Formal logic and mental logic
@Logic, then, comprises the science of reasoning, as well as an art, founded on that science. - J.S.Mill@
How is logic related to reasoning?
- How we actually reason vs. how we ought to reason.
- Knowledge of logic helps explain:
- Our problem-solving abilities - minimal logic seems necessary for survival
- Our knowledge of logical connections
@1. It is raining and it is sunny.@
2. It is raining or it is sunny.
3. It is raining.
@4. Joe believes that Po speaks Cantonese.@
5. Everything Joe believes is true.
6. It is true that Po speaks Cantonese.
Theory: Reasoning = construction of mental proofs
- Thoughts are compositional representations. See The language of thought hypothesis.
- Mental processes are structure-sensitive rules operating on those representations.
- Mental proofs = step-by-step application of logical rules to thoughts.
- Modus ponens. P→Q. P. Therefore, Q. demo
- Logic in AI - LISP, Prolog
Objection #1: People often do not reason logically
@All kakas are dadas.@
Peter is a dada.
Therefore Peter is a kaka.
- Maybe they use the wrong logical principle.
- Maybe they use the correct rules but apply them wrongly.
- Do we have modus tollens? P→Q. ¬Q. Therefore, ¬P.
- Or do we rely on: Assume P. Therefore Q. Contradiction. Therefore ¬P.
Objection #2: Logic cannot explain inductive reasoning
- Logic only explains deductive reasoning.
- We also need probability theory, Bayesian principles
- Reply: People do not use principles of probability
Objection #3: People use domain-specific heuristics rather than abstract logic
- People do poorly at the Wason's selection task but they do well at a similar task with the same logical structure.
- Evolutionary psychology - We have a "cheater detection module". In the drinking problem, violators of the rule are cheaters who do not have to pay the cost. This possibility of cheaters engages the module.
@Our neural circuits were designed by natural selection to solve problems that our ancestors faced during our species' evolutionary history.@
- Reply #1: But surely it can't be all general heuristics?
- Reply #2: The logical form of the rules are different. See Jerry Fodor. (2000). Why we are so good at catching cheaters. Cognition. Apr 14; 75(1):29-32.
@Data that appear to exhibit a 'cheater detection' effect on performance in the Wason Selection task are widely interpreted as implying that deontic reasoning is effected by a domain specific, cognitive module. The 'cheater detection module' is said to offer a clear example of an effect of evolutionary selection on human cognitive architecture. This interpretation depends critically on assuming that deontic conditionals and their indicative controls are identical in structure; hence that the asymmetries in S's performance must be effects of content variables. I argue that this assumption is untrue and that structural features of deontic conditionals predict the 'cheater detection' data without assumptions about either the architecture or the history of cognition. According to this analysis, the putative cheater detection effect on the Wason task is actually a materials artifact.@
See also: Buller DJ. (2005). Adapting Minds : Evolutionary Psychology and the Persistent Quest for Human Nature. Bradford Books, New York.
Objection #4: The theory of mental models provide a better alternative
- Given a statement, construct a mental model of the situation described by the statement.
- Deduction: Check whether a description is true of the model. If so, then the description is a logical consequence of the statement.
- A visual example:
@A fork is to the left of a spoon. A knife is to the right of the spoon.@
How is the fork related to the knife?
- Still relies on compositional representations.
- Rules for manipulating models?
- Complexity of models varies with difficulty of problem?
- Applicable only when mental imagery is used?
Three types of rules:
- Formal rules
- Content rules
- Mental models