they select a conditional that captures those possibilities, hard to account for according to defective truth table account. If they select a conditional that captures those possibilities, hard to account for according to defective truth table account.

Problem 1 Imagine that you are playing cards, and you know are interested in the hand of cards that one of your opponents has. You know that he has one of the following hands (there are only two cards in his hand): Imagine that you are playing cards, and you know are interested in the hand of cards that one of your opponents has. You know that he has one of the following hands (there are only two cards in his hand): Hand 1: aceking Hand 2: not aceking Hand 3: not acenot king Please indicate which of the following statements describes what you know about his hand of cards. You can select as many statements as you think apply. Please read each statement carefully. _______ i. If there is an ace in his hand, there is a king in his hand. _______ ii. If there is not an ace in his hand, there is a king in his hand. _______ iii. If there is an ace in his hand, there is not a king in his hand. _______ iv. If there is not an ace in his hand, there is not a king in his hand. _______ v. If there is king in his hand, there is an ace in his hand. _______ vi. If there is not a king in his hand, there is an ace in his hand. _______ vii. If there is a king in his hand, there is not an ace in his hand. _______ viii. If there is not a king in his hand, there is not an ace in his hand.

Problem 2 Imagine that you are at a fancy restaurant which has a large number of options on the menu. You are scanning the menu and you notice that you can order one of the following kinds of dish (there are no other possibilities): Imagine that you are at a fancy restaurant which has a large number of options on the menu. You are scanning the menu and you notice that you can order one of the following kinds of dish (there are no other possibilities): Dish 1: beef potatoes Dish 2: not beef potatoes Dish 3: not beef not potatoes Please indicate which of the following statements describes what kind of dish you can order. You can select as many statements as you think apply. Please read each statement carefully. _______ i. If there are potatoes in the dish, there is beef in the dish. _______ ii. If there are no potatoes in the dish, there is beef in the dish. _______ iii. If there are potatoes in the dish, there is no beef in the dish. _______ iv. If there are no potatoes in the dish, there is no beef in the dish. _______ v. If there is beef in the dish, there are potatoes in the dish. _______ vi. If there is no beef in the dish, there are potatoes in the dish. _______ vii. If there is beef in the dish, there are no potatoes in the dish. _______ vii. If there is no beef in the dish, there are no potatoes in the dish.

Percentage exactly correct according to basic conditional account (80 undergraduates) Chance: ? < 50% Chance: ? < 50%

Study 3: Summary Ss readily selected just those conditionals that correctly described the possibilities consistent with a basic conditional. Ss readily selected just those conditionals that correctly described the possibilities consistent with a basic conditional. Conditional treated as equivalent to contra-positive (if not q then not q) in this abstract context. Conditional treated as equivalent to contra-positive (if not q then not q) in this abstract context. How do they do this, according to defective truth table accounts? How do they do this, according to defective truth table accounts? Suggests that sets of possibilities are important to the representation of the conditional: conditional statements can be inferred from them. Suggests that sets of possibilities are important to the representation of the conditional: conditional statements can be inferred from them.

Conclusions Data from all three studies consistent with MMT’s account of the basic conditional. Data from all three studies consistent with MMT’s account of the basic conditional. Challenges to conditional probability/ defective truth table views: Challenges to conditional probability/ defective truth table views: If conditionals are probabilistic, why do people qualify them with probabilistic terms for probabilistic data patterns? If conditionals are probabilistic, why do people qualify them with probabilistic terms for probabilistic data patterns? If the defective truth table account is correct; how do people describe possibilities consistent with MMT’s basic conditional, using the correct conditional? If the defective truth table account is correct; how do people describe possibilities consistent with MMT’s basic conditional, using the correct conditional?

Thanks Phil Johnson-Laird Phil Johnson-Laird Jeremy Boyd Jeremy Boyd Paula Rubio Paula Rubio Sunny Khemlani Sunny Khemlani

SPECIFIC: SPECIFIC: If you get ticketed, then it is a rainy day. If you get ticketed, then it is a rainy day. If you have a cigarette, then you are anticipating giving a public talk. If you have a cigarette, then you are anticipating giving a public talk. GENERAL: GENERAL: If a therapy for depression is effective, then it includes some element of drug treatment. If a therapy for depression is effective, then it includes some element of drug treatment. If a country has the virus, then it has an annual per person income of less than $500. If a country has the virus, then it has an annual per person income of less than $500.

Is the basic conditional probabilistic? Geoff Goodwin University of Pennsylvania.

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