9.1 The findings obtained across experiments ... 263 9.1.1 Coherent responses to MT ... 263 9.1.2 Changing responses to AC and DA ... 266 9.1.3 Conditionals, or-introduction, and the conjunction fallacy ... 266 9.1.4 Comparing above-chance coherence between inferences ... 267 9.1.5 The effect of an explicit inference task and working memory ... 268 9.1.6 Certain vs. uncertain premises, probabilistic vs. binary paradigm instructions ... 269 9.1.7 Factors with no systematic effect on above-chance coherence ... 269 9.1.8 The precision of people's degrees of belief ... 270 9.1.9 The variance of belief distributions ... 271 9.1.10 P-validity matters over and above coherence ... 272 9.2 Conclusions ... 273 9.3 Implications for belief bias and dual-component theories ... 273 9.4 Limits of deduction and dynamic reasoning ... 275 9.5 Where next? ... 276 9.5.1 Dynamic reasoning ... 276 9.5.2 Counterfactuals, generals, and universals ... 277 9.5.3 Coherence and rationality ... 279
References ... 280
Appendices ... 299 Appendix A. Jeffrey tables for the probability biconditional... 299 Appendix B. The materials used in Experiment 1.. ... 300 Appendix C. The materials used in Experiments 3 and 4 ... 311
38 Appendix D. The materials used in Experiment 5 ... 330 Appendix E. The materials used in Experiments 6 and 7 ... 332 Appendix F. The materials used in Experiment 9 ... 335 Appendix G. The materials used in Experiment 10 ... 337
39 List of tables and figures
Numbered on the basis of the chapter in which they occur.
List of tables
Caption Page
Table 1.1. The truth tables for conjunction, disjunction, and the conditional in classical logic.
49
Table 2.1. The truth table for the material conditional. 57
Table 2.2. Examples of inferences that are invalid in the modal logical systems of Stalnaker (1968) and Lewis (1973), but valid in classical logic.
64
Table 2.3. Truth table for the example about a coin flip. 67
Table 3.1. Probability preservation properties of inferences, based on Adams (1996). 79 Table 3.2. The de Finetti table for the probability conditional. 81 Table 3.3. The Jeffrey table for the probability conditional. 82
Table 5.1. The inferences used in Experiment 1. 112
Table 5.2. The inferences used in Experiment 2. 127
Table 5.3. The four inferences of the experiment embedded in the Linda scenario. 128 Table 5.4. The inferences investigated in Experiments 3 and 4. 134 Table 5.5. The coherence intervals for the four conditional syllogisms. 135 Table 5.6. The Jeffrey table for the probability biconditional that results from adding the converse, if q then p, to the original conditional.
153
Table 5.7. The coherence intervals for biconditional AC (Bic AC) and biconditional DA (Bic DA).
154
Table 5.8. Variances of conclusion probability judgments in Experiment 3, separately for each group, inference, and premise probability condition.
157
Table 5.9. Variances of conclusion probability judgments in Experiment 4, separately for each group, inference, and premise probability condition.
165
Table 6.1. The inferences investigated in Experiments 5 to 7. 171 Table 6.2. The conclusion probabilities used in Experiment 5 for each inference and premise probability condition.
174
Table 7.1. The inferences used in Experiment 8. 225
Table 8.1. P-valid inferences with categorical conclusions and their p-invalid counterparts with conditional conclusions. Taken from Edgington (1995).
248
40 properties.
Table 9.1. The inferences investigated in the 10 experiments of the thesis. 264
List of figures
Caption Page
Figure 5.1. Observed and chance rate coherence for the eight inferences of Experiment 1, separately for each group. Error bars represent 95% CIs.
117
Figure 5.2. Above-chance coherence for the eight inferences of Experiment 1, separately for each group. Error bars show 95% CIs.
117
Figure 5.3. Above-chance coherence for the four inferences and the two task conditions of Experiment 2. Error bars show 95% CIs.
129
Figure 5.4. Distribution of the proportion of correct responses to the memory task in Group 3. Upper panel: for Experiment 3; lower panel: for Experiment 4.
137
Figure 5.5. Coherence intervals for the four conditional syllogisms: MP, MT, AC, and DA, as a function of their premise probabilities. The shaded areas in the graphs represent the coherence intervals.
138
Figure 5.6. Premise and conclusion probabilities in Experiment 3 for the inferences of type A, separately for each group and premise probability condition. Error bars show 95% CIs.
143
Figure 5.7. Premise and conclusion probabilities in Experiment 3 for the inferences of type B, separately for each group and premise probability condition. Error bars show 95% CIs.
144
Figure 5.8 Premise and conclusion probabilities in Experiment 3 for the inferences of type C, separately for each group and premise probability condition. Error bars show 95% CIs.
144
Figure 5.9. Coherence information for the IfOr inference and the OrIf inference of Experiment 3. "P(prem)" = premise probability, "P(concl)" = conclusion probability, "Observed" = observed coherence, "Chance" = chance-rate coherence, and "Above" = above-chance coherence. Error bars show 95% CIs.
145
Figure 5.10. Mean values of observed and above-chance coherence for the 12 inferences of Experiment 3, separately for each group and for three levels of measurement precision (see text for details). The black horizontal line represents a coherence rate of 0% in the panels for observed coherence, and it represents the chance rate of a coherent response in the panels for above-chance coherence. Error bars show 95% CIs.
41 Figure 5.11. Above-chance coherence for the inferences IfOr, MT, AC and DA of Experiment 3, separately for each group and premise probability condition. The horizontal line in the panels represents the chance rate of a coherent response. Error bars show 95% CIs.
151
Figure 5.12. Coherence intervals for biconditional AC and biconditional DA as a function of their premise probabilities.
155
Figure 5.13. Above-chance coherence for AC and DA of Experiment 3, in the original version of the inferences (left), in a version in which the conditional premise is substituted with a biconditional (middle), and in a version in which the conditional premise is substituted with its converse, if q then p (right). Error bars show 95% CIs.
155
Figure 5.14. Mean values of observed and above-chance coherence for the 12 inferences of Experiment 4, separately for each group and for three levels of measurement precision. The black horizontal line represents a coherence rate of 0% in the panels for observed coherence, and it represents the chance rate of a coherent response in the panels for above-chance coherence. Error bars show 95% CIs.
162
Figure 6.1. The conclusion probabilities used in Experiment 5 for each inference. The dots represent the conclusion probabilities, and the vertical lines represent the coherence intervals for each premise probability condition.
172
Figure 6.2. Proportion of "yes" and "no" responses to the inference p, therefore not-p with a premise probability of 1, observed during the practice trials.
177
Figure 6.3. Observed and above-chance coherence for the 12 inferences investigated in Experiment 5. Error bars show 95% CIs.
177
Figure 6.4. Above-chance coherence for Experiment 5 as a function of premise probability and whether the probability of the conclusion was at the edge of the coherence interval or clearly on one side of it. Error bars show 95% CIs.
181
Figure 6.5. Above-chance coherence for Experiment 5 as a function of premise probability and whether the probability of the conclusion was inside or outside of the interval. Error bars show 95% CIs.
190
Figure 6.6. Observed and above-chance coherence for the inferences in Experiment 6, excluding the data from the condition in which premise probability was 1 and the question was whether the probability of the conclusion could be higher. Error bars show 95% CIs.
200
Figure 6.7. Above-chance coherence for Experiment 6, separately for each premise probability and question condition. Higher: question of whether the probability of the conclusion can be higher than that of the premise (resp. for the two-premise inferences, whether it can be higher than .5). Lower: question of whether the probability of the
42 conclusion can be lower than that of the premise (resp. for the two-premise inferences, whether it can be lower than .5). Error bars show 95% CIs.
Figure 6.8. Observed and above-chance coherence for the 12 inferences of Experiment 7. Error bars show 95% CIs.
215
Figure 6.9. Above-chance coherence for binary instructions (Exp. 7) and probabilistic instructions (Exp. 6) when the question was whether the probability of the conclusion can be lower than the probability of the premise (resp. for the two premise inferences, whether it can be lower than 50%). The lower left corner of each panel shows the premise probability condition in Exp. 6 with which the data from Exp. 7 was compared to. Error bars show 95% CIs.
216
Figure 6.10. Above-chance coherence for binary instructions (Exp. 7) and probabilistic instructions (Exp. 6) when the question was whether the probability of the conclusion can be higher than the probability of the premise (resp. for the two premise inferences, whether it can be higher than 50%). The lower left corner of each panel shows the premise probability condition in Exp. 6 with which the data from Exp. 7 was compared to. Error bars show 95% CIs.
217
Figure 7.1. Coherence intervals for MP (upper row), DA (middle row), and and-to-if (lower row) as a function of premise probabilities. The shaded areas in the graphs represent the coherence intervals.
226
Figure 7.2. Distribution of conclusion probability judgments for MP as a function of premise probabilities. The horizontal lines beneath the distributions indicate the location of the respective coherence intervals.
232
Figure 7.3. Distribution of conclusion probability judgments for DA as a function of premise probabilities. The horizontal lines beneath the distributions indicate the location of the respective coherence intervals.
232
Figure 7.4. Distribution of conclusion probability judgments for &If as a function of premise probabilities. The horizontal lines beneath the distributions indicate the location of the respective coherence intervals.
233
Figure 7.5. Distribution of conclusion probability judgments for inferences 4 and 5: DM and nDM, as a function of premise probabilities. The horizontal lines below the distributions show the location of the coherence interval for the condition that matches their colour.
233
Figure 7.6. Observed and above-chance coherence for the 5 inferences of the Experiment. Error bars show 95% CIs.
235
Figure 7.7. Conclusion probability judgments for MP, DA, and &If, as a function of premise probabilities. Error bars show 95% CIs.
43 Figure 7.8. Conclusion probability judgments for DM and nDM as a function of premise probabilities. The three lines in each panel display the three repetitions of each premise probability condition for these inferences. Error bars show 95% CIs.
236
Figure 7.9. Mean judgments of response confidence for MP, DA, and &If, as a function of premise probabilities. Error bars show 95% CIs.
238
Figure 7.10. Mean judgments of response confidence for DM and nDM as a function of premise probabilities. The three lines in each panel display the three repetitions of each premise probability condition for these inferences. Error bars show 95% CIs.
238
Figure 7.11. Mean probability judgments and judgments of response confidence for the conditions of Experiment 9. Error bars show 95% CIs.
243
Figure 8.1. Density curves showing the distribution of conclusion probability judgments. The shaded area represents the coherence interval for each inference and premise probability.
254
Figure 8.2. Mean values of observed and above-chance coherence for the probabilistically informative inferences. Error bars show 95% CIs.
256
Figure 8.3. Judgments of inference quality for the inferences investigated. The error bars show 95% CIs, and the grey lines in the background show the individual participant values.
44 PART 1. INTRODUCTION
45 CHAPTER 1. INTRODUCTION Contents 1.1 Types of reasoning 1.2 Types of statements 1.3 Research questions 1.4 Outline of the thesis
46 TYPES OF REASONING
The present thesis is concerned with a specific form of thinking: reasoning. To reason is to produce one mental representation, called a conclusion, from one or more other mental representations, called premises. One says that the conclusion follows, or is inferred, from the premises, and the process of doing this is called drawing, or making, an inference. Reasoning can be distinguished from other forms of thinking, like associative thinking, in being directed: the relations between mental representations that it establishes are not always symmetrical. Reasoning can be distinguished from other forms of directed thinking, like creating a story, by the fact that it can be judged by epistemic norms: it makes sense to say that a reasoning outcome is correct or incorrect, whereas one could not say this of a story. An invented story can be of high or low artistic quality, realistic or unrealistic, interesting or boring, but not correct or incorrect. The premises and conclusion of an inference represent pieces of information, and the information in the conclusion is warranted to a higher or lower degree given the combined information in the premises.
Different forms of reasoning can be distinguished based on the type of relation established between premises and conclusion. The most common distinction made is that between deductive and inductive reasoning. An inference is deductive if and only if in each case in which the premises are true, or have a probability of 1, the conclusion is also true, or has a probability of 1. Said in another way, an inference is deductive if and only if it would be inconsistent for the premises to be true but the conclusion false. This feature of inferences has been called certainty preservation (Adams, 1996). An example of a deductive inference is
Modus Ponens (MP): "If it is raining, the road will be muddy. It is raining. Therefore, the road
will be muddy." Whenever you are certain that the two premises of this inference are true, you can also be certain that the conclusion is true, so that your certainty is preserved when going from the premises to the conclusion.
An inference is inductive if and only if it does not have the above property. That is, if it is not inconsistent for the premises to be true but the conclusion false. A conclusion can then still be likely given the truth of the premises, but it does not necessarily follow from the premises. An example of an inductive inference is: "It is raining. Therefore, the road will be muddy." The presence of rain can make it more likely that the road will be muddy, but does not necessarily imply that it will be muddy. After all, the road could be paved.
Inductive reasoning can be distinguished further into abductive reasoning and other forms of inductive reasoning. An inference is abductive if the truth of its conclusion is a good explanation for the truth of its premises. This explanation is often such that it postulates a causal link between premises and conclusion (Douven, 2011; Evans, Handley, Hadjichristidis, Thompson, Over, & Bennett, 2007; Oaksford & Chater, 2016; Oberauer, Weidenfeld, & Fischer, 2007). An example of an abductive inference is "The road is muddy. Therefore, it
47 rained". Other forms of inductive reasoning include category induction, e.g. "If a relatively large object moves steadily and swiftly along the road in the distance, it is probably a car ", enumerative induction, e.g. "On each occasion in which it has rained in the past, the road was muddy. Therefore, every time it rains, the road will be muddy", and analogical reasoning, e.g. "Cars are to roads like boats are to rivers".
This thesis will concern reasoning from declarative statements as premises to declarative statements as conclusions. These are statements providing information about events, e.g. "It is raining", as opposed to other speech acts, such as asking questions, e.g. "Is it raining?" But inferences concerning information about events can be of two kinds. They can be concerned with whether an event is the case, given that some other event is the case, or with whether an event should be the case, given that some other event is the case. Typically only the former are called declarative, whereas the latter are deontic. Deontic statements are statements about what is permissible and obligatory, e.g. in the context of a moral judgment or a legal rule. In this context, it is correct to infer that if a person should perform some action, then the person is allowed to perform the action. And if the person is not allowed to perform the action, then it is false that she should perform it. These inferences can be judged correct or incorrect without committing oneself to any specific moral or legal ideas, and it is in this sense that they can be viewed as declarative. This thesis is concerned only with declarative statements in the narrow sense: statements about what is the case or not, like "It is raining".