Natural deduction and arbitrary objects

Journal of Philosophical Logic -     

Natural deduction rules for English

A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as the c, a (an) c, some c, any c, and every c, where c represents a common noun. These latter expressions are called quantifiers, and other expressions of the form that c or that c itself, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), and the, and also for any which, every which, and so on, as well as rules for some other concepts. One outcome of these rules is that Every man loves some woman is implied by, but does not imply, Some woman is loved by every man, since the latter is taken to mean the same as Some woman is loved by all men. Also, Jack knows which woman came is implied by Some woman is known by Jack to have come, but not by Jack knows that some woman came.     

Tools for connectionist modeling: The dynamical systems methodology

Some understanding of dynamical systems is essential to achieving competency in connectionist models. This mathematical background can be acquired either through a rigorous set of upper undergraduate and/or graduate formal courses or via disciplined self-teaching. As part of developing a course in connectionism, we feel that although certain very basic mathematical tools are most appropriately learned in their “pure” form (i.e., from mathematics textbooks and courses), more advanced exposure to dynamical systems theory can be given in the context of an introduction to connectionism. Students thus learn to write connectionist simulations by first writing programs for simulating arbitrary dynamical systems, then using them to learn some aspects of dynamical systems in general by simulating some special cases, and finally applying this technique to connectionist models of increasing complexity.     

A deduction theorem schema for deductive systems of propositional logics

We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) L and p, A(p, q) _s q.     

Sensitivity analysis in connectionist modeling

The application of sensitivity analysis methodology to connectionist modeling is discussed. Sensitivity analysis examines the functional relationship of parameters in dynamic systems to specify the interactions of parameters and the boundary conditions of dynamic system models. Sensitivity analysis is contrasted with parameter estimation. Reasons to favor sensitivity analysis in connectionist modeling are discussed. Six steps for sensitivity analysis are detailed with examples from the modeling of short- and long-term memory retrieval.     

Catastrophic forgetting in connectionist networks

All natural cognitive systems, and, in particular, our own, gradually forget previously learned information. Plausible models of human cognition should therefore exhibit similar patterns of gradual forgetting of old information as new information is acquired. Only rarely does new learning in natural cognitive systems completely disrupt or erase previously learned information; that is, natural cognitive systems do not, in general, forget ‘catastrophically’. Unfortunately, though, catastrophic forgetting does occur under certain circumstances in distributed connectionist networks. The very features that give these networks their remarkable abilities to generalize, to function in the presence of degraded input, and so on, are found to be the root cause of catastrophic forgetting. The challenge in this field is to discover how to keep the advantages of distributed connectionist networks while avoiding the problem of catastrophic forgetting. In this article the causes, consequences and numerous solutions to the problem of catastrophic forgetting in neural networks are examined. The review will consider how the brain might have overcome this problem and will also explore the consequences of this solution for distributed connectionist networks.     

Relevant deduction

This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between relevance and validity. It is argued that irrelevant arguments are, although formally valid, nonsensical and even harmful in practical applications. The basic idea is this: a valid deduction is relevant iff no subformula of the conclusion is replaceable on some of its occurrences by any other formula salva validitate of the deduction. The paper first motivates the approach by showing that four paradoxes seemingly very distant from each other have a common source. Then the exact definition of relevant deduction is given and its logical properties are investigated. An extension to relevance of premises is discussed. Finally the paper presents an overview of its applications in philosophy of science, ethics, cognitive psychology and artificial intelligence.This paper is based on the first part of my habilitation Schurz (1989). For various helps and comments I am indebted to Paul Weingartner. Andrzej Wrónski, Georg Kreisel, David Miller, Kit Fine, Terence Parsons and Peter Woodruff.     

Parameterization of connectionist models

We present a method for estimating parameters of connectionist models that allows the model’s output to fit as closely as possible to empirical data. The method minimizes a cost function that measures the difference between statistics computed from the model’s output and statistics computed from the subjects’ performance. An optimization algorithm finds the values of the parameters that minimize the value of this cost function. The cost function also indicates whether the model’s statistics are significantly different from the data’s. In some cases, the method can find the optimal parameters automatically. In others, the method may facilitate the manual search for optimal parameters. The method has been implemented in Matlab, is fully documented, and is available for free download from the Psychonomic Society Web archive atwww.psychonomic.org/archive/.     

Logic programs and connectionist networks

One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic programs can be computed by feedforward connectionist networks, and that the same semantic operators for first-order normal logic programs can be approximated by feedforward connectionist networks. Turning the networks into recurrent ones allows one also to approximate the models associated with the semantic operators. Our methods depend on a well-known theorem of Funahashi, and necessitate the study of when Funahashi's theorem can be applied, and also the study of what means of approximation are appropriate and significant.     

Harmonising natural deduction

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented.     

Parameterization of connectionist models.

We present a method for estimating parameters of connectionist models that allows the model's output to fit as closely as possible to empirical data. The method minimizes a cost function that measures the difference between statistics computed from the model's output and statistics computed from the subjects' performance. An optimization algorithm finds the values of the parameters that minimize the value of this cost function. The cost function also indicates whether the model's statistics are significantly different from the data's. In some cases, the method can find the optimal parameters automatically. In others, the method may facilitate the manual search for optimal parameters. The method has been implemented in Matlab, is fully documented, and is available for free download from the Psychonomic Society Web archive at www.psychonomic.org/archive/.     

Aspects of analytic deduction

Let ? be the ordinary deduction relation of classical first-order logic. We provide an “analytic” subrelation ?³ of ? which for propositional logic is defined by the usual “containment” criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq Atom(\Gamma ),$$ whereas for predicate logic, ?^a is defined by the extended criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq ' Atom(\Gamma ),$$ where Atom(?) $ \subseteq '$ Atom(Γ) means that every atomic formula occurring in ? “essentially occurs” also in Γ. If Γ, ? are quantifier-free, then the notions “occurs” and “essentially occurs” for atoms between Γ and ? coincide. If ? is formalized by Gentzen's calculus of sequents, then we show that ?^a is axiomatizable by a proper fragment of analytic inference rules. This is mainly due to cut elimination. By “analytic inference rule” we understand here a rule r such that, if the sequent over the line is analytic, then so is the sequent under the line. We also discuss the notion of semantic relevance as contrasted to the previous syntactic one. We show that when introducing semantic sequents as axioms, i.e. when extending the pure logical axioms and rules by mathematical ones, the property of syntactic relevance is lost, since cut elimination no longer holds. We conclude that no purely syntactic notion of analytic deduction can ever replace successfully the complex semantico-syntactic deduction we already possess.