Logic Modelling of Cognitive Reasoning

1. Introduction

In this paper we present and discuss elements or a logic mod­ elling framework for the study of human reasoning. There are three assumptions involved in developing logic modelling. Firstly, by using the concept of modelling we want to emphasize the computational hypothesis about cognitive processing. Also, computational studies of deductive reasoning are seldom seen, though such studies might throw further light on issues both in AI and cognitive psychology. Secondly, as denoted by using the concept of logic we intend to study and ana,.. lyze cognitive reasoning in a computational framework which is close to formal logic. Thus, we use logic as a modelling language. Thirdly, we assume that reasoning is a rational process. This is a controversial hypothesis.

Basically, there are three competing hypotheses about the rela,.. tion between logic and human reasoning [1[. They rest on different interpretations of the following observation. Suppose we have a set of premises, {Pt , ... , Pm }. We give the set to a theorem-prover which generates an inference, it · We also give the set to a subject who gen­ erates another inference, i2 · Suppose it implies that i2 is false. This might indicate that the subject bas understood the task, but uses a faulty reasoning strategy [2, 3]. This is the "ill-logical" hypothesis [2[. Alternatively, the subject may not use valid rules of inference at all. In this interpretation, the subject is "noo-logical" [2, 4[ The third interpretation means that i 2 is made because the subject's un­ derstanding is different [1, 3[. Thus, the inference is perfectly valid, but generated from another set or premises, {Pt ·. .. ,p,.}.

As pointed out by Smedsluod [5[ it is impossible to tell whether or not a subject is "logical" without assuming that he/she bas understood the task correctly, and vice versa. We know fairly well what it means to be logical, but, what does it mean to understand correctly! As m!ntioned, it means that it and i 2 both are valid only if they were derived from different sets or premises. rr the sets were identical, then the subject's "logic" is "ill" or possibly not there at all.

In our view, understanding means to construct a mental set or the external premises as they are perceived and interpreted. Obviously, such a construction might not be a direct one-to-one mapping or the external premises. Thus, a model might be "incorrect" in the sense or not being a mapping, but "correct" in the sense or being consistent to the reasoner's knowledge.

*The research reported herein was sponl!Ored by the Swedish Board for Technical Development (STU).

Logic modelling is an attempt to explore such a relation between task environment and mental model within a computational frame­ work. In the next sections, we present logic modelling in some detail and introduce the notion of propositional models as a major compo­ nent in reasoning.

:1. Constructing and running propositional modela

The following notation is used throughout the discussion. We will assume an implicit V quantification and use A, V, -+, +-+, and , for conjunction, disjunction, conditional, biconditional, and negation. Variables will be denoted by .z, 11. z, etc. and constants by a beginning upper case letter. A propoliliort is either an turerliofl, e.g., P(.z, 11) or a rtde (we also refer to it as an a11ampliora), e.g., P(.z, 11) A Q(ll, z)-+ R( .z, z ). A model is a set of assertions and rules.

The mechanisms behind the construction process are knowledge and attention. The goal is to select a set or relevant propositions that correspond to the external reasoning task. These propositions can enter from memory either as assertions or as assumptions. Ir the construction is driven by knowledge the reasoning effort can be reduced compared to the cases where the construction cannot rely on domain know ledge. The construction of the perceived premises is continued until the model is acceptable (or executable).

The only logical aspect or importance in the construction process is consistency. That is, we assume that an acceptable model is also a consistent model. Naturally, a model can be revised and made inconsistent with an earlier version .

Let us give a very simple illustration. Consider the following situation in the blocks world, where R is a red, B is a blue, Y is a yellow, G is a green, and W is a white block.

A propositional model or this scene could be constructed as fol­ lows,

In this model ( 1) to ( 4) correspond to the representation of the objects, whereas (5) and (6) are introduced rules (or assumptions) that might be relevant for the scene. For example, the rule in (5) states that an object, .z, is to the left of another object, .z, if .z is to the left or an object, II. which is to the left or z.

The following propositions constitute a different model,

This model indicates quite another understanding or the scene as shown in (7) and (8). Notice that the two models are not isomorphic. If the assumption (8) was revised to,

Model 3

they would be isomorphic but still quite different. That is, there is a one-t()-one map between model 1 and 3 but not between 1 and 2, and vice versa. Also, there is no direct map between 2 and 3 despite the fact that they share the same "view".

We assume that a constructed model is executed within a control structure based on logic. There are several important design issues involved in such an interpretation. For instance, how propositions are matched (e.g., unification), how they are manipulated (e.g., Modus ponens and Modus tolleos), when they are manipulated (e.g. , depth­ or breadth-first), etc .. In the following we assume an interpretation that is a mixture or the control round in production systems [6] and logic programming [7] . Let us exemplify how a model can be run.

Suppose the following question (or goal) is included in the example models. I· the blue block to the left of the white bloclcf The matching process will produce the following list or applicable instantiated rules for model 1 in a first cycle,

An obvious solution to this decision problem is simply to execute all the instantiations in parallel. However, this is not very realistic from a cognitive point or view. We assume a control that (i) uses unification , (ii) is basically forward driven and (iii) executes a model in a depth-first manner.

Let us continue tbe examples. As can be verified, our example models (1, 2 and 3) can conclude that the blue block is to the left or the white. Even though the models differ markedly , they can conclude the very same thing. It is trivial that two reasoners can make the same conclusion with or without the same understanding.

Suppose we give another goal. l1 the red block to the left of the green bloclcf Using a forward inference strategy or some combination, it can be verified that model I and 3 will succeed and conclude that it is true that the red block is to the left or the green block. The second model cannot resolve tbe goal and will conclude tbe negation or the statement by failure to prove. Naturally, this outcome means that the understanding is quite different. However, and most important, the inferences are all valid.

This discussion puts the issue about logic and reasoning in yet another perspective. For example, suppose that two subjects con­ struct isomorphic models. Their answers to abstract questions (e.g., I· there a relation between th.e red and th.e greell bloclcf) should then be isomorphic too. If they are not, we can conclude that at least one or them cannot be modelled by logic. However, it is not clear bow we can study such a perspective. Let us start by analyzing reasoning tasks that have been empirically studied.

3. Form and content In reaaonlng about a rule

A classical task in studies or deductive reasoning is the four­ card problem invented by Wason [8] and also extensively investigated by others. Several versions have heeo studied, but we Cocus on tbe rollowiog two.

You are presented with (our cards showing, respectively, A, D, 4, 7 , and you know Crom previous experience that every card, or wbicb these are a subset, bas a letter on one side and a number on tbe other side. You are then given this rule about the Cour cards in Crunt or you: If a card h.a1 a vowel Oil Olle oide, thell it ha1 an nell llamber on the other oide. You are told that the task is to tell wbicb or tbe cards tbaL need to be turned over in order to find out whether the rule is true or Calse. The most Crequent answers are "A and 4" and "only A". In one study only 5 out or 128 subjects gave tbe answer A and 7 which is regarded as the correct answer. [3].

In another version the objects are Cour envelopes: a sealed, an unsealed, an envelope with a 4p stamp, and an envelope with a 5p stamp. The task is to test the rule: If all erutelope io oeti/ed, th.ell it ha1 a 5p llamp Oil it. In one study, 22 out oC 25 subjects picked the sealed envelope and the envelope with a 4p stamp QD it , which is considNed to be the correct answer [3] . Thus, tbe results are in sharp contrast .

Ir the two tasks are isomorphic, the result might, on the surrace, indicate that human reasoners do not use rules or ioCerence, but only domain-specific operators. In tbe Collowing we try to show how the results could be interpreted in a logic modelling Cramework.

Let Card(x, 11) denote an object in tbe "card" task with x as tbe perceived symbol or value and with y as the hidden symbol. Let Env(x , y) be the corresponding proposition Cor tbe "envelope" task. Thus, we can represent the "card" task as composed or Cour rea­ soning objects , e.g. , Card( A, turn( A)). Tbe "envelope" task can be represented analogously, e.g., Env(Sealed, turn( Sealed)), where turn is a (unction that gives the hidden value in both cases.

Suppose instead that we represent a card as, Card(x, 1/, z), where x is the perceived value on the side 1/, that is shown, and z is the hidden value on the other side. Likewise, Env(x , 1/, z) could represent the objects in the "envelope" task . For example, Env(Sealed, Rever&e, turn( Sealed)) represents the envelope wbicb has its reverse side turned up. There is no ambiguity in this repre­ sentation since every reasoner knows that to see iC an envelope is sealed one has to get the reverse side. Similarly, starnps are always placed on the Crootside. Consider now tbe "card" task. Should we represent the A as Card(A, Front, turn( A)), Card( A, Back, turn(A)), Card( A, Front , turn( Front)), etc.! Thus, it is possible to argue that this representation shows that the two tasks do not share the same Corm since we would have,

In order to analyze the rules to be tested we choose the first representation which makes the tasks isomorphic. The rules in the tasks can be expressed as Collows,

The rules in (4) and (6) represent those cases in which tbe reasoner bas to think or possible hidden values. A reasoner who only chooses to turn A might have excluded (4) Crom his/her model. Ira reasoner chooses to introduce the Collowiog rule instead or (4) he/she will turn A and 4,

In the "envelope" task the relevant assumptions are more easily introduced because or the oatu'ral relation between the values or each object. In other words, knowledge about the domain Cacilitate tbe introduction or propositions. For example, an envelope can only be sealed or unsealed but not both. Ir there is a 5p_stamp it does not matter whether it is sealed or not. Thus, it is straight-Corward to recognize that iC it is not a 5p_stamp on it, then it must not be sealed, Env(x,u)-+ {-.(x = 5p-·tamp)-+ -.(11 =Sealed)} (8)

This assumption is equivalent to (6) but expressed as a cootraposi­ tive and it is straight-Corward to apply. In contrast, the cootrapositive to (4) is not easy to recognize. It might be easier to introduce th con· verse premise as shown in (7).

In short, the postulated isomorphism is not straight-forward, In general, the ambiguity in a reasoning task can trigger the construction of very different models. Inferences that on the surface seem to be invalid can be completely valid if the particular mental model from which they were derived is taken into account.

4- Towards the study of cognitive model construction

In this section we briefty present an attempt to study the reason­ ing process within the logic modelling framework . Also, we show bow the two processes, construction and execution, interact in a conlfruct - run - refine - cue/e.

Consider the following reasoning task !OJ,

If A p/41/1 4W41/ 4{14inrt B, A will/ore. If A p/41/1 4t h.ome 4{14inrt C, A will not lore. Now, A llill not p/4JI &W&J agaiut B. Wl4t en th.ell be 14itlV

Figure 1 shows a complete protocol given by a subject who bad never seen this task before and who had never studied logic.

The fragments are divided into three episodes corresponding to three reasoning attempts. Each episode terminates with an inference and the next episode starts with a smaller or larger revision of the earlier model.

In the first episode there are two inferences made,

In the first episode, S attends to the first and third premise (see F6-F8). The following propositional model is inferred from the pro­ tocol,

S excludes the second premise and introduces two assumptions (E13 and EH). The inference in F4 follows naturally from this model if it is executed forwardly.

• In #2, S concludes that "A should win if they played at home against B". This is the same inference as in the first episode. The model is not revised, but the fragments FlO and Fll indicate that S is beginnin« to revise it. ln #3 the model is modified,

S has changed his understanding of the task so that be now at­ tends to all the premises in the task. Also, be makes a case analysis (OR-elimination) in which two alternatives are tried. Let us give an example run of this model,

In step 5 and 7 we find the inferences that S makes in the third episode (Fl7 and FlO).

This brief analysis shows that a reasoning process can be viewed as interactions or constructions and executions. It also shows bow it is possible to study reasoning models !OJ.

6. Concluding remarks

In this paper we have presented elements or an approach to the study of human reasoning that is called logic modelling. We pro­ posed that a reasoning process is composed of two processes, called the construction and execution process. Knowledge and attentional mechanisms directs the construction of a propositional model. An acceptable model is executed within a logical control structure. The basics of this structure are unification, forward and backward infer­ encing, and rules of reasoning. A few examples have been put forward to illustrate some aspects or logic modellin,. The major arguments in the analyses are (i) that inferences should be evaluated against the models from which they were derived and {ii) that models seldom are one-to-one maps or external premises.

In short, we think that logic modelling is a framework which can contribute to the general study of human reasoning. However, as this paper has indicated, we need to study and analyze the processes or construction and execution in much more detail. For example, "when and why" are implicit assumptions introduced, "when and bow" are a model modified, "what" inference rules are used, "how" are reasoning rules designed, and so on.

G. Acknowledgments

We thank Sten-Ake Tarnlund for many valuable comments.

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