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Geurts, B. (2003). Reasoning with quantifiers. Cognition, 86, 223-251
@Article{Geurts2003,
author = {Bart Geurts},
title = {Reasoning with Quantifiers},
journal = {Cognition},
year = {2003},
volume = {86},
pages = {223—251},
}
Author of the summary: Stephanie McKeen, 2012, slmckeen@xplornet.ca
Cite this paper for:
· Syllogistic inference: a restricted form of reasoning with quantifiers. [224]
· For determining performance on syllogistic tasks, logical validity is a key factor. [227]
· Conversion: two statements can be used interchangeably to have the same meaning; i.e “Some A are B” has the same meaning as “Some B are A”. [228]
· Figural effects: The cause of errors in illicit conversions due to the assumption that lower numbered figures are easier than higher numbers. Geurst argues that this is less substantial than claimed and are linguistic in nature. [228-230]
· “...All approaches to syllogistic reasoning are ad hoc from the vantage point of language understanding.” [235]
· The first rule of inference: “any expression α occurring in an upward entailing position may be replaced with any expression β that is implied by α, and any expression α occurring in a downward entailing position may be replaced with any expression β that implies α.” [242]
· The second rule of inference: based on symmetry, it is the conversion rule or CONV. For example, “some A are B” is converted to “some B are A”.[242]
· The third rule of inference or The NO/ALL-NOT rule: this rule is similar to the conversion rule and takes care of the rest of the valid syllogisms that MON and CONV cannot. For example, “no A are B” converts to “all A are not B”. [242]
· There is a “...connection between the meaning of an expression and valid arguments which make essential use of that expression.” [250]
Inference and interpretation in logic are closely tied. [223]
“φ & ψ” is false unless both ψ is true and φ is true. [224]
The general framework for quantification in the field of natural-language semantics is relevant to the psychology of syllogistic inference. [224]
There are four types or moods of syllogistic language:
Mnemonic
1. All A are B – universal affirmative (A)
2. Some A are B – particular affirmative (I)
3. No A are B – universal negative (E)
4. Some A are not B – particular negative (O)
Of the 256 syllogistic arguments, 24 are valid according to traditional syllogistic logic. Of the 24 that are valid, only 15 are valid in modern predicate logic. This is because of differences in “all” and “no”. [225] In traditional logic, the following is valid:
All A are B No A are B ___________ ______________
Some A are B Some A are not B [225]
This traditional notion of validity is the same in psychological literature and Guerts uses it as his definition of validity. [226]
In experiments in syllogistic reasoning, in most cases participants were given two premises and were asked to choose the conclusion from a list of possibilities. This type of experiment uses an evaluation paradigm. [226]
According to the data from Chater and Oakford’s experiments on syllogism, valid syllogisms are chosen 51% of the time on average. Errors tended to occur in the vicinity of valid argument forms. [227]
Overall, people are good at syllogistic reasoning in that they often recognize valid arguments. [228]
In syllogistic reasoning experiments, a large part of the errors are due to illicit conversions. Geurts references Dickstein who suggests that this is because there is a general preference for symmetric relations (a figural effect). Like questions on a test the assumption is the earlier the figure (i.e Figure 4 vs. Figure 5), the easier the syllogism. This is significant in terms of performance within a subset of syllogisms. [228]
Geurts states that the figural effects described above are less substantial then they seem and are linguistic rather than inferential. [230]
There are three families of syllogistic reasoning: [230]
1. Logic-based
2. Mental-model theory
3. Heuristics theory
Logic based reasoning uses full predicate logic that generalizes more easily beyond its syllogistic argument forms. However, it has its limits in terms of quantifiers such as “most” or “at least half of” since they can’t be expressed in predicate logic. [232]
The mental model theory has quantified propositions represented by arbitrary individuals. This theory is criticized because it lacks an explicit semantics making it a bit unclear exactly what it is or how it’s represented because it has gone through many revisions. [233-234]
The heuristics theory is based on Chater and Oakford’s probabilistic semantics (Charter & Oaksford, 1999; Oaksford & Chater, 2001). They stated that humans reason probabilistically rather than logically. Quantifiers would be interpreted with probability. For example, “no A are B” would say that the probability of B given A is 0. [234]
Though the heuristics theory is good for a lot of quantifiers, it makes all quantifiers proportional which does not work for quantifiers such as “some” which do not really have a definite proportion. [235]
Symmetrical quantifiers, which include words like “some”, “no”, and “three” can be distinguished from non-symmetric quantifiers with there-sentences. For example, if you have a sentence “there are _____ lawyers on the beach”, you could only use the symmetric quantifiers listed above and not words like “all”, “each”, or “most”. [237]
Non-symmetrical quantifiers are universal quantifiers like “all”, “each”, and “every” or proportional ones like “most” or “half of the”. [237]
Monotonicity is the upward (a positive expression) or downward entailment (a negative expression) where the first sentence entails the other. [238]
The first rule of inference: “any expression α occurring in an upward entailing position may be replaced with any expression β that is implied by α, and any expression α occurring in a downward entailing position may be replaced with any expression β that implies α.” This rule is also called the monotonicity or MON rule. [241-242]
Insert other stuff (the second rule and about the processing model that Geurts creates)
The second rule: based on symmetry, it is the conversion rule or CONV. For example, “some A are B” is converted to “some B are A”.[242]
Q = “some” or “no”
Q(A, B) ________
Q(B, A)
The NO/ALL-NOT rule: this rule is similar to the conversion rule and takes care of the rest of the valid syllogisms that MON and CONV cannot. For example, “no A are B” converts to “all A are not B”. [242]
No(A, B) ___________
All(A, not B)
Geurts conducted an experiment that found that there were no differences between the quantifiers “some” and “at least”. He also found that it is generally more difficult to process quantifiers like “at most” because it has a negative element. For example, “at most n A are B” is represented by “not more than n A are B”. [246]
Geurts states that a processing model that is based on reasoning with quantifiers done in terms of sets rather than in terms of individuals can be successful. [249]
Summary author’s notes:
· None
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