C81COG: Cognitive Psychology 1
SYLLOGISTIC REASONING
Dr. Alastair D. Smith
Room B22 – School of Psychology
Aims & Objectives
Aims
• This lecture will introduce conditional syllogisms and some of the research which has looked at people’s syllogistic reasoning.
Learning Objectives
• Give examples of two valid types of syllogism - modus ponens and modus tollens and two invalid types - affirmation of the consequent and denial of the antecedent.
• Describe belief bias and give examples of research on syllogistic reasoning which demonstrate it.
• Describe some of the key features of abstract rule and mental models accounts of reasoning.
Conditional Reasoning
Conditional Syllogism
• If it is sunny then Cedric will miss the lecture.
• It is sunny.• Therefore Cedric will miss the
lecture.
Modusponens
Propositional Calculus is a branch of formal logic which looks at rules for determining when arguments are valid. Much of it dates back to Aristotle’s work on syllogisms. These are examples of deductive reasoning.
We will focus on a subset of syllogisms which look at the conditional term “if”.
Logical structure
• If P then Q• P• Therefore Q
Antecedent
Consequent
Modus Ponens & Modus Tollens
Conditional Syllogism
• If it is sunny then aliens will invade Earth today.
• It is sunny.• Therefore aliens will invade Earth
today.
Logical structure
• If P then Q• P• Therefore Q
ModusPonens
(this example is implausible but valid)
• If it is sunny then Cedric will miss the lecture.
• Cedric has not missed the lecture.
• Therefore it is not sunny.
• If P then Q• Not-Q• Therefore not-P
Modustollens
Two invalid syllogisms
Conditional Syllogism
• If it is sunny then Cedric will miss the lecture.
• Cedric is not at the lecture.• Therefore it is sunny.
Logical structure
• If P then Q• Q• Therefore P
Affirmation of the consequent
• If it is sunny then Cedric will miss the lecture.
• It is not sunny.• Therefore Cedric will attend
the lecture.
• If P then Q• Not-P• Therefore not-Q
Denial of the antecedent
Affirmation of the consequent• If you drop the glass of beer
then the floor will be wet• The floor is wet.• Therefore you dropped the
glass of beer.
Denial of the antecedent
• If you drop the glass of beer then the floor will be wet.
• You did not drop the glass of beer.
• Therefore the floor is not wet.
Modus ponens• If you drop the glass of beer
then the floor will be wet• You dropped the glass of beer.• Therefore the floor will be wet.
Modus tollens• If you drop the glass of beer
then the floor will be wet.• The floor is not wet.• Therefore you did not drop
the glass of beer.
Two invalid syllogisms
• These are each common and often plausible and useful forms of reasoning, but logically they are both incorrect.
• Normal human thought and problem solving is often subject to illogical reasoning
(Typical data from Rips & Marcus, 1977)
Normal reasoning performance
– People find Modus Ponens reasoning very easy and straightforward.
– They find Modus Tollens substantially more difficult.
– They also often make incorrect inferences, generally either affirmation of the consequent or denial of the antecedent.
• Given series of conditional statements and premises (e.g. “If she gets up early she will go for a run” and “She gets up early”) and asked to draw a conclusion if possible:
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ModusPonens
ModusTollens
Affirmationof the
Consequent
Denial of theAntecendent
Reasoning with Abstract Rules (1)
Abstract Rule Theory (Braine, 1978).
• We have a series of simple mental rules which we can apply logically. For example modus ponens (but not modus tollens).
• Where we can use these rules directly reasoning is perfect.
• Errors often come about through failures of comprehension (we didn’t understand the terms in the problem properly).
• So we may represent the premises incorrectly and then reason logically to an invalid conclusion.
Reasoning with Abstract Rules (2)
Abstract Rule Theory (Braine, 1978).
• “If you mow the lawn I will give you £5” normally implies “If you don’t mow the lawn I won’t give you £5” (e.g. Geis & Zwicky, 1971).
• So we interpret “If P then Q”, as “If not-P then not-Q”.
• Reasoning logically with the revised premises will lead to illogical conclusions (denial of the antecedent or affirmation of the consequent)
• There are some problems with an abstract rule account:
– Underspecification: Most of the predictions are made on the basis of comprehension processes rather than reasoning ones, but no complete theory of comprehension is provided.
– Generality: Abstract rule theory seems to only apply to this type of propositional reasoning rather than other types.
Appraisal of abstract rules
Reasoning with Mental Models
The lamp is on the right of the pad
The book is on the left of the pad
The clock is in front of the book
The vase is in front of the lamp
The lamp is on the right of the pad
The book is on the left of the lamp
The clock is in front of the book
The vase is in front of the pad
book pad lamp
clock vase
“the clock is to the left of the vase”
book pad lamp
clock vase
pad book lamp
vase clock
• Mental Models (Johnson-Laird, 1983):
– We build up world models based on the information in the problem and look at these models to see whether the conclusion is justified.
– Errors often come about because we fail to build all the possible models that could describe the information in the problem.
– The more models are required, the more difficult the problem is.
Reasoning with Mental Models
Mental Models and syllogisms
1. Some of the artists are beekeepers
2. All of the beekeepers are chemists
3. None of the artists are beekeepers
artist beekeeper
artist beekeeper
…
[beekeeper] chemist
[beekeeper] chemist
…
[artist]
[artist]
[beekeeper]
[beekeeper]
…
Mental Models and syllogisms
Comprehension combines two premises in a single model:
• Some of the artists are beekeepers• All of the beekeepers are chemists
artist [beekeeper] chemist
artist [beekeeper] chemist
…
• Therefore some of the artists are chemists– This is consistent with the possibility that there may be other
artists who are not chemists
Mental Models and syllogisms
Conclusions must be validated by searching for other models that are consistent with the premises but not with the conclusion
• All of the artists are beekeepers• Some of the beekeepers are chemists
[artist] beekeeper chemist
[artist] beekeeper chemist
…
[artist] beekeeper
[artist] beekeeper
beekeeper chemist
beekeeper chemist
…
• Therefore all of the artists are chemists
• Therefore there is no valid conclusion
Characteristics of mental models
• The mental model theory has been extensively tested and the experiments have corroborated several tell-tale signs of the use of mental models:
– A mental model represents one possibility, capturing what is common to all the different ways in which the possibility may occur.
– Mental models represent explicitly what is true, but not what is false.
– The greater the number of models that a task elicits, and the greater the complexity of individual models, the poorer performance is.
– Procedures for reasoning with mental models rely on counterexamples to refute invalid inferences; they establish validity by ensuring that a conclusion holds over all the models of the premises.
Belief Bias – Evans, Barston & Pollard (1983)
• One of the commonest reasons for giving incorrect conclusions to syllogisms
• We tend to select conclusions which are believable and reject conclusions which are unbelievable (belief bias).
• This effect is particularly strong for invalid syllogisms (i.e. we accept conclusions which are logically incorrect if they appear to be true in the real world).
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unbelievable believable
valid
invalid
No addictive things are inexpensive
Some cigarettes are inexpensive
-----------------------------------------------
Some cigarettes are not addictive
• Why should believability be more important for invalid syllogisms than valid ones?• One possibility is that coming up with a believable model stops you generating
further models which might invalidate the conclusion.• Newstead et al. (1992) tested this idea and found some evidence that it might be
true.
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unbelievable believable
valid
invalid
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unbelievable believable
valid
invalid
Multiple Model SyllogismsSingle Model Syllogisms
Mental models and believability
Adding ‘irrelevant’ terms greatly affects reasoning
(Data from Byrne, 1989)
Comprehending syllogisms
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ModusPonens
ModusTollens
Affirmationof the
Consequent
Denial of theAntecendent
Normal Form
Alternative Antecendent
Additional Antecendent• If it is raining then she will get wet.
• If it is snowing then she will get wet.
(an alternative antecedent)
• She gets wet...
• If she has an essay to write then she will go to the library.
• If the library is open then she will go to the library.
(an additional antecedent)
• She has an essay to write...
• Problems with mental models– In principle there may be no difference between a mental model theory and an abstract rule theory (in psychological terms
the two theories do appear to make different predictions)– Underspecification: Like the abstract rule theory there is a lack of specificity in deciding exactly how the initial terms are
comprehended.
• Advantages of mental models– Generality: The mental models framework has be extended to include a wide variety of reasoning tasks (not just syllogisms).– Testability: The approach generates interesting predictions about which types of problem will be easiest and hardest, and
generally these predictions have been found to be accurate.
Mental models and abstract rules
Conclusions
• Deductive reasoning using syllogisms shows that people are frequently illogical in the conclusions they draw.
• Belief bias effects frequently show that knowledge about the real world affects the way we reason (even though it technically shouldn’t).
• The way we frame a reasoning question affects the answer we get. Thus adding ‘irrelevant’ information changes the conclusions people give.
• An abstract rule approach to reasoning presumes that our reasoning is logical, but limited by factors such as comprehension.
• The mental models approach suggests that reasoning is based on a less formal process of modelling possible situations. Failures occur because incorrect or incomplete models are induced by the task.
Additional reading
• Garnham & Oakhill (1994). Thinking and reasoning. Chapters 5 & 6.
• Evans, Newstead & Byrne (1993). Human reasoning: The psychology of deduction. Chapters 2 & 3.
Sample exam questions
• Which of the following are valid logical inferences:– Modus ponens and modus tollens– Denial of the antecedant and moduls tollens– Denial of the antecedant and affirmation of the consequent– Modus ponens and affirmation of the consequent
• According to Johnson-Laird reasoning errors occur when:– The reasoner has a low IQ– Multiple mental models are required– Mental models are required– The problems concern probability
