An ACT-R Model of Syllogistic Inference

We describe SYLLOG, a model of syllogistic inference, built with in the ACT-R framework. Its construction was guided by data obtained from 88 subjects performing the syllogistic inference task. The model’s inference engine uses the PC inference rule Hypothetical Syllogism (HS) and a set of modal logic transformation rules either to create a representation of the premises that allows an appropriate use of HS, or to check the appropriateness of having already used HS to obtain a putative conclusion. SYLLOG models the task of generating a necessarily true conclusion for the 64 pairs of categorical syllogistic premises (four “moods” for the first premise x four “moods” for the second premise x 4 figures). It can be described as comprising two stages. The initial one produces an interpretation of the input given to the model. This interpretative phase creates a declarative blueprint for building a production corresponding to each of the premises. This interpretation represents quantification over possible worlds instead of over objects. For instance, for “Some As are Bs”, a procedural interpretation would, if the top goal were a represented object with property A, modify this goal chunk to represent that there would be a possible world or circumstance in which the corresponding object would have both property A and property B. The second stage is the inference engine of the system and it searches the problem space for a valid conclusion. In this stage, SYLLOG uses the information yielded at the last declarative level of the interpretation process of the premises. These declarative structures contain the __________ 1 E.g. “all As are Bs” “Some As are Bs” “No As are B” and “Some As are not Bs”. 2 The “figure” of a syllogistic problem refers to the position of the terms in the pair of premises. If the terms were A, B and C, we would have the four figures AB-CB, BA-BC, AB-BC and BA-CB. 3 SYLLOG does compile a production that instantiates the instruction conveyed by the premise: a Dependency is created, which, when popped, yields the corresponding production. Having a premise rendered as a Dependency signals that the interpretation is complete and prompts SYLLOG to continue to the next step in the process of solving the syllogistic problem. However, Only the declarative blueprint for the premise’s interpretation is used necessary information to compile productions corresponding to each of the quantification “moods”. This information can be notated as follows, using modal operators (M for the Possibility operator and L for the Necessity operator): (1) Lj (A(xj) ⊃ B(xj)) (All As are Bs) (2) Lj (A(xj) ⊃ Mj B(xj)) (Some As are Bs) (3) Lj (A(xj) ⊃ ~ B(xj)) (No As are Bs) (4) Lj (A(xj) ⊃ Mj ~ B(xj)) (Some As are not Bs) Our use of modal operators is guided by the work of Venema & Marx (1999), on the modalization of first-order logic. The semantics for the modal formalism offers a far more interesting ground to explore from the psychological point of view than the standard model-theoretic construal of quantification in first order logic. Some of the subjects’ verbalizations suggest that alternative states of affairs are being considered while interpreting a syllogistic problem, and it is common to encounter the use of expressions such as “can be” or “may be”, as in “an A may be C”, for which the model-theoretical approach in natural language semantics also proposes a rendering that uses modal logic. Our current approach yields a correlation coefficient r=0.814 among the predicted and empirical difficulty rankings of the 27 syllogistic problems that support valid conclusions. From now on we will refer to (1) simply as A ⊃ B, and simplify (2)-(4) accordingly. We will assume that SYLLOG is dealing with a pair of premises of the figure AB-BC. We will consider the other figures under the heading “premise conversion”. Generating Putative Conclusions The task of producing a conclusion for a syllogistic problem is taken by SYLLOG as corresponding to building a blueprint for a production that integrates the condition of the interpreted first premise with the action of the interpreted second premise. The validity of such a conclusion depends on finding representations for the two premises corresponding to productions that will, in all circumstances, fire sequentially. Thus, SYLLOG attempts to transform the representations of the premises until they support an interpretation in which applying the instruction conveyed by the first premise always yields a chunk that always from this point on, because it is available for inspection and modifications, unlike the corresponding production. requires the firing of the production that interprets the second premise. This amounts to transforming the representation of the premises until the use of the PC rule Hypothetical Syllogism is allowed. In fact, HS is the sole inference rule upon which SYLLOG’s inference engine is based. (5) A ⊃ B B ⊃ C ______ A ⊃ C When dealing with the first premise while trying to generate a valid conclusion, SYLLOG may identify as obstacles to the use of HS the operators in the consequent of each of the three implications: (6) A ⊃ Mj B (7) A ⊃ ~ B (8) A ⊃ Mj ~ B For each of these, SYLLOG uses one of the following equivalences, namely (9) (valid) to deal with (6), either (10) (invalid) or (11) (valid) to deal with (7), and any combination (9)/(10) or (9)/(11) to deal with (8). (9) (A ⊃ Mj B) ≡ Mj (A ⊃ B) (10) (A ⊃ ~ B) ≡ ~ (A ⊃ B) (11) (A ⊃ ~ B) ≡ (A ⊃ (B ⊃ ~ (A & B)) Rebutting Putative Conclusions SYLLOG’s use of HS may be mislead. For instance, in the case of the pair “All of the As are Bs. Some of the Bs are Cs”, the immediate use of HS upon A ⊃ B and B ⊃ Mj C to obtain A ⊃ Mj C (Some of the As are Cs) doesn’t provide a valid conclusion: in the set of circumstances compatible with the premises, the conclusion A ⊃ Mj C is possible, but not necessarily true. It fails to be true if none of the Bs that are As is also C, which is a circumstance compatible with both premises. This contingency becomes apparent if the second premise is represented as an implication that holds only in some possible worlds, Mj (B ⊃ C). The representation of the second premise may therefore be reconstructed in a way that blocks the use of HS, meaning that the production that interprets the second premise won’t necessarily fire in sequence with the production that interprets the first premise. Negation may be treated in a similar manner. If ~ (B ⊃ C) is obtained by using the invalid equivalence (10), the putative conclusion under scrutiny is also rejected, although erroneously. If (11) is used instead, disposal of the putative conclusion is __________ 4 This invalid transformation allows SYLLOG to generate most of the invalid conclusions found in the data set for problems in which the first premise contains a negation. 5 This equivalence is valid in the direction Mj (A ⊃ B) ⊃ (A ⊃ Mj B), used for creating a structure from which SYLLOG can read a conclusion after applying HS, if a transformation in the opposite direction has previously taken place. This is always the case in SYLLOG. considered, but eventually rejected. When used to check the validity of a putative conclusion, the productions that instantiate all of the above transformations have to compete with the productions that read off final conclusions. This makes them more unlikely to fire in this conclusion rebuttal stage then when they are used for creating a context for the use of HS, by transforming the representation of the first premise. This asymmetry follows closely the answer patterns found in the data set. Premise Conversion The use of HS requires that the other three syllogistic figures besides AB-BC are dealt with by some premise conversion procedure that yields the AB-BC figure. SYLLOG most simple conversion procedure amounts to directly swapping the terms of the premise. This yields two invalid conversions, namely. “All of the As are Bs” being converted in “All of the Bs are As”, and “Some of the As are not Bs” being converted in “Some of the Bs are not As”. A subsequent, more sophisticated, conversion procedure may be applied. If reached, this stage involves “noticing” that the new structure is not true in all possible worlds, but only in those where the previous conditional had been true. To yield a valid outcome, SYLLOG has to signal this limitation. It does so by introducing a Possibility operator. This further procedure may then rectify the conversion of A ⊃ B in B ⊃ A, which would now turn into B ⊃ Mj A. It also allows the detection of the impossibility of a valid conversion of A ⊃ Mj ~ B and introduces a still valid, but weaker version of the previously valid conversion of A ⊃ ~ B, B ⊃ Mj ~ A. The stronger conversion B ⊃ ~ A may still be recovered, if another production notices that the correction is not called for in this instance. Otherwise, the conclusions that SYLLOG will derive from the converted premise will reflect the loss introduced by the conversion.