Rediscovering Human Logical Reasoning

According to state-of-the art science there is no human logical reasoning. Today’s psychological literature shows that people don’t interpret even the basic components of logic according to the dictates of logic. Today’s science of logic claims that modern, mathematic logics are axiomatic and independent of human reasoning, and cannot be used to describe it. However, in the psychological literature people’s inferences haven’t been tested in a systematic way. In fact, it can be demonstrated that people’s basic logical inferences can be described in terms of logic. Through an ancient, very central abstraction error in logic, which can be also found, unchanged, in mathematical logics, it can be also demonstrated that the roots of modern mathematical logics are non-axiomatic, and are based on human reasoning. The suggested solution has many benefits. For example, it solves several puzzles regarding logic and human reasoning, and links logic and learning to each other.

Keywords: logical reasoning; conditional; everyday reasoning; biconditional; fallacy; scholastic logic

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Introduction When thinking about whether living beings other than humans are also capable of logical reasoning, the phylogenetic and ontogenetic origins of logical reasoning, and how this bears on our philosophical understanding of human thought, first of all one should clarify how to interpret logical reasoning. Only then can we assess whether people reason logically at all. With regard to the first question, the science of logic has existed for about 2,400 years, so in the scientific world it would seem reasonable to assume that by logical reasoning one can understand thought based on rules determined by the science of logic. This is, however, not a self-evident approach. In many branches of science, particularly in psychology, the expression ‘logical reasoning’ is often used in such a way that no attempt is made to make it compatible with a concrete logical system. In such cases the word ‘logical’ is used merely as a synonym for ‘consistent’, ‘rational’, and other such concepts. This approach to logical reasoning reflects everyday jargon, where it is also often said that something is logical or not without making any attempt to describe it with the actual tools of the science of logic. This way of approaching logical reasoning is inappropriate for scientific endeavours because it makes it impossible to define how the term ‘logical’ should be understood. With this approach, the definitions become circular: if someone is asked to define logical reasoning, they would at best be able to describe it using some other everyday synonyms, such as saying that it is some rational thought that adheres to some rules; but if they were then asked to define the rational thought that adheres to some rules,

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then a few synonyms sooner or later, they would finally define it as a kind of logical reasoning.

The state-of-the art science It seems therefore obvious that by logical reasoning one should understand thought that can be described using the tools of the science of logic. However, this seems to pose several serious problems. The science of logic has evolved very intensively over the past 100 years. Since antiquity until the beginning of the last century – that is, for about 2,000 years – only one logic, Greco-Roman or scholastic logic, prevailed, so it was possible to speak about ‘the’ logic. Since the beginning of the last century, that is, after the appearance and strengthening of mathematical logics, a plethora of logics have appeared. Some of the newer ones have been designed specifically to describe human or human-like reasoning (see e. g. the various ‘natural logics’ the ‘logics of contextual reasoning’ and ‘non-monotonic logics’). Which of these should therefore be chosen as a normative reference point for the analysis of human logical reasoning? An obvious solution would be to investigate with psychological experiments how people interpret the basic building blocks of logic – the negation, the ‘and’, the ‘or’, and the ‘if–then’ connectives of propositional logic and the ‘none’, ‘some’, and ‘all’ quantifiers of syllogisms. How do people interpret, for instance, the meaning of the if–then connective in the ‘if it is raining, then I take a coat’ sentence, or how do they interpret the meaning of the ‘some’ quantifier in the ‘some cats are curious’ statement? To date, all logical systems have been built on the definitions of these building blocks. If a logic can be found that is consistent with how people interpret these basic building blocks in everyday life, then obviously this would be the correct 3

logic for the characterisation of people’s logical reasoning. In which case this logic could be used to further explore people’s logical reasoning. In psychology, people’s syllogistic inferences have been investigated for about 100 years (see e.g. Eidens, 1929; Wilkins, 1928; Woodworth & Schlosberg, 1954), and their inferences on the connectives for about 50 years (see e.g. Morf, 1957; Wason, 1966, 1968; O’Brien 1972) – both quite intensively. Looking at the data in the literature it seems obvious that the inference people draw on the building blocks of logic cannot be described with the tools of any logical system (see e.g. Evans, Newstead & Byrne, 1993; Evans & Over, 2004). In a relatively recent study Stenning and van Lambalgen (2008) posit that some principles of modern logic can be used to explain the experimental results. However, they also assume that they can identify various heuristics, biases and other non-logical components even in people’s inferences on these basic logical building blocks. In addition, the authors explain only a few of the results of the main experimental tasks. In the past ten to fifteen years a paradigm shift has taken place in the literature, and more and more researchers hold the opinion that human inferences cannot be described in terms of logic, but can be approached only in terms of probabilities (see e.g. Oaksford & Chater, 1998; Evans & Over, 2004; Manktelow, 2012). It is commonly accepted that people’s inferences cannot be described in terms of logic. The situation in the science of logic seems to be similarly clear. When founding modern logic, logicians made it absolutely clear that modern logics are not for the description of people’s inferences, that human inferences and logical inferences are fundamentally different (see e.g. Frege 1918-19/1984; Husserl, 1913/2001; Copi, 1982; Burgess, 2009).

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How and in what framework is it possible then to investigate people’s logical reasoning? If ‘logical’ is only used as a synonym of a thought that is ‘consistent’, ‘adheres to some rules’, or ‘rational’, then only circular definitions can be obtained that are unsuitable for scientific research. On the other hand, with the state-of-the-art stances of psychology and logic in view, it is even unscientific to speak about human logical reasoning.

The problem with the state-of-the-art science The current stance of these sciences, however, gives rise to serious concerns. Psychology has been detached from philosophy along with logic, and both view themselves to an extent as sciences of inferences (and hence, in a sense, of thought). Consequently, they have developed strongly opposing views. The result of this opposition is that logicians have ruled out psychological approaches, the socalled psychologism, and psychologists have ruled out logical approaches, the socalled logicism, in their discipline. This opposition still generates extremely paradoxical situations that fundamentally undermine the investigation of human logical reasoning. After all, the investigation of people’s logical reasoning is in the investigation of the link between knowledge in psychology about human inferences and knowledge in logic and philosophy about logical constructs.

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Beyond the state-of-the-art psychology When psychology detached itself from philosophy at the beginning of the last century, empiricism was chosen as its methodological backbone. Experiments still play a very important role in this discipline. The literature investigating human logical reasoning is also fundamentally experimental. Despite this, in this literature even some of the basic principles of experimental methodology are violated. The literature began in the 1960–70s with the papers of Wason (e.g. 1966, 1968) and his two students, Johnson-Laird (e.g. Johnson-Laird & Wason, 1970) and Evans (e.g. 1972). It mainly focuses on research on the if–then connective, that is, on the conditional statement. The reason for this is that of the four basic connectives of logic, the negation, the ‘and’, the ‘or’, and the ‘if–then’, the basic logical abstractions and people’s actual inferences deviate markedly only in the case of the latter. In the literature, four main task types and three main types of negatives are used for the investigation of the conditional. This means twelve main experimental settings altogether. However, despite the fact that these four tasks and three negatives have been in use in the literature for almost 30 years, until 2007 only half of these twelve main experimental settings had been tested, as can be seen in Table 1.

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Table 1. The most frequent responses on the main abstract task-types with the three types of negatives for the abstract* conditional statement

Implicit negatives

Explicit negatives

Dichotomous negatives

Selection task

P&Q

P&Q

P&Q

Truth table task

Defective truth

Defective truth

Biconditional (83%)

†

table

‡

table/ †

Biconditional? Simple

inference

?

?

?

?

?

Biconditional (92%)

task Inference

‡

production task

*

The abstract conditional statement means that the if–then connective connects letters and numbers,

which are independent from one another in everyday life. For this reason content and context effects do not influence the interpretation of this conditional statement. Consequently, the abstract conditional statement reveals the relationship that the if–then connective evokes in itself. †

Evans, Clibbens, and Rood (1996) – The authors did not report the complete response patterns but

only the rate of individual inferences. The defective truth table is still obvious with implicit negatives, but in case of explicit negatives, the most frequent response could be both the defective truth table and the biconditional. ‡

George (1992)

The response deemed logically correct for the conditional would be the ‘P & not-Q’ or ‘conditional’ response. It is clearly visible in Table 1 that this response did not appear in any of the tested tasks. However, if the remaining five experimental settings are also tested, then, as shown in Table 2, almost all of the missing tasks reveal equivalent (in other words, biconditional) inference patterns.

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*

†

Table 2. The results obtained by Veszelka (2007) and Wagner-Egger (2007) on the abstract conditional statement with the main abstract task types and the three types of negatives

Implicit negatives

Explicit negatives

Dichotomous negatives

*

Selection task

P&Q

P&Q

Truth table task

Defective truth *

Simple

inference

task

*

Defective truth

*

Biconditional (50%)

*

*

table

table

Defective truth

Defective truth

*

table / Biconditional (48%)

P&Q

Biconditional

*

*

table

(42%) /

†

Inference

Biconditonal

production task

(73.3%)

Biconditional (52%)

*

Biconditional (60%)

†

Biconditional (67%)

*

*

The equivalence/biconditional is also a response pattern that is well-known and easily identified in terms of logic. It is an interesting phenomenon from the literature that for decades has advanced the idea that logic cannot be employed for the description of human inferences, such that the results of those tasks that reveal obvious logical inferences are missing. When running the missing experiments it is not only obvious that the most frequent answer (in five or six out of the twelve main tasks) is the equivalent/biconditional inference. With this equivalent inference pattern it is also possible to derive results that seemingly deviate from this pattern (Veszelka, 2014). With psychological experiments it can be empirically demonstrated that the tasks that do not reveal equivalent responses (the selection task in particular) simply distort the results because of their layout (Wagner-Egger, 2007; Veszelka, 2007; 2012; 2014).

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There are some similar additional methodological flaws in the psychological literature (Veszelka, 2014). By correcting these, a completely different picture can be obtained to that on which the literature has been built for several decades. By making the research methodologically correct and systematic, it becomes clear that people’s inferences on the basic components of logic can be described in terms of logic. The theories in the literature, including the current paradigm switch towards probabilistic accounts, are all based on these methodologically questionable, incomplete data. For this reason, and also because of the inconsistent formulation of the theories (see again Veszelka, 2014), none of these theories and ‘paradigms’ are substantiated. Characteristically, in the literature the traditional practice is to forcefully implement theories popular at the time in other fields onto the experimental results. Currently, probabilistic approaches have some momentum. In addition, it seems that in the general attempt in the literature to prove that not even the inferences people draw on the basic building blocks of logic can be described in terms of logic, the data that contradict this assumption, such as the missing basic experimental tasks mentioned above, have either simply not been published, or if on the few occasions they get published (e.g. George, 1992; Wagner-Egger, 2007) they are completely neglected thereafter.

Beyond the state-of-the-art logic The approach of today’s logicians gives rise to similar concerns. As mentioned, Greco-Roman or scholastic logic was the dominant logic before modern logics for more than 2,000 years, and it should be noted that this logic has its roots in the

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human sciences. According to modern logicians, however, Greco-Roman or scholastic logic is completely insignificant, and it had nothing to do with today’s modern, mathematical logics (e.g. Tarski, 1946/1995). As is generally maintained, mathematical logics have been created in themselves, independently of anything else, in an axiomatic way. As it is argued, this is one of the reasons why they cannot be employed to describe human reasoning, which is fallible, prone to biases, and non-axiomatic (see e.g. Frege, 1918-19/1984; Husserl, 1913/2001; Burgess, 2009). It is also easy to see here, however, that scientists approach truth in a somewhat flexible way. It can be clearly demonstrated that Aristotle (1928, 167b1ff) made a logical mistake in the abstraction of the if–then connective, that is, in the abstraction of the conditional, and it has been taken up by logicians ever since (see later in this paper or in Veszelka, 2014). This is the reason why modern psychological experiments (in case of systematic testing) do not reveal the inference pattern traditionally equated with the conditional, but the equivalent/biconditional inferences. By fixing Aristotle’s abstraction error, it can be logically demonstrated that the correct logical interpretation of the conditional statement is the equivalent, biconditional response (Veszelka, 2014). The error made by Aristotle was relatively small. However, since it occured in the roots of logic, it has huge consequences on today’s logics, and precisely on the area of concern here: on the relation between logic and psychology, that is, on the relation between logic and human reasoning. It can, for example, be clearly demonstrated that mathematical logics have been also built on ancient, scholastic logic, because they completely took this erroneous abstraction over from scholastic logic.

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But why then do mathematical logicians claim precisely the opposite? Probably because scholastic logic was clearly not a mathematical logic, but a logic based on everyday insights, and related to the social sciences. That is, the rules of this logic have followed the rules of human reasoning. Anyone can look back and see that the ancient logicians deduced and explained classical logical abstractions with simple, everyday example sentences, instead of doing it in an axiomatic way. However, modern logicians wanted to make logic the foundations of mathematics, and although this wasn’t finally fully successful (see e.g. Nagel, Newman, Hoftadter, 2008), they could still very markedly implement logic into mathematics. The situation has been further complicated by the fact that the mathematical logicians who founded modern logic had a dispute specifically with logicians who interpreted logic as a part of psychology and who wanted to employ it to describe human reasoning (see. e.g. Kusch, 1995; Husserl, 1913/2001). For this reason mathematical logicians probably did not and still do not welcome the argument that the roots of logic lie in the human sciences. Since human reasoning is mainly studied today in psychology, this would mean that the foundations of logic, and, hence, a great part of mathematics, are psychological. Neither does psychology welcome the idea that the foundations of human reasoning, and hence that of psychology, lie in logic. In fact, behaviourists, the founding fathers of ‘modern psychology’, specifically argued against such ‘armchair’ theorising, and preferred a very reductionist experimental methodology where all references to mental states, such as ‘thought’, ‘rationality’, ‘logic’, or ‘reasoning’ were completely exorcized. As a late consequence of these founding principles, the assumption that human reasoning follows logical rules is still not welcome in either of

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these two disciplines. In fact, such an assumption is usually seen in both disciplines as overly naïve, simplistic, or simply unscientific.

The fundamental abstraction error in the roots of logic The abstraction error of the conditional

With regard to the ancient, Grego-Roman logic, ancient logicians merely thought over, with insight, what the basic components of logic, for example the ‘if–then’ connective of the conditional statement, meant in everyday language, and what relation they created between the terms they connected. In the case of the conditional statement, the ancient logicians said, for example, that when taking the ‘if it is a cat, it is curious’ example sentence, from ‘cat’ it can be deduced that it is ‘curious’ (modus ponens inference), and from ‘not being curious’ it can be deduced that ‘it is not a cat’ (modus tollens inference). However, they argued that if a living being is ‘curious’ it does not follow that ‘it is a cat’ (that is, the affirmation of the consequent inference does not follow) and similarly, if a living being is ‘not a cat’, it does not follow that it is ‘not curious’ (that is the denial of the antecedent inference does not follow either). According to the only explanation available (see e.g. Jevons, 1906), the two latter inferences do not follow because other living beings can be also curious. It is not only scholastic logicians that defined the interpretation of the conditional this way: today’s mathematical logicians use this same argument when they want to illustrate, with an everyday example, the interpretation of the conditional statement in mathematical logics.

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However, there is an easily identifiable abstraction error in this definition. In the example above the relation between ‘cat’ and ‘curious’ was analysed, but during the analysis an additional component, the ‘other living beings’ was also taken into account – without being denoted. Yet the most fundamental principle of logic, which is even more fundamental than the abstraction of the connectives, is that if a component is taken into account during the analysis, it should be denoted, or if a component is not denoted, it shouldn’t be taken into account either. It is clear that if some components are arbitrarily denoted and others are not, the complete construction of logic loses it sense. This is the fundamental logical mistake that has been committed in the abstraction of the conditional from antiquity until today. It looks like a minor error, but since it relates to one of the most basic components of logic, it is in fact similar to saying that in mathematics ‘2=5’ is a correct equation because one could add ‘3’ to ‘2’, without denoting the ‘3’. In case of the conditional this same error is made, because instead of the ‘if it is a cat, it is curious’, the more compound ‘if it is a cat or possibly some other living being, it is curious’ statement is analysed, without denoting the additional ‘or’ connective (!) and the ‘possibly some other living being’ term.

Naturally, in this more compound

statement, from ‘curious’ it indeed does not follow that it is a cat – because it can be some other living being as well. So as described above, the affirmation of the consequent is not a valid inference here. But this relation is already the result of a more compound statement. It is obvious that no working, useful mathematical system could be built if the addition and subtraction had been defined in a similar way, for instance if the definition of addition also included a subtraction, without denoting this subtraction. For this reason, there is a good chance that by respecting the fundamental logical

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principle of denoting the components taken into account in abstractions, logic could have much more benefits than are assumed to be possible today. As a first step, it can be deduced that the correct logical abstraction of the conditional is the equivalent/biconditional inference pattern (see Veszelka, 2014) – precisely that inference pattern that is revealed by the psychological experiments – if the experimental investigation is made in a systematic way.

The abstraction error of the universal affirmative statement

The same result can be obtained from the other main branch of ancient logic: syllogisms. The logical interpretation of syllogisms and people’s syllogistic inferences also markedly deviate from one another (see e.g. Johnson-Laird & Bara, 1984). One of the four basic statements in syllogisms, the universal affirmative statement, is, however, seen as equivalent to the conditional. Furthermore, not only can the erroneous abstraction of the conditional be traced back to Aristotle (see Aristotle, 1928, 167b1ff), he is also generally known as the founding father of syllogisms. Finally, the abstraction of the universal affirmative statement is explained with example sentences just like that of the curious cat above (see e.g. Brennan, 1961), except that the ‘if it is a cat, it is curious’ statement is replaced with the ‘all cats are curious’ universal affirmative statement. For this reason, when, in line with the above, the abstraction error in the conditional statement is also fixed in the universal affirmative statement, people’s inferences can also be immediately described in terms of this new logic 80–100% of the time (Veszelka, 2014). That is, there is demonstrably a logic that people follow when drawing inferences from the basic

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components of logic. This is not an accident. The meanings of the quantifiers in syllogisms and of the connectives in sentential logic were extracted by ancient logicians from everyday, natural language, and were taken up in unchanged form by modern, mathematical logicians. This is obvious in case of syllogisms, and as discussed, it can be also demonstrated in case of the connectives, for instance in case of the if-then connective. So naturally, if the meanings of these quantifiers and connectives are extracted correctly, then they should match our everyday usage.

New results and new opportunities Beyond the experimental results and the fact that the abstraction error in the basic construct of ancient logic is easy to explain and understand, there is additional evidence that this corrected logic could be used to describe human reasoning. With the help of this logic, several traditional philosophical dilemmas can be immediately solved, such as Hempel’s Raven paradox, the counterfactuals, the paradox of the conditional, and many other problems that have been raised in the past 100 years in philosophy, linguistics, and psychology to demonstrate the incompatibility of logic and human reasoning (Veszelka, 2014). In addition, this corrected logic opens the door to several additional, new opportunities. It can, for example, be deduced that in this new logic the basic logical abstractions and the basic learning processes are inseparably connected to one another (Veszelka, 2014). Connecting inferences and learning in a common, abstract, logical system could be very important in the future – not only in making psychology a more exact discipline, but also in research into artificial intelligence.

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In order to create a machine capable of real thinking, one must be able to program an abstract set of rules into a computer through which both learning and reasoning could take place. The approach illustrated in this study can lay the foundations for this. Also, the central concept of the ‘scientific logic’ of the beginning of the 20th century was the ‘conditional’. The central concept of the ‘scientific psychology’ of the beginning of the 20th century was ‘conditioning’. (The similar word-forms of ‘conditional’ and ‘conditioning’ are certainly not an accident.) Conditioning was seen as the fundamental organizing component of learning in ‘scientific psychology’ (note that learning, as mentioned above, is once again in the picture), and it was extensively tested with animal experimentation. Consequently, the comparison of the empirical results on conditioning with the – methodologically correctly obtained – experimental results on people’s conditional inferences also gives an empirically and theoretically well-established, accurate standing point from which to investigate whether animals are also capable of logical reasoning. There is a third discipline where the conditional is ‘perhaps the most important concept’ (Swinnen, 2005, p 63). This is computer programing, where, as mentioned, in

terms

of

artificial

intelligence,

the

unified

interpretation

of

the

conditional/conditioning could have the most important benefits.

Literature Aristotle (1928). De sophisticis elenchis. In: W. D. Ross. (ed), The works of Aristotle, Oxford: Oxford University Press. Brennan J. G. (1961). A handbook of logic. Second edition. New York: Harper and Row.

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Copi, I. M. (1982). Introduction to logic. New York: Macmillan. Eidens, Hermann 1929. Experimentelle Untersuchungen über den Denkverlauf bei unmittelbaren Folgerungen. Arch ges Ps, 71, 1-66. Evans, J. St. B. T. (1972). Interpretation and matching bias in a reasoning task. Quarterly Journal of Experimental Psychology, 24, 193-199. Evans, J. St. B. T., Clibbens, J., & Rood, B. (1996). The role of implicit and explicit negation in conditional reasoning bias. Journal of Memory & Language 35(3), 392-409. Evans, J. St. B. T., Newstead, S. E, & Byrne, R. M. J. (1993). Human reasoning: The psychology of deduction. Hove, UK: Lawrence Erlbaum Associates Ltd. Evans, J. St. B. T., & Over, D. E. (2004). If. Oxford: Oxford University Press Frege, G. (1918-19/1984). Logical investigations. In B. McGuinness (ed), Collected papers on mathematics, logic, and philosophy (pp. 351-407). Oxford: Basil Blackwell. George, C. (1992). Rules of inference in the interpretation of the conditional connective. Cahiers de Psychologie Cognitive/ Current Psychology of Cognition, 12(2), 115139. Husserl, E. (1913/2001). Logical investigations. Vol 1. New York: Routledge Jevons, W. S. (1906). Logic. London: Macmillan. Johnson-Laird, P. N., & Bara, B. G. (1984), Syllogistic inference, Cognition, 16, 1-61. Johnson-Laird, P. N., & Wason, P . C . ( 1970). Insight into a logical relation. Quarterly Journal of Experimental Psychology, 22, 49-61. O’Brien, T., C. (1972). Logical thinking in adolescents. Educational Studies in

Mathematics, Vol

4(4)., pp 401-428. Kusch, M. (1995). Psychologism. A case study in the sociology of philosophical knowledge. London: Routledge. Manktelow, K. (2012). Thinking and reasoning. An introduction to the psychology of reason, judgment and decision making. New York: Psychology Press. Morf, Albert (1957). Les relations entre la logique et lb langage lors du passage du raisonnement concret au raisonnement formel. In Apostol, L., B. Mandelbrot, and A. Morf. Logique, language et theorie de l’information. Etudes D’epistemologie Genetique, Vol. 3. Paris: Presses Univ.; 173-204.

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Nagel, E., Newman, J, Hofstadter, D. R. (2008). Gödel’s proof. NYU Press. Oaksford, M., & Chater, N. (1998). Rationality in an uncertain world: Essay on the cognitive science of human reasoning. Hove, UK: Psychology Press. Stenning K., & van Lambalgen M. (2008). Human reasoning and cognitive science. Cambridge, MA: MIT Press. Swinnen, G., (2005). Apprendre à programmer avec Python.(An open access book on the Internet) Tarski, A., (1946/1995). Introduction to logic and to the methodology of deductive sciences. Mineola, NY: Dover Publications. Wagner-Egger, P. (2007). Conditional reasoning and the Wason selection task: Biconditional interpretation instead of a reasoning bias. Thinking & Reasoning, 13(4), 484-505. Wason, P. C. (1966). Reasoning. In B. M. Foss (ed), New horizons in psychology (pp. 135151). Harmondsworth, UK: Penguin Books. Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20, 273-281. Veszelka, A. (2007). A feltételes állítás kísérleti vizsgálata: mégis van benne logika? [The experimental investigation of the conditional statement: Does it have logic?]. Magyar Pszichológiai Szemle 62(4), 475-488., English version available upon request. Veszelka, A. (2012). The experimental investigation of the updated traditional interpretation of the conditional statement. In: Jakub Szymanik and Rineke Verbrugge (eds.): Proceedings of the Logic & Cognition Workshop at European Summer School in Logic, Language and Information (ESSLLI) 2012, Opole, Poland, 13-17 August, 2012, CEUR Workshop Proceedings, vol. 883, pp. 100-117 Veszelka, A. (2014). Everyday abstract conditional reasoning - in light of a 2,400 year-old mistake in logic. Kecskemét, Hungary: Pellea Humán Kutató és Fejlesztő Bt. Wilkins, Minna C. (1928). The effect of changed material on ability to do formal syllogistic reasoning. Arch. Psychol., 16, 1-83. Woodworth, R. S., & Schlosberg, H. (1954). Experimental psychology, 2nd ed., New York: Holt, Rinehart & Winston.

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According to state-of-the art science there is no human logical reasoning. Today’s psychological literature shows that people don’t interpret even the basic components of logic according to the dictates of logic. Today’s science of logic claims that modern, mathematic logics are axiomatic and independent of human reasoning, and cannot be used to describe it. However, in the psychological literature people’s inferences haven’t been tested in a systematic way. In fact, it can be demonstrated that people’s basic logical inferences can be described in terms of logic. Through an ancient, very central abstraction error in logic, which can be also found, unchanged, in mathematical logics, it can be also demonstrated that the roots of modern mathematical logics are non-axiomatic, and are based on human reasoning. The suggested solution has many benefits. For example, it solves several puzzles regarding logic and human reasoning, and links logic and learning to each other.

Keywords: logical reasoning; conditional; everyday reasoning; biconditional; fallacy; scholastic logic

1

Introduction When thinking about whether living beings other than humans are also capable of logical reasoning, the phylogenetic and ontogenetic origins of logical reasoning, and how this bears on our philosophical understanding of human thought, first of all one should clarify how to interpret logical reasoning. Only then can we assess whether people reason logically at all. With regard to the first question, the science of logic has existed for about 2,400 years, so in the scientific world it would seem reasonable to assume that by logical reasoning one can understand thought based on rules determined by the science of logic. This is, however, not a self-evident approach. In many branches of science, particularly in psychology, the expression ‘logical reasoning’ is often used in such a way that no attempt is made to make it compatible with a concrete logical system. In such cases the word ‘logical’ is used merely as a synonym for ‘consistent’, ‘rational’, and other such concepts. This approach to logical reasoning reflects everyday jargon, where it is also often said that something is logical or not without making any attempt to describe it with the actual tools of the science of logic. This way of approaching logical reasoning is inappropriate for scientific endeavours because it makes it impossible to define how the term ‘logical’ should be understood. With this approach, the definitions become circular: if someone is asked to define logical reasoning, they would at best be able to describe it using some other everyday synonyms, such as saying that it is some rational thought that adheres to some rules; but if they were then asked to define the rational thought that adheres to some rules,

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then a few synonyms sooner or later, they would finally define it as a kind of logical reasoning.

The state-of-the art science It seems therefore obvious that by logical reasoning one should understand thought that can be described using the tools of the science of logic. However, this seems to pose several serious problems. The science of logic has evolved very intensively over the past 100 years. Since antiquity until the beginning of the last century – that is, for about 2,000 years – only one logic, Greco-Roman or scholastic logic, prevailed, so it was possible to speak about ‘the’ logic. Since the beginning of the last century, that is, after the appearance and strengthening of mathematical logics, a plethora of logics have appeared. Some of the newer ones have been designed specifically to describe human or human-like reasoning (see e. g. the various ‘natural logics’ the ‘logics of contextual reasoning’ and ‘non-monotonic logics’). Which of these should therefore be chosen as a normative reference point for the analysis of human logical reasoning? An obvious solution would be to investigate with psychological experiments how people interpret the basic building blocks of logic – the negation, the ‘and’, the ‘or’, and the ‘if–then’ connectives of propositional logic and the ‘none’, ‘some’, and ‘all’ quantifiers of syllogisms. How do people interpret, for instance, the meaning of the if–then connective in the ‘if it is raining, then I take a coat’ sentence, or how do they interpret the meaning of the ‘some’ quantifier in the ‘some cats are curious’ statement? To date, all logical systems have been built on the definitions of these building blocks. If a logic can be found that is consistent with how people interpret these basic building blocks in everyday life, then obviously this would be the correct 3

logic for the characterisation of people’s logical reasoning. In which case this logic could be used to further explore people’s logical reasoning. In psychology, people’s syllogistic inferences have been investigated for about 100 years (see e.g. Eidens, 1929; Wilkins, 1928; Woodworth & Schlosberg, 1954), and their inferences on the connectives for about 50 years (see e.g. Morf, 1957; Wason, 1966, 1968; O’Brien 1972) – both quite intensively. Looking at the data in the literature it seems obvious that the inference people draw on the building blocks of logic cannot be described with the tools of any logical system (see e.g. Evans, Newstead & Byrne, 1993; Evans & Over, 2004). In a relatively recent study Stenning and van Lambalgen (2008) posit that some principles of modern logic can be used to explain the experimental results. However, they also assume that they can identify various heuristics, biases and other non-logical components even in people’s inferences on these basic logical building blocks. In addition, the authors explain only a few of the results of the main experimental tasks. In the past ten to fifteen years a paradigm shift has taken place in the literature, and more and more researchers hold the opinion that human inferences cannot be described in terms of logic, but can be approached only in terms of probabilities (see e.g. Oaksford & Chater, 1998; Evans & Over, 2004; Manktelow, 2012). It is commonly accepted that people’s inferences cannot be described in terms of logic. The situation in the science of logic seems to be similarly clear. When founding modern logic, logicians made it absolutely clear that modern logics are not for the description of people’s inferences, that human inferences and logical inferences are fundamentally different (see e.g. Frege 1918-19/1984; Husserl, 1913/2001; Copi, 1982; Burgess, 2009).

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How and in what framework is it possible then to investigate people’s logical reasoning? If ‘logical’ is only used as a synonym of a thought that is ‘consistent’, ‘adheres to some rules’, or ‘rational’, then only circular definitions can be obtained that are unsuitable for scientific research. On the other hand, with the state-of-the-art stances of psychology and logic in view, it is even unscientific to speak about human logical reasoning.

The problem with the state-of-the-art science The current stance of these sciences, however, gives rise to serious concerns. Psychology has been detached from philosophy along with logic, and both view themselves to an extent as sciences of inferences (and hence, in a sense, of thought). Consequently, they have developed strongly opposing views. The result of this opposition is that logicians have ruled out psychological approaches, the socalled psychologism, and psychologists have ruled out logical approaches, the socalled logicism, in their discipline. This opposition still generates extremely paradoxical situations that fundamentally undermine the investigation of human logical reasoning. After all, the investigation of people’s logical reasoning is in the investigation of the link between knowledge in psychology about human inferences and knowledge in logic and philosophy about logical constructs.

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Beyond the state-of-the-art psychology When psychology detached itself from philosophy at the beginning of the last century, empiricism was chosen as its methodological backbone. Experiments still play a very important role in this discipline. The literature investigating human logical reasoning is also fundamentally experimental. Despite this, in this literature even some of the basic principles of experimental methodology are violated. The literature began in the 1960–70s with the papers of Wason (e.g. 1966, 1968) and his two students, Johnson-Laird (e.g. Johnson-Laird & Wason, 1970) and Evans (e.g. 1972). It mainly focuses on research on the if–then connective, that is, on the conditional statement. The reason for this is that of the four basic connectives of logic, the negation, the ‘and’, the ‘or’, and the ‘if–then’, the basic logical abstractions and people’s actual inferences deviate markedly only in the case of the latter. In the literature, four main task types and three main types of negatives are used for the investigation of the conditional. This means twelve main experimental settings altogether. However, despite the fact that these four tasks and three negatives have been in use in the literature for almost 30 years, until 2007 only half of these twelve main experimental settings had been tested, as can be seen in Table 1.

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Table 1. The most frequent responses on the main abstract task-types with the three types of negatives for the abstract* conditional statement

Implicit negatives

Explicit negatives

Dichotomous negatives

Selection task

P&Q

P&Q

P&Q

Truth table task

Defective truth

Defective truth

Biconditional (83%)

†

table

‡

table/ †

Biconditional? Simple

inference

?

?

?

?

?

Biconditional (92%)

task Inference

‡

production task

*

The abstract conditional statement means that the if–then connective connects letters and numbers,

which are independent from one another in everyday life. For this reason content and context effects do not influence the interpretation of this conditional statement. Consequently, the abstract conditional statement reveals the relationship that the if–then connective evokes in itself. †

Evans, Clibbens, and Rood (1996) – The authors did not report the complete response patterns but

only the rate of individual inferences. The defective truth table is still obvious with implicit negatives, but in case of explicit negatives, the most frequent response could be both the defective truth table and the biconditional. ‡

George (1992)

The response deemed logically correct for the conditional would be the ‘P & not-Q’ or ‘conditional’ response. It is clearly visible in Table 1 that this response did not appear in any of the tested tasks. However, if the remaining five experimental settings are also tested, then, as shown in Table 2, almost all of the missing tasks reveal equivalent (in other words, biconditional) inference patterns.

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*

†

Table 2. The results obtained by Veszelka (2007) and Wagner-Egger (2007) on the abstract conditional statement with the main abstract task types and the three types of negatives

Implicit negatives

Explicit negatives

Dichotomous negatives

*

Selection task

P&Q

P&Q

Truth table task

Defective truth *

Simple

inference

task

*

Defective truth

*

Biconditional (50%)

*

*

table

table

Defective truth

Defective truth

*

table / Biconditional (48%)

P&Q

Biconditional

*

*

table

(42%) /

†

Inference

Biconditonal

production task

(73.3%)

Biconditional (52%)

*

Biconditional (60%)

†

Biconditional (67%)

*

*

The equivalence/biconditional is also a response pattern that is well-known and easily identified in terms of logic. It is an interesting phenomenon from the literature that for decades has advanced the idea that logic cannot be employed for the description of human inferences, such that the results of those tasks that reveal obvious logical inferences are missing. When running the missing experiments it is not only obvious that the most frequent answer (in five or six out of the twelve main tasks) is the equivalent/biconditional inference. With this equivalent inference pattern it is also possible to derive results that seemingly deviate from this pattern (Veszelka, 2014). With psychological experiments it can be empirically demonstrated that the tasks that do not reveal equivalent responses (the selection task in particular) simply distort the results because of their layout (Wagner-Egger, 2007; Veszelka, 2007; 2012; 2014).

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There are some similar additional methodological flaws in the psychological literature (Veszelka, 2014). By correcting these, a completely different picture can be obtained to that on which the literature has been built for several decades. By making the research methodologically correct and systematic, it becomes clear that people’s inferences on the basic components of logic can be described in terms of logic. The theories in the literature, including the current paradigm switch towards probabilistic accounts, are all based on these methodologically questionable, incomplete data. For this reason, and also because of the inconsistent formulation of the theories (see again Veszelka, 2014), none of these theories and ‘paradigms’ are substantiated. Characteristically, in the literature the traditional practice is to forcefully implement theories popular at the time in other fields onto the experimental results. Currently, probabilistic approaches have some momentum. In addition, it seems that in the general attempt in the literature to prove that not even the inferences people draw on the basic building blocks of logic can be described in terms of logic, the data that contradict this assumption, such as the missing basic experimental tasks mentioned above, have either simply not been published, or if on the few occasions they get published (e.g. George, 1992; Wagner-Egger, 2007) they are completely neglected thereafter.

Beyond the state-of-the-art logic The approach of today’s logicians gives rise to similar concerns. As mentioned, Greco-Roman or scholastic logic was the dominant logic before modern logics for more than 2,000 years, and it should be noted that this logic has its roots in the

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human sciences. According to modern logicians, however, Greco-Roman or scholastic logic is completely insignificant, and it had nothing to do with today’s modern, mathematical logics (e.g. Tarski, 1946/1995). As is generally maintained, mathematical logics have been created in themselves, independently of anything else, in an axiomatic way. As it is argued, this is one of the reasons why they cannot be employed to describe human reasoning, which is fallible, prone to biases, and non-axiomatic (see e.g. Frege, 1918-19/1984; Husserl, 1913/2001; Burgess, 2009). It is also easy to see here, however, that scientists approach truth in a somewhat flexible way. It can be clearly demonstrated that Aristotle (1928, 167b1ff) made a logical mistake in the abstraction of the if–then connective, that is, in the abstraction of the conditional, and it has been taken up by logicians ever since (see later in this paper or in Veszelka, 2014). This is the reason why modern psychological experiments (in case of systematic testing) do not reveal the inference pattern traditionally equated with the conditional, but the equivalent/biconditional inferences. By fixing Aristotle’s abstraction error, it can be logically demonstrated that the correct logical interpretation of the conditional statement is the equivalent, biconditional response (Veszelka, 2014). The error made by Aristotle was relatively small. However, since it occured in the roots of logic, it has huge consequences on today’s logics, and precisely on the area of concern here: on the relation between logic and psychology, that is, on the relation between logic and human reasoning. It can, for example, be clearly demonstrated that mathematical logics have been also built on ancient, scholastic logic, because they completely took this erroneous abstraction over from scholastic logic.

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But why then do mathematical logicians claim precisely the opposite? Probably because scholastic logic was clearly not a mathematical logic, but a logic based on everyday insights, and related to the social sciences. That is, the rules of this logic have followed the rules of human reasoning. Anyone can look back and see that the ancient logicians deduced and explained classical logical abstractions with simple, everyday example sentences, instead of doing it in an axiomatic way. However, modern logicians wanted to make logic the foundations of mathematics, and although this wasn’t finally fully successful (see e.g. Nagel, Newman, Hoftadter, 2008), they could still very markedly implement logic into mathematics. The situation has been further complicated by the fact that the mathematical logicians who founded modern logic had a dispute specifically with logicians who interpreted logic as a part of psychology and who wanted to employ it to describe human reasoning (see. e.g. Kusch, 1995; Husserl, 1913/2001). For this reason mathematical logicians probably did not and still do not welcome the argument that the roots of logic lie in the human sciences. Since human reasoning is mainly studied today in psychology, this would mean that the foundations of logic, and, hence, a great part of mathematics, are psychological. Neither does psychology welcome the idea that the foundations of human reasoning, and hence that of psychology, lie in logic. In fact, behaviourists, the founding fathers of ‘modern psychology’, specifically argued against such ‘armchair’ theorising, and preferred a very reductionist experimental methodology where all references to mental states, such as ‘thought’, ‘rationality’, ‘logic’, or ‘reasoning’ were completely exorcized. As a late consequence of these founding principles, the assumption that human reasoning follows logical rules is still not welcome in either of

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these two disciplines. In fact, such an assumption is usually seen in both disciplines as overly naïve, simplistic, or simply unscientific.

The fundamental abstraction error in the roots of logic The abstraction error of the conditional

With regard to the ancient, Grego-Roman logic, ancient logicians merely thought over, with insight, what the basic components of logic, for example the ‘if–then’ connective of the conditional statement, meant in everyday language, and what relation they created between the terms they connected. In the case of the conditional statement, the ancient logicians said, for example, that when taking the ‘if it is a cat, it is curious’ example sentence, from ‘cat’ it can be deduced that it is ‘curious’ (modus ponens inference), and from ‘not being curious’ it can be deduced that ‘it is not a cat’ (modus tollens inference). However, they argued that if a living being is ‘curious’ it does not follow that ‘it is a cat’ (that is, the affirmation of the consequent inference does not follow) and similarly, if a living being is ‘not a cat’, it does not follow that it is ‘not curious’ (that is the denial of the antecedent inference does not follow either). According to the only explanation available (see e.g. Jevons, 1906), the two latter inferences do not follow because other living beings can be also curious. It is not only scholastic logicians that defined the interpretation of the conditional this way: today’s mathematical logicians use this same argument when they want to illustrate, with an everyday example, the interpretation of the conditional statement in mathematical logics.

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However, there is an easily identifiable abstraction error in this definition. In the example above the relation between ‘cat’ and ‘curious’ was analysed, but during the analysis an additional component, the ‘other living beings’ was also taken into account – without being denoted. Yet the most fundamental principle of logic, which is even more fundamental than the abstraction of the connectives, is that if a component is taken into account during the analysis, it should be denoted, or if a component is not denoted, it shouldn’t be taken into account either. It is clear that if some components are arbitrarily denoted and others are not, the complete construction of logic loses it sense. This is the fundamental logical mistake that has been committed in the abstraction of the conditional from antiquity until today. It looks like a minor error, but since it relates to one of the most basic components of logic, it is in fact similar to saying that in mathematics ‘2=5’ is a correct equation because one could add ‘3’ to ‘2’, without denoting the ‘3’. In case of the conditional this same error is made, because instead of the ‘if it is a cat, it is curious’, the more compound ‘if it is a cat or possibly some other living being, it is curious’ statement is analysed, without denoting the additional ‘or’ connective (!) and the ‘possibly some other living being’ term.

Naturally, in this more compound

statement, from ‘curious’ it indeed does not follow that it is a cat – because it can be some other living being as well. So as described above, the affirmation of the consequent is not a valid inference here. But this relation is already the result of a more compound statement. It is obvious that no working, useful mathematical system could be built if the addition and subtraction had been defined in a similar way, for instance if the definition of addition also included a subtraction, without denoting this subtraction. For this reason, there is a good chance that by respecting the fundamental logical

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principle of denoting the components taken into account in abstractions, logic could have much more benefits than are assumed to be possible today. As a first step, it can be deduced that the correct logical abstraction of the conditional is the equivalent/biconditional inference pattern (see Veszelka, 2014) – precisely that inference pattern that is revealed by the psychological experiments – if the experimental investigation is made in a systematic way.

The abstraction error of the universal affirmative statement

The same result can be obtained from the other main branch of ancient logic: syllogisms. The logical interpretation of syllogisms and people’s syllogistic inferences also markedly deviate from one another (see e.g. Johnson-Laird & Bara, 1984). One of the four basic statements in syllogisms, the universal affirmative statement, is, however, seen as equivalent to the conditional. Furthermore, not only can the erroneous abstraction of the conditional be traced back to Aristotle (see Aristotle, 1928, 167b1ff), he is also generally known as the founding father of syllogisms. Finally, the abstraction of the universal affirmative statement is explained with example sentences just like that of the curious cat above (see e.g. Brennan, 1961), except that the ‘if it is a cat, it is curious’ statement is replaced with the ‘all cats are curious’ universal affirmative statement. For this reason, when, in line with the above, the abstraction error in the conditional statement is also fixed in the universal affirmative statement, people’s inferences can also be immediately described in terms of this new logic 80–100% of the time (Veszelka, 2014). That is, there is demonstrably a logic that people follow when drawing inferences from the basic

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components of logic. This is not an accident. The meanings of the quantifiers in syllogisms and of the connectives in sentential logic were extracted by ancient logicians from everyday, natural language, and were taken up in unchanged form by modern, mathematical logicians. This is obvious in case of syllogisms, and as discussed, it can be also demonstrated in case of the connectives, for instance in case of the if-then connective. So naturally, if the meanings of these quantifiers and connectives are extracted correctly, then they should match our everyday usage.

New results and new opportunities Beyond the experimental results and the fact that the abstraction error in the basic construct of ancient logic is easy to explain and understand, there is additional evidence that this corrected logic could be used to describe human reasoning. With the help of this logic, several traditional philosophical dilemmas can be immediately solved, such as Hempel’s Raven paradox, the counterfactuals, the paradox of the conditional, and many other problems that have been raised in the past 100 years in philosophy, linguistics, and psychology to demonstrate the incompatibility of logic and human reasoning (Veszelka, 2014). In addition, this corrected logic opens the door to several additional, new opportunities. It can, for example, be deduced that in this new logic the basic logical abstractions and the basic learning processes are inseparably connected to one another (Veszelka, 2014). Connecting inferences and learning in a common, abstract, logical system could be very important in the future – not only in making psychology a more exact discipline, but also in research into artificial intelligence.

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In order to create a machine capable of real thinking, one must be able to program an abstract set of rules into a computer through which both learning and reasoning could take place. The approach illustrated in this study can lay the foundations for this. Also, the central concept of the ‘scientific logic’ of the beginning of the 20th century was the ‘conditional’. The central concept of the ‘scientific psychology’ of the beginning of the 20th century was ‘conditioning’. (The similar word-forms of ‘conditional’ and ‘conditioning’ are certainly not an accident.) Conditioning was seen as the fundamental organizing component of learning in ‘scientific psychology’ (note that learning, as mentioned above, is once again in the picture), and it was extensively tested with animal experimentation. Consequently, the comparison of the empirical results on conditioning with the – methodologically correctly obtained – experimental results on people’s conditional inferences also gives an empirically and theoretically well-established, accurate standing point from which to investigate whether animals are also capable of logical reasoning. There is a third discipline where the conditional is ‘perhaps the most important concept’ (Swinnen, 2005, p 63). This is computer programing, where, as mentioned, in

terms

of

artificial

intelligence,

the

unified

interpretation

of

the

conditional/conditioning could have the most important benefits.

Literature Aristotle (1928). De sophisticis elenchis. In: W. D. Ross. (ed), The works of Aristotle, Oxford: Oxford University Press. Brennan J. G. (1961). A handbook of logic. Second edition. New York: Harper and Row.

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Copi, I. M. (1982). Introduction to logic. New York: Macmillan. Eidens, Hermann 1929. Experimentelle Untersuchungen über den Denkverlauf bei unmittelbaren Folgerungen. Arch ges Ps, 71, 1-66. Evans, J. St. B. T. (1972). Interpretation and matching bias in a reasoning task. Quarterly Journal of Experimental Psychology, 24, 193-199. Evans, J. St. B. T., Clibbens, J., & Rood, B. (1996). The role of implicit and explicit negation in conditional reasoning bias. Journal of Memory & Language 35(3), 392-409. Evans, J. St. B. T., Newstead, S. E, & Byrne, R. M. J. (1993). Human reasoning: The psychology of deduction. Hove, UK: Lawrence Erlbaum Associates Ltd. Evans, J. St. B. T., & Over, D. E. (2004). If. Oxford: Oxford University Press Frege, G. (1918-19/1984). Logical investigations. In B. McGuinness (ed), Collected papers on mathematics, logic, and philosophy (pp. 351-407). Oxford: Basil Blackwell. George, C. (1992). Rules of inference in the interpretation of the conditional connective. Cahiers de Psychologie Cognitive/ Current Psychology of Cognition, 12(2), 115139. Husserl, E. (1913/2001). Logical investigations. Vol 1. New York: Routledge Jevons, W. S. (1906). Logic. London: Macmillan. Johnson-Laird, P. N., & Bara, B. G. (1984), Syllogistic inference, Cognition, 16, 1-61. Johnson-Laird, P. N., & Wason, P . C . ( 1970). Insight into a logical relation. Quarterly Journal of Experimental Psychology, 22, 49-61. O’Brien, T., C. (1972). Logical thinking in adolescents. Educational Studies in

Mathematics, Vol

4(4)., pp 401-428. Kusch, M. (1995). Psychologism. A case study in the sociology of philosophical knowledge. London: Routledge. Manktelow, K. (2012). Thinking and reasoning. An introduction to the psychology of reason, judgment and decision making. New York: Psychology Press. Morf, Albert (1957). Les relations entre la logique et lb langage lors du passage du raisonnement concret au raisonnement formel. In Apostol, L., B. Mandelbrot, and A. Morf. Logique, language et theorie de l’information. Etudes D’epistemologie Genetique, Vol. 3. Paris: Presses Univ.; 173-204.

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Nagel, E., Newman, J, Hofstadter, D. R. (2008). Gödel’s proof. NYU Press. Oaksford, M., & Chater, N. (1998). Rationality in an uncertain world: Essay on the cognitive science of human reasoning. Hove, UK: Psychology Press. Stenning K., & van Lambalgen M. (2008). Human reasoning and cognitive science. Cambridge, MA: MIT Press. Swinnen, G., (2005). Apprendre à programmer avec Python.(An open access book on the Internet) Tarski, A., (1946/1995). Introduction to logic and to the methodology of deductive sciences. Mineola, NY: Dover Publications. Wagner-Egger, P. (2007). Conditional reasoning and the Wason selection task: Biconditional interpretation instead of a reasoning bias. Thinking & Reasoning, 13(4), 484-505. Wason, P. C. (1966). Reasoning. In B. M. Foss (ed), New horizons in psychology (pp. 135151). Harmondsworth, UK: Penguin Books. Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20, 273-281. Veszelka, A. (2007). A feltételes állítás kísérleti vizsgálata: mégis van benne logika? [The experimental investigation of the conditional statement: Does it have logic?]. Magyar Pszichológiai Szemle 62(4), 475-488., English version available upon request. Veszelka, A. (2012). The experimental investigation of the updated traditional interpretation of the conditional statement. In: Jakub Szymanik and Rineke Verbrugge (eds.): Proceedings of the Logic & Cognition Workshop at European Summer School in Logic, Language and Information (ESSLLI) 2012, Opole, Poland, 13-17 August, 2012, CEUR Workshop Proceedings, vol. 883, pp. 100-117 Veszelka, A. (2014). Everyday abstract conditional reasoning - in light of a 2,400 year-old mistake in logic. Kecskemét, Hungary: Pellea Humán Kutató és Fejlesztő Bt. Wilkins, Minna C. (1928). The effect of changed material on ability to do formal syllogistic reasoning. Arch. Psychol., 16, 1-83. Woodworth, R. S., & Schlosberg, H. (1954). Experimental psychology, 2nd ed., New York: Holt, Rinehart & Winston.

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