The Scandal of Deduction

This article provides the first comprehensive reconstruction and analysis of Hintikka’s attempt to obtain a measure of the information yield of deductive inferences. The reconstruction is detailed by necessity due to the originality of Hintikka’s contribution. The analysis will turn out to be destructive. It dismisses Hintikka’s distinction between surface information and depth information as being of any utility towards obtaining a measure of the information yield of deductive inferences. Hintikka is right to identify the failure of canonical information theory to give an account of the information yield of deductions as a scandal, however this article demonstrates that his attempt to provide such an account fails. It fails primarily because it applies to only a restricted set of deductions in the polyadic predicate calculus, and fails to apply at all to the deductions in the monadic predicate calculus and the propositional calculus. Some corollaries of these facts are a number of undesirable and counterintuitive results concerning the proposed relation of linguistic meaning (and hence synonymy) with surface information. Some of these results will be seen to contradict Hintikka’s stated aims, whilst others are seen to be false. The consequence is that the problem of obtaining a measure of the information yield of deductive inferences remains an open one. The failure of Hintikka’s proposal will suggest that a purely syntactic approach to the problem be abandoned in favour of an intrinsically semantic one.     

The Logical Deduction of Doctrine

The idea that Roman Catholic doctrines for which there is no early testimony can be explained as logical deductions from undoubtedly early teachings is usually dismissed as obviously false. By invoking the logical properties of doctrines expressed as explicit generalizations, however, and by distinguishing deductions in which all the assumptions represent Apostolic doctrine from those in which all the doctrinal assumptions are Apostolic, a way is found to deduce the disputed doctrines while leaving the immutability of doctrine intact. Although a theory of theological development is thus not needed to justify doctrinal additions, developments in theology nevertheless often motivate the authoritative pronouncements cited by doctrinal deductions. Finally, it is argued that a correct understanding of such deductions improves the prospects for reunion between those whose doctrinal axioms coincide even if differing historical information has rendered them incapable of following the same chain of deductions.     

Deduction without awareness

We investigated whether two basic forms of deductive inference, Modus Ponens and Disjunctive Syllogism, occur automatically and without awareness. In Experiment 1, we used a priming paradigm with a set of conditional and disjunctive problems. For each trial, two premises were shown. The second premise was presented at a rate designed to be undetectable. After each problem, participants had to evaluate whether a newly-presented target number was odd or even. The target number matched or did not match a conclusion endorsed by the two previous premises. We found that when the target matched the conclusion of a Modus Ponens inference, the evaluation of the target number was reliably faster than baseline even when participants reported that they were not aware of the second premise. This priming effect did not occur for any other valid or invalid inference that we tested, including the Disjunctive Syllogism. In Experiment 2, we used a forced-choice paradigm in which we found that some participants were able to access some information on the second premise when their attention was explicitly directed to it. In Experiment 3, we showed that the priming effect for Modus Ponens was present also in subjects who could not access any information about P(2). In Experiment 4 we explored whether spatial relations (e.g., "a before b") or sentences with quantifiers (e.g., "all a with b") could generate a priming effect similar to the one observed for Modus Ponens. A priming effect could be found for Modus Ponens only, but not for the other relations tested. These findings show that the Modus Ponens inference, in contrast to other deductive inferences, can be carried out automatically and unconsciously. Furthermore, our findings suggest that critical deductive inference schemata can be included in the range of high-level cognitive activities that are carried out unconsciously.     

Contextual Deduction Theorems

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT??a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case.     

Models of Deduction*

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful. *Since the text of this talk was written in 1999, the author has published several papers about related matters (see ‘Identity of proofs based on normalization and generality’, The Bulletin of Symbolic Logic 9 (2003), 477–503, corrected version available at: http://arXiv.org/math.LO/0208094; other titles are available in the same archive).     

The Second Step of the B‐Deduction

This paper offers a new interpretation of Kant's puzzling claim that the B‐Deduction in the Critique of Pure Reason should be considered as having two main steps. Previous commentators have tended to agree in general on the first step as arguing for the necessity of the categories for possible experience, but disagree on what the second step is and whether Kant even needs a second step. I argue that the two parts of the B‐Deduction correspond to the two aspects of a priori cognition: necessity and universality. The bulk of the paper consists of support for the second step, the universality of the categories. I show that Kant's arguments in the second half of the B‐Deduction aim to define the scope of that universality for possible experience by considering the possibilities of divine intellectual intuition, of non‐human kinds of sensible intuition, and of apperception of the self. In these ways Kant delimits the boundaries of the applicability of the categories and excludes any other possible experience for human beings.     

A Double Deduction System for Quantum Logic Based On Natural Deduction

The author presents a deduction system for Quantum Logic. This system is a combination of a natural deduction system and rules based on the relation of compatibility. This relation is the logical correspondant of the commutativity of observables in Quantum Mechanics or perpendicularity in Hilbert spaces.Contrary to the system proposed by Gibbins and Cutland, the natural deduction part of the system is pure: no algebraic artefact is added. The rules of the system are the rules of Classical Natural Deduction in which is added a control of contexts using the compatibility relation.The author uses his system to prove the following theorem: if propositions of a quantum logical propositional calculus system are mutually compatible, they form a classical subsystem.     

Deduction from Uncertain Premises

We investigate how the perceived uncertainty of a conditional affects a person's choice of conclusion. We use a novel procedure to introduce uncertainty by manipulating the conditional probability of the consequent given the antecedent. In Experiment 1, we show first that subjects reduce their choice of valid conclusions when a conditional is followed by an additional premise that makes the major premise uncertain. In this we replicate Byrne (1989). These subjects choose, instead, a qualified conclusion expressing uncertainty. If subjects are given a third statement that qualifies the likelihood of the additional premise, then the uncertainty of the conclusions they choose is systematically related to the suggested uncertainty. Experiment 2 confirms these observations in problems that omit the additional premise and qualify the first premise directly. Experiment 3 shows that the qualifying statement also affects the perceived probability of the consequent given the antecedent of the conditional. Experiment 4 investigates the effect of suggested uncertainty on the fallacies and shows that increases in uncertainty reduce the number of certain conclusions that are chosen while affirming the consequent but have no effect on denying the antecedent. We discuss our results in terms of rule theories and mental models and conclude that the latter give the most natural account of our results.     

Deduction from if-then personality signatures

Personality signatures are sets of if-then rules describing how a given person would feel or act in a specific situation. These rules can be used as the major premise of a deductive argument, but they are mostly processed for social cognition purposes; and this common usage is likely to leak into the way they are processed in a deductive reasoning context. It is hypothesised that agreement with a Modus Ponens argument featuring a personality signature as its major premise is affected by the reasoner's own propensity to display this personality signature. To test this prediction, Modus Ponens arguments were constructed from conditionally phrased items extracted from available personality scales. This allowed recording of (a) agreement with the conclusion of these arguments, and (b) the reasoner's propensity to display the personality signature, using as a proxy this reasoner's score on the personality scale without the items used in the argument. Three experiments (N = 256, N = 318, N = 298) applied this procedure to Fairness, Responsive Joy, and Self-Control. These experiments yielded very comparable effects, establishing that a reasoner's propensity to display a given personality signature determines this reasoner's agreement with the conclusion of a Modus Ponens argument featuring the personality signature.     

Deduction chains for common knowledge

Deduction chains represent a syntactic and in a certain sense constructive method for proving completeness of a formal system. Given a formula , the deduction chains of are built up by systematically decomposing into its subformulae. In the case where is a valid formula, the decomposition yields a (usually cut-free) proof of . If is not valid, the decomposition produces a countermodel for . In the current paper, we extend this technique to a semiformal system for the Logic of Common Knowledge. The presence of fixed point constructs in this logic leads to potentially infinite-length deduction chains of a non-valid formula, in which case fairness of decomposition requires special attention. An adequate order of decomposition also plays an important role in the reconstruction of the proof of a valid formula from the set of its deduction chains.     

Deduction,inference and illation

From the standpoint of the theory of medicine, a formulation is given of three types of reasoning used by physicians. The first is deduction from probability models (as in prognosis or genetic counseling for Mendelian disorders). It is a branch of mathematics that leads to predictive statements about outcomes of individual events in terms of known formal assumptions and parameters. The second type is inference (as in interpreting clinical trials). In it the arguments from replications of the same process (‘data’) lead to conclusions about the parameters of a system, without calling into question either the probabilistic model or the criteria of evidence. The third is illation (as in the elucidation of symptoms in a patient). It is a process whereby, in the light of the total evidence and the conclusions from the other types of reasoning, one may modify, expand, simplify or demolish a conceptual framework proposed for deductions, and modify the nature of the evidence sought, the criteriology, the axioms, and the surmised complexity of the scientific theory. (The process of diagnosis as applied to a patient may in extreme cases lead to the discovery of an entirely new disease with its own, quite new, set of diagnostic criteria. This course cannot be accommodated inside either of the other two types of reasoning.) Illation has something of the character of Kuhn's ‘scientific revolution’ in physics; but it differs in that it is the nature, not the degree or frequency of change that distinguishes it from Kuhn's ‘normal science.’     

Natural Deduction for Dual-intuitionistic Logic

We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.     

Natural Deduction and Curry's Paradox

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.     

Self-understanding in Kant's Transcendental Deduction

I argue that §§15–20 of the B-Deduction contain two independent arguments for the applicability of a priori concepts, the first an argument from above, the second an argument from below. The core of the first argument is §16's explanation of our consciousness of subject-identity across self-attributions, while the focus of the second is §18's account of universality and necessity in our experience. I conclude that the B-Deduction comprises powerful strategies for establishing its intended conclusion, and that some assistance from empirical psychology might well have produced a completely successful argument.