The Content of Deduction

For deductive reasoning to be justified, it must be guaranteed to preserve truth from premises to conclusion; and for it to be useful to us, it must be capable of informing us of something. How can we capture this notion of information content, whilst respecting the fact that the content of the premises, if true, already secures the truth of the conclusion? This is the problem I address here. I begin by considering and rejecting several accounts of informational content. I then develop an account on which informational contents are indeterminate in their membership. This allows there to be cases in which it is indeterminate whether a given deduction is informative. Nevertheless, on the picture I present, there are determinate cases of informative (and determinate cases of uninformative) inferences. I argue that the model I offer is the best way for an account of content to respect the meaning of the logical constants and the inference rules associated with them without collapsing into a classical picture of content, unable to account for informative deductive inferences.     

On the emergence of the discriminative mode for transitive-inference

Transitive-inference tasks play increasing roles in many areas of cognitive science. However, discrepancies between the two competing paradigms (classical-Piagetian and IP-paradigm) imply that one paradigm fails to index its target (deductive) competence. Here, potential flaws of one were addressed using considerations from the other. Children of 5–7 years (N=216) solved 3-term transitive problems, some implying a single largest item, and some not. Findings indicate a discriminative competence (a Transitive-Switch) at least 2 years before competence in actually deducing correct solutions. Interestingly, although the transitive-switch was constant across age with both memory and inference increasing with age, it was nevertheless a much stronger predictor of transitive performance than either memory or age. A predictive relationship between memory and inference held for the 5- and 6-year-olds but not the 7-year-olds; suggesting as it approaches maturity, transitive-inference becomes independent of memory. Coincidental with this latter finding, only the 7-year-olds’ transitive performance approximated a deductive competence. As the initial validation of the IP-paradigm rested on acceptance of its age estimate of 4 years, its claim to index deduction must now be revisited.     

Monotonicity in Practical Reasoning

Classic deductive logic entails that once a conclusion is sustained by a valid argument, the argument can never be invalidated, no matter how many new premises are added. This derived property of deductive reasoning is known as monotonicity. Monotonicity is thought to conflict with the defeasibility of reasoning in natural language, where the discovery of new information often leads us to reject conclusions that we once accepted. This perceived failure of monotonic reasoning to observe the defeasibility of natural-language arguments has led some philosophers to abandon deduction itself (!), often in favor of new, non-monotonic systems of inference known as `default logics'. But these radical logics (e.g., Ray Reiter's default logic) introduce their desired defeasibility at the expense of other, equally important intuitions about natural-language reasoning. And, as a matter of fact, if we recognize that monotonicity is a property of the form of a deductive argument and not its content (i.e., the claims in the premise(s) and conclusion), we can see how the common-sense notion of defeasibility can actually be captured by a purely deductive system.     

社会认知中归纳推论－演绎推论的不对称性现象

在社会知觉中,人们会较多的根据个体的行为表现推论其所具有的人格特质（归纳推论）,较少根据人格特质来推论个体的行为表现（演绎推论）,Maass将这一现象命名为归纳推论－演绎推论的不对称性,并在西方文化下验证了这一现象。刻板印象在这一不对称性现象中起着调节作用,行为信息与刻板印象是否一致影响了归纳推论,对与刻板印象一致的信息更易做出归纳推论,而对演绎推论没有影响。在不同的文化背景下,归纳推论－演绎推论的不对称性表现存在差异,东方文化下的个体对两种推论过程没有表现出明显的倾向性。本文介绍了这一现象的表现,并总结了产生这一社会现象的内在原因及相关的研究进展。     

Peirce and the autonomy of abductive reasoning

Essential to Peirce's distinction among three kinds of reasoning, deduction, induction and abduction, is the claim that each is correlated to a unique species of validity irreducible to that of the others. In particular, abductive validity cannot be analyzed in either deductive or inductive terms, a consequence of considerable importance for the logical and epistemological scrutiny of scientific methods. But when the full structure of abductive argumentation — as viewed by the mature Peirce — is clarified, every inferential step in the process can be seen to dissolve into familiar forms of deductive and inductive reasoning. Specifically, the final stage is a special type of practical inference which, if correct, is deductively valid, while the creative phase, surprisingly, is not inferential at all. In neither is abduction a type of inference to the best explanation. The result is a major reassessment of the relevance of Peirce's views to contemporary methodological studies.     

Proof-Theoretic Semantics, Self-Contradiction, and the Format of Deductive Reasoning

From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a crucial advantage over natural deduction, where substitution is built into the general framework.     

The effect of negation on deductive inferences

Research shows that negation can suppress the activation of propositions presented explicitly in text, but does negation have a similar effect on propositions that can be inferred? That is, does negation inhibit the inference process? Four experiments investigated whether a deductive inference that produces a negated conclusion (therefore not a) is made as readily as a similar inference form that yields an affirmative conclusion (therefore a). A combination of naming latencies, verification times, and reading times indicate that negation does not affect the deductive inference process itself, although it may inhibit the activation of inferred concepts.     

Belief in reasoning

Reasoning has three types, deduction, induction, and abduction, of which we perceive deduction to be necessarily true, induction plausibly true, and abduction only hypothetically true. Syllogistic is a theory of deductive reasoning, introducing three figures of inferencing, of which figure-1 is obviously true, figure-3 and figure-2 are increasingly less transparent. We argue that the three figures of syllogistic and the three types of reasoning are related and their truth perceptions can be explained as different degrees of belief. We suggest that the source of this difference can be found in the conversion of a premise required by syllogistic processing. Experimental results illustrating our theory are included.     

Getting one step closer to deduction: Introducing an alternative paradigm for transitive inference

Transitive inference is claimed to be “deductive”. Yet every group/species ever reported apparently uses it. We asked 58 adults to solve five-term transitive tasks, requiring neither training nor premise learning. A computer-based procedure ensured all premises were continually visible. Response accuracy and RT (non-discriminative nRT) were measured as is typically done. We also measured RT confined to correct responses (cRT). Overall, very few typical transitive phenomena emerged. The symbolic distance effect never extended to premise recall and was not at all evident for nRT; suggesting the use of non-deductive end-anchor strategies. For overall performance, and particularly the critical B?D inference, our findings indicate that deductive transitive inference is far more intellectually challenging than previously thought. Contrasts of our present findings against previous findings suggest at least two distinct transitive inference modes, with most research and most computational models to date targeting an associative mode rather than their desired deductive mode. This conclusion fits well with the growing number of theories embracing a “dual process” conception of reasoning. Finally, our differing findings for nRT versus cRT suggest that researchers should give closer consideration to matching the RT measure they use to the particular conception of transitive inference they pre-held.     

The epistemic significance of valid inference

The traditional picture of logic takes it for granted that ??valid arguments have a fundamental epistemic significance??, but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments and acts of inferences and that we have to reconsider the latter notion to arrive at the desired explanation. More precisely, the notions should be developed so that the following relationship holds: one gets in possession of a ground for a conclusion by inferring it from premisses for which one already has grounds, provided that the inference in question is valid. The paper proposes explications of the concepts of ground and deductively valid inference so that this relationship holds as a conceptual truth. Logical validity of inference is seen as a special case of deductive validity, but does not add anything as far as epistemic significance is concerned??it resides already in the deductively valid inferences.     

Demonstrative Induction and the Skeleton of Inference

It has been common wisdom for centuries that scientific inference cannot be deductive; if it is inference at all, it must be a distinctive kind of inductive inference. According to demonstrative theories of induction, however, important scientific inferences are not inductive in the sense of requiring ampliative inference rules at all. Rather, they are deductive inferences with sufficiently strong premises. General considerations about inferences suffice to show that there is no difference in justification between an inference construed demonstratively or ampliatively. The inductive risk may be shouldered by premises or rules, but it cannot be shirked. Demonstrative theories of induction might, nevertheless, better describe scientific practice. And there may be good methodological reasons for constructing our inferences one way rather than the other. By exploring the limits of these possible advantages, I argue that scientific inference is neither of essence deductive nor of essence inductive.     

Why Is a Valid Inference a Good Inference?

True beliefs and truth‐preserving inferences are, in some sense, good beliefs and good inferences. When an inference is valid though, it is not merely truth‐preserving, but truth‐preserving in all cases. This motivates my question: I consider a Modus Ponens inference, and I ask what its validity in particular contributes to the explanation of why the inference is, in any sense, a good inference. I consider the question under three different definitions of ‘case’, and hence of ‘validity’: (i) the orthodox definition given in terms of interpretations or models, (ii) a metaphysical definition given in terms of possible worlds, and (iii) a substitutional definition defended by Quine. I argue that the orthodox notion is poorly suited to explain what's good about a Modus Ponens inference. I argue that there is something good that is explained by a certain kind of truth across possible worlds, but the explanation is not provided by metaphysical validity in particular; nothing of value is explained by truth across all possible worlds. Finally, I argue that the substitutional notion of validity allows us to correctly explain what is good about a valid inference.     

Epistemic closure and commutative, nonassociative residuated structures

K-axiom-based epistemic closure for explicit knowledge is rejected for even the most trivial cases of deductive inferential reasoning on account of the fact that the closure axiom does not extend beyond a raw consequence relation. The recognition that deductive inference concerns interaction as much as it concerns consequence allows for perspectives from logics of multi-agent information flow to be refocused onto mono-agent deductive reasoning. Instead of modeling the information flow between different agents in a communicative or announcement setting, we model the information flow between different states of a single agent as that agent reasons deductively. The resource management of the database of agent states for the deductive reasoning fragment in question is covered by the residuated structure that encodes the nonassociative Lambek Calculus with permutation, bottom, and identity: NLP ₀₁ .     

Mathematical Method and Proof

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.     

Expressive Power and Incompleteness of Propositional Logics

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this.     

Generalized Logical Consequence: Making Room for Induction in the Logic of Science

We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures in contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented.     

Analyticity and Justification in Frege

That there are analytic truths may challenge a principle of the homogeneity of truth. Unlike standard conceptions, in which analyticity is couched in terms of “truth in virtue of meanings”, Frege’s notions of analytic and a priori concern justification, respecting a principle of the homogeneity of truth. Where there is no justification these notions do not apply, Frege insists. Basic truths and axioms may be analytic (or a priori), though unprovable, which means there is a form of justification which is not (deductive) proof. This is also required for regarding singular factual propositions as a posteriori. A Fregean direction for explicating this wider notion of justification is suggested in terms of his notion of sense (Sinn)—modes in which what the axioms are about are given—and its general epistemological significance is sketched.     

The preservation of coherence

It is argued that the preservation of truth by an inference relation is of little interest when premiss sets are contradictory. The notion of a level of coherence is introduced and the utility of modal logics in the semantic representation of sets of higher coherence levels is noted. It is shown that this representative role cannot be transferred to first order logic via frame theory since the modal formulae expressing coherence level restrictions are not first order definable. Finally, an inference relation, calledyielding, is introduced which is intermediate between the coherence preservingforcing relation introduced elsewhere by the authors and the coherence destroying, inference relation of classical logic.     

WHEN DEDUCTION LEADS TO BELIEF

The paper questions the common assumption that rational individuals believe all propositions which they know to be logical consequences of their other beliefs: although we must acknowledge the truth of a proposition which is a deductive consequence of our beliefs, we may not genuinely believe it. This conclusion is defended by arguing that some familiar counterexamples to the claim that knowledge is justified true belief fail because they involve propositions which are not really believed. Beliefs guide conduct or issue in assertion by answering questions which arise in the course of deliberation and conversation, but the troublesome cases present propositions which do not present the agent's answer to any question. The paper concludes by sketching the conditions under which the deductive consequences of our beliefs can be believed.¹     

Methodological Patterns In A Structuralist Setting

A new approach to analyze scientific methods as patternsof state transitions is proposed and exemplified by the two mostimportant, general methods: induction and deduction. Though only`local' states of science are considered in this paper, includinghypotheses, data, approximation and degree of fit, the approach caneasily be extended to more comprehensive kinds of states. Two `pure'forms of induction are distinguished, enumerative and hypothesisconstruction induction. A combination of these two forms is proposedto yield a more adequate picture of induction. While the pure forms ofinduction are clearly distinct from the deductive pattern, the patternof the combined form of induction is very similar to the latter. Thepresent account of scientific methods not only points out thedifferences between different methods but – in contrast to usualdiscussions of methodology – also clarifies what they have in common.