The syllogistic and transitive reasoning PDF

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Categorical syllogism consists of two premises and a conclusion, in which a new connection is established between propositions through the middle. The premise containing the subject of the conclusion is called the minor premise, and the one containing the predicate of the conclusion, major premise. ...

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THE SYLLOGISTIC AND TRANSITIVE REASONING Introduction Psychological research on deductive reasoning aims to study the cognitive processes that subjects use when solving a reasoning task and explain the mistakes they make. By importing the logical model as a normative model, the study of deductive reasoning has taken advantage of the criterion of logical validity. The deductive argument comprises a conclusion that is necessarily followed by the premises, and the experimental tasks have adopted this structure. A deductive argument is often presented as an experimental task, and subjects have to evaluate whether the argument is valid or not, select the valid conclusion from a set of alternatives, or draw their own conclusions. Subject responses are evaluated as correct or incorrect according to the validity criterion of the logical model. The deductive arguments that have been studied have focused on the types of inferences, which depend on logical operators (denial, disjunction, conditional, biconditional, internal relations between propositions, such as transitivity and propositions with quantifiers). Depending on the deductive argument type, investigations can be grouped into 3 types: Sylogistic reasoning: propositions with quantifiers. Transitive reasoning: propositions with internal relationships. Propositional reasoning: propositions with conditional, biconditional, denial and disjunction. Sylogistic reasoning Categorical syllogism consists of two premises and a conclusion, in which a new connection is established between propositions through the middle. The premise containing the subject of the conclusion is called the minor premise, and the one containing the predicate of the conclusion, major premise. E.g., "all wise men avoid tobacco" (minor premise) "all doctors are wise men" (major premise) Consequently, "all doctors avoid tobacco".

The position of the middle ground in each of the premises is also taken into account. There are 4 positions that give rise to the 4 figures of categorical syllogism: From the combination of the 4 types of statements, in each of the premises and the conclusion, and of the 4 types of figures, 256 syllogisms are obtained, of which only 24 are valid. Jonson-laird and steedman, find that subjects do not always consider the second premise as the subject of the conclusion, with conclusions in the 2 senses: c-a and a-c. Thus, the number of syllogisms would amount to 512 of which 48 will be valid. Valid syllogisms include all valid conclusions, whether strong or weak (a particular conclusion is obtained when a universal conclusion is allowed).

Assumptions of the atmosphere of the premises It is not a theory but an explanation to explain the performance of the subjects, which is based on the trend observed in their responses, and which seems determined by the structural aspects of the syllogisms. Woodworth and Sells presented a task of categorical syllogisms, in which subjects had to assess the validity of the arguments. They found that the subjects were wrong, and that errors could be explained by the manner of the premises. For example: when the mode of premises was the same, subjects show a tendency to choose the conclusion in the same way to the structural aspect of syllogism creates an "atmosphere" that led to the choice of a conclusion of the same type. Begg and Denny also took into account the amount (universal, particular) and polarity or quality (affirmative, negative) of the premises, and extended the effect for premises that were heterogeneous so that when the premises contain at least one premise in particular, then the conclusion is particular, and otherwise universal, and where the conclusion is negative is negative, and otherwise affirmative. This effect was called the "atmosphere effect". Sells suggests that there may be a general principle of prudence (universal conclusions are less prudent than individual ones), which would explain the subjects' preference for them. However, subjects produce few particular conclusions - negative (or). The non-logical trend of the atmosphere effect can predict an acceptable percentage of responses (43%). However, this effect only explains the tendency to error, but it is not itself a theory of systologistic reasoning. In addition, when the

effect is found to be different for valid syllogisms than for disabled people, there should be a process of inference in addition to the response bias described. On the other hand, the hypothesis always predicts a conclusion, however, the subjects often respond that there is no valid conclusion when the task is production.

Top theories and models to explain sylogistical reasoning Models based on the interpretation of the premises Chapman and chapman explain the results of their experiments for the illicit conversion of premises and probabilistic inference. Principle of probabilistic inference: Subjects show an internal consistency in their reasoning, but with a wrong rule. This rule leads them to consider that entities that share certain properties will be more likely to share other relationships. Example: Some a are b ("some fish are carnivores") Some c are b (some plants are carnivorous") As a result, some c are a ("some plants are fish") Hypothesis of illicit conversion: This is an error of interpretation of the universal affirmative and the particular refusal (without alluding to the atmosphere effect). It occurs when the affirmative universal premise (a) "all a is b" is interpreted to be equivalent to "all b are a"; and that the particular negative premise (or) "some a are not b" is equivalent to "some b are not a". This conversion is lawful for premises in the negative universal mode (i) and particular affirmative (e). Example: (a): "All men are mortals" is not equivalent to "all mortals are men." (or): "Some men are not athletes" is not equivalent to "some athletes are not men". (e): "No man is an immortal being" is equivalent to "no immortal being is a man." (i): "Some men are athletes" is equivalent to "some athletes are men".

Revlis, formulates a proposal based on the conversion of premises in all syllogisms. In the model proposed by Revlis there are two representations of syllogism: a version of the original premises (as presented) and a version of the converted premises. According to this model: 1o.- the premises are encoded and converted with both versions represented. 2o.- A composite representation of the 2 premises is prepared and the confinement is passed on. 3o.- a comparison process is carried out between the joint representation of the 2 premises and the representation of the conclusion. If both representations are congruent, syllogism is accepted as valid. If they are incongruous, the representations in their original versions are compared until a valid argument is found. If the valid argument is still not obtained or the time limit has been exceeded, you can proceed either with a model for the random selection of one of the alternatives presented, or with a model of trait selection (quantity and quality), whose predictions are almost my than those of the atmosphere effect. Original version Some fish are carnivores Some plants are carnivorous Conclusion: some plants are fish Converted version Some carnivores are fish Some carnivores are plants Conclusion: some plants are fish. The representation of the conclusion in both versions is a question of linking the terms a and c through the middle ground b, which is common to both premises. When we provide content to this syllogism, the subject warns, in both versions, that it is invalid since its structure does not follow a conclusion. However, in its abstract version it is easy for the subject to get it wrong and admit the validity of the original version, while in the converted version it can detect that a conclusion is not followed.

Effect of belief content: Thematic content can facilitate systologistic reasoning, since the conversion process can be blocked, when the content results in premises that conflict with the subject's knowledge (e.g., the conversion of "all cats are felines" in "all felines are cats", would result in a premise contrary to the subject's knowledge). However, the effect of content or beliefs continues to be manifested even in syllogisms where conversion does not alter the validity of the argument. Experimental results have shown that subjects tend to consider an argument to be valid if they agree with its conclusion, and that it is false if they are not. Specifically, subjects accepted more credible conclusions (80%) incredible (33%), regardless of its logical validity. But, the difference between accepting credible and incredible invalid conclusions was greater (61%) that the difference between credible and incredible valid conclusions (33%). To explain the interaction between logical validity and credibility, Evans and Cols, proposed 2 models to show how the conclusions are evaluated and evaluated: Selective counting model: Subjects focus first on the conclusion of the argument, and if it is consistent with their belief system, they simply accept it (they don't bother doing a logical analysis). Only when the conclusions are incredible is it that the logical analysis of syllogism is carried out. It is a proposal based on the results of the analysis of verbal protocols, which identified 3 types of strategies for the solution of syllogisms ("conclusion only", "from conclusion to premises", and "premises to conclusion"). It would correspond to the "conclusion only" strategy (during the resolution of syllogism aloud, the subjects only mention the conclusion without alluding to the premises) or the "conclusion and premises" (the subjects first mention the conclusion and then the premises). The subjects responding using the 2 strategies mentioned were the ones who had the greatest tendency to respond according to the credibility of the conclusion. Model based on an incorrect interpretation of the "logical need": It argues that the subjects do not actually understand the "logical necessity" (a deductive argument is valid only if it is impossible for their conclusion to be false as long as their premises are true). According to the model, when there is an argument with possible conclusions, but not logically necessary (indeterminate invalid syllogism), the subjects respond according to the credibility of the conclusions.

This is a distinction between: Indeterminate invalid syllogism: it would lead to possible but unnecessary conclusions, and subjects would find themselves influenced by credibility. Determined invalid syllogism: no conclusion is followed and credibility has no influence Both models leave the reasoning process unexplained, focusing on possible explanations of errors for some syllogisms, such as indeterminate invalids. Begg and Harris: According to them, errors of systologistic reasoning are found in the interpretation of premises, but not by a process of conversion, but because the subjects interpret quantifiers according to the logic of linguistic conventions, and not with the meaning proper to logic. Subjects interpret quantifiers only in 3 ways: "none" as exclusion, "some" as an intersection, and "all" as identity. The interpretation of "some" as "all" and "some not" as "none" is psychologically useless. According to the maximum conversational amount, as much information as possible should be transmitted, and not deliberately conceal some of the information (Grice). From this perspective, the logical meaning of "some" would be a transgression of the conversational maxims, when it is known that "everything" is also true and this quantifier is the one that transmits the most information. Begg shows that the interpretation of some quantifiers is governed by conversational implications, not logic. These theoretical perspectives defend rationality from a model of linguistic conventions, in which the rules of language give an intuitive and natural logic. However, the data do not suggest that Grice's interpretations are the main source of errors in systologistic reasoning. Models based on the representation of sets Erickson proposed a sylogistic almistic reasoning theory based on Euler's circles. Euler Circle Method: It is based on the use of circles for the representation, on a Euclidean plane, of the relationships between the sets covered by the premises of the arguments To know whether a conclusion is valid or not, consider all possible ways in which diagrams representing all possible combinations of the 2 premises are combined.

According to their model, the subjects represent each premise in a similar way to the spatial representation of Euler's circles. 3 stages: Interpretation of the premises, influenced by the content of the argument. The different possible interpretations of the premises are associated with probabilistic parameters that determine their selection. Combination of the representations of the premises into a single representation, to evaluate or produce a conclusion. It is assumed that the subject does not perform a thorough analysis of all possible combinations, but selects a combination according to probabilistic parameters. Selection of a verbal label that corresponds to the description of the combination obtained in the previous stage, to give the answer. Reasoning errors can occur in any of the three stages, assuming, moreover, that subjects can be prone to the "atmosphere effect". Problem: The biggest problem is the large number of different ways diagrams can be combined. To solve this, Erickson uses the porbabilistic parameters, but these parameters have been determined a posteriori based on empirical data from previous research. Guyote and Sternberg transitive chain model: They also propose a 3-stage model based on Euler's circles, but error-free as it is an exhaustive representation. At the combination stage the interpretations of the awards are integrated into "transitive chains" (links between the first and last term of the premises through the middle ground). Errors can occur at this stage due to limited short-term memory capacity. At the comparison stage, a conclusion is chosen consistent with the combined representations. At this stage errors may occur due to the lack of consideration of any conclusion consistent with the combination of premises and also the "atmosphere effect". Finally, the answer is given that corresponds to the selected conclusion. Criticism: It assumes that there are no errors in the first stage of interpretation of the premises, when there is experimental data that they do.

As in Ericsson's model, the effects of the figure are not contemplated and the atmosphere effect is used as a response bias without psychological explanation. Models based on Euler circles predict that the difficulty of syllogismwill will depend on the number of diagrams required for the interpretation and combination of premises. However, the data show that some of the syllogisms that subjects solve without difficulty require a greater number of diagrams than others that are more difficult for the subject (although they require fewer diagrams). Newell proposes a model based on Venn diagrams. Venn diagrams– A logical technique for representing relationships between sets. Syllogism is presented in a single diagram. Circles are overlapping in a way that represents the relationships between the syllogism sets. This model aims to accommodate categorical syllogisms, to the framework designed by Newel and Simon for problem solving. They understand that sylogistic reasoning is a process of searching through problem space, and that that space is represented in a similar way to Venn diagrams. It makes no predictions about errors or provides empirical data on the viability of the model. Johnson-LairdSet Mental Models: Today it is a general theoretical framework for deductive reasoning. According to this model, deductive reasoning consists of 3 stages: Understandingstage: the subject uses his knowledge of language and his knowledge in general to understand the premises. A mental model of the content of the premises of the argument is constructed. Description stage: the subject generates a description, as simple as possible, of the models built from the premises. This description should state something that is not explicit in the premises, and is the tentative conclusion of the argument. Stage of searching for alternative models of premises that may false the tentative conclusion. If you do not find them, the conclusion will be valid. If you find them, return to stage 2 to continue testing conclusions, by finding counterexamples. It is at this stage that the "deductive" aspect of reasoning is located. The difficulty of the problems explains in terms of amount of processing, and errors by limiting memory in the consideration of all combinations of relevant representations.

The higher the number of mental models needed to find a valid conclusion, the more difficult the problem will be and the greater the probability of error when running operations on the mcp, with limited capacity. The figure of the syllogism also increases the difficulty, and the probability that a conclusion is of type c-a or a-c. This is because the conclusions are read in the mental model in the same direction in which they have been represented, and when the representations of the prizes do not meet in the same direction, it is necessary an operation that reverses the order in order to be able to perform the combination of mental models Content or beliefs may also be a source of error, which influences interpretation, combination of models, and the process of acceptance or rejection of the tentative conclusion. If the content influences, it would be reasoned with misinterpreted models, which could shorten the process of finding alternative models, if the tentative conclusion is compatible with the belief system, even if it is invalid. It could also happen that the subject looked for alternative models, when the conclusion was contrary to his belief system. Therefore, the effect of the content influences the purely deductive aspect of the theory: the search for counterexamples. By affecting the construction of alternative models, one would expect that the content would not influence the syllogisms with a single mental model, but this is not the case, since, an incredible conclusion could also act as a filter in the evaluation process, making the replace it with a credible one, or who respond that there is no conclusion. Advantages of the model: Explain, under a single theoretical framework, both the process and the errors of reasoning in general. Problems: Although it admits the influence of knowledge, it does not analyze this influence but assumes it, and its explanations and predictions are based on the semantic interpretation of logical operators. This theory self-defines as semantics, but does not explain how stored knowledge is linked to the construction of mental models, in operational memory. The theory does not explain how mental models are constructed, nor the order in the construction of the sequence of possible interpretations, does not compromise with the type of representation, nor with the order of the combination of mental models

However, this theory is a strong theoretical perspective, supported by a wide range of experimental works, although the debate with other theories has not concluded, nor is it accepted as a single model. Transitive reasoning Also called linear syllogism, linear order problem or 3-term series problem. They study the inferences that depend on transitivity relationships. Transitivity Ratio– Property of any scale or dimension, according to which objects are compared or sorted. It is a syllogism consisting of 2 premises in which 3 terms are related based on a transitive property, and 1 conclusion that establishes the relationship between the 2 non-adjacent terms. Example: "(a) is greater than (b)" "(b) is greater than (c)" Accordingly, "(a) is greater than (c)" Negative relationships can also be entered in one or both premises (premises of denied equality) "(a) not as small as (b)" "(b) is not as small as (c)" Consequently, "(a) is not as small as (c)" In total there would be 32 sets of premises: 8 affirmative...