cognitive science - formal logic and analogical reasoning

what does a formal (axiomatic) system consist of?

axioms, inference rules, theorems, and reasoning

what are axioms?

statements assumed to be true

what are inference rules?

procedures for deriving new statements

what are theorems?

statements derived from axioms using rules

what does the dream of mechanized reasoning say is possible if reasoning is formal?

thought is symbol manipulation and rules guarantee truth preservation, so a machine could implement reasoning

what are the two central properties required in a perfect formal system?

consistency and completeness

what is consistency in a system?

no contradictions can derived, can’t prove both P and -P

what is completeness in a system?

all true statements in the system can be derived

what is Godel’s incompleteness theorem?

for any sufficiently powerful formal system, you can’t have consistency and completeness

why does Godel say you can’t have both consistency and completeness?

if statement G says it can’t be proven in the system, if it is provable, the system proves a falsehood, so it’s inconsistent, and if it’s not provable, it’s true but unprovable, so it’s incomplete

why does the incompleteness theorem matter?

shows there are limits on mechanized reasoning, no system can capture all truths, formal proof doesn’t equate to truth

what is deductive reasoning?

if premises are true, then the conclusion must be true, truth-preserving

what is inductive reasoning?

premises support conclusion probabilistically, conclusion may be false even if premises are true

what are the valid deductive forms?

modus ponens and modus tollens

what is modus ponens?

if P → Q, P, therefore Q

what is an example of modus ponens?

if it rains, the ground gets wet, it is raining, therefore the ground gets wet

what is modus tollens?

if P → Q, not Q, therefore not P

what is an example of modus tollens?

if it rains, the ground gets wet, the ground is not wet, therefore it is not raining

what does modus ponens do?

affirms the antecedent

what does modus tollens do?

denies the consequent

what is the rule of the Wason selection task?

if a card has a vowel on 1 side, then it has an even number on the other

what is the Wason selection task?

given the cards A, K, 2, and 7, which cards must be turned over to test the rule?

what do most participants choose in the Wason selection task and why?

A and 2 because people tend to seek confirmation and formal logic is difficult for humans

what is the correct answer in the Wason selection task?

A and 7

which deductive form is used in the correct answer to the Wason selection task?

modus tollens to falsify the rule

what is belief bias?

people are more likely to accept believable conclusions even if invalid and reject unbelievable conclusions even if valid

what did the Evans experiment do?

demonstrated belief bias effect experimentally

what did Gampa’s experiment show?

political ideology influences evaluation of logically valid arguments

what is the mental models theory?

people reason by constructing mental representations simulating possible states of the world and drawing conclusions from them

which models do people often fail to represent when given if P → Q?

not P → Q, not P → not Q

what happens when alternative possibilities are not constructed?

errors arise

how do people do on the 2-4-6 Wason task when they only construct one mental model?

errors arise, don’t try to falsify their model

what is the dual-goal version of the 2-4-6 task?

participants instructed to confirm/disconfirm and consider alternative hypotheses

how do people perform in the dual-goal version of the 2-4-6 task vs the OG?

reasoning and performance improve

what does inductive thinking consist of?

analogical transfer and analogical transference

what is analogical transfer?

solving a problem in one domain based on solution in another domain

what is analogical transference?

generalizing properties/relations from one domain to another

what is special about thinking?

it’s structure-sensitive and flexible in the way in which knowledge is accessed and used

what do reasoning, problem solving, and learning depend on?

a capacity to represent and manipulate relational knowledge

what do complex structures of thinking emerge from?

systematic recombination of more primitive elements

how do healthy controls and patients with anterior temporal damage perform on non-relational and relational problems?

about the same

how do patients with prefrontal damage perform on non-relational and relational problems compared to healthy controls and patients with anterior temporal damage?

substantially worse, especially at 2-relation problems

when told the radiation problem and no hints are given, what proportion of participants will give you the right answer?

20%

when told the army story and the radiation problem and no hints are given, what proportion of participants will give you the right answer?

30%

when told the army story and the radiation problem and given a hint, what proportion of participants will give you the right answer?

75%

what does the army/radiation problem show?

that noticing an analogy is a separate step from constructing the analogy

what are the steps of analogical transfer?

recognition, abstraction, and mapping

what is (analogical transfer) recognition?

identify a potential analog or base domain

what is (analogical transfer) abstraction?

abstract general principle from base problem

what is (analogical transfer) mapping?

apply principle to target

what is an example of analogical inference?

an atom (target) is like the solar system (base)

how can relations be represented?

as a proposition which specifies which elements fill the roles of the predicate

what are higher-order relations?

relations that are nested within other relations

what is an example of a structured relational representation of an attribute?

big(sun)

what is an example of a structured relational representation of a lower-order relation?

bigger(sun, planets)

what is an example of a structured relational representation of a higher-order relation?

Cause [bigger(sun, planets), revolves around (planet, sun)]

what is an analogy?

when two conceptual domains share relational similarity

what is the structure-mapping theory of analogy?

comparisons involve an alignment of relational structures

what is an example of one-to-one mapping?

sun → nucleus

what is an example of parallel connectivity?

sun → nucleus, planets → electrons

what is systematicity?

idea that deeply nested relational structures make better analogies

what are the types of comparisons?

literal similarity, analogy, abstraction, anomaly, and mere appearance

what is a comparison of literal similarity?

many attributes and relations in common, ex: milk is like water

what is a comparison of analogy?

few attributes, many relations in common, ex: heat is like water

what is a comparison of abstraction?

few attributes, many relations in common, ex: heat flow is a through-variable

what is a comparison of anomaly?

few attributes and relations in common, ex: coffee is like the solar system

what is a comparison of mere appearance?

many attributes, few relations in common, ex: the glass tabletop gleamed like water

what were the results of Spellman and Holyoak’s experiment?

subjects’ preferred policy was significantly more interventionist when scenario contained WWII features than when it contained Vietnam features

do people map objects across two different situations on the basis of surface similarity or structural similarity between the two?

usually surface similarity, but if forced to compare the two scenes, then based on structural similarity

when given triads that show the same relational pattern across different dimensions, how do kids perform at recognizing the pattern?

with great difficulty

what helped to foster relational representations in kids?

comparison of highly similar before less similar items fosters re-representations of the relevant relations

what is near transfer?

apply knowledge from a closely related base domain to the target domain

what is far transfer?

apply knowledge from seemingly distant base domain to target domain

how do LLMs perform on letter-string analogies in Latin alphabet (near transfer)?

at or above the level of children

how do LLMs perform on letter-string analogies in the Greek alphabet (near transfer) compared to Latin?

performance reduces somewhat

how do LLMs perform on letter-string analogies in the Symbol alphabet (far transfer)?

fail catastrophically

why can’t LLMs generalize when solving (far) letter-string analogies?

solving these problems requires an abstraction of the relations previous and next

what was the goal of the research in the Dunbar reading?

to identify the points in time at which creative scientific thinking occurred

when were scientists in the Dunbar reading most likely to use analogies?

when the goal was to explain, usually methodological issues

when were non-biological or distant analogies used by scientists in the Dunbar reading?

rarely and to explain a concept

what kinds of analogies were scientists likely to draw when designing or fixing an experiment in the Dunbar reading?

same organism or different organism

which analogies did scientists draw when formulating hypotheses in the Dunbar reading?

analogies to other organisms

how did scientists draw other organism analogies in the Dunbar reading?

using homology

how were analogies generally used by scientists in the Dunbar reading?

as a scaffolding in the construction of new theories and methodologies that is discarded once they have been advanced

did the Dunbar reading find a difference in how scientists reacted to expected and unexpected findings?

yes, there is much more reasoning for unexpected findings

did the Dunbar reading find a difference in if the scientists attended to unexpected findings based on whether it was consistent with their hypothesis?

no, scientists attended to the findings regardless of whether they were consistent with their hypothesis

when did the Dunbar reading find that scientists ignore unexpected findings?

when it occurs early and it is not a core hypothesis in the field

when did the Dunbar reading find that scientists attend to unexpected findings?

when it occurs late in the research or when it occurs early and is unexpected relative to the central assumptions of the field

how are the findings in the Dunbar reading about distributed reasoning different from findings about brainstorming?

brainstorming groups perform no better than brainstorming individuals, but groups of scientists do generate new concepts