Introduction Rules Research Articles

Reconciliation and Revival: James R. Mann and the House Republicans in the Wilson Era

Preface Introduction Rules Battles and Legislative Successes, November 1908-March 1911 Leading the Minority: Damage Control in the Sixty-second Congress, April 1911-March 1913 Starting Back, 1913 Election Year, 1914 The Politics of Peace and Preparedness, December 1914-June 1916 Election, 1916 War, 1917 Victory and Defeat, April 1917-March 1919 Aftermath, March 1919-November 1922, and in Retrospect Selected Bibliography Index

Structuring Resolution Proofs by Introducing New Lemmata

Resolution proofs are unstructured by their very nature, since they cannot use substantial lemmata. To impose structure on given resolution proofs and thereby improve their readability we will introduce new lemmata in a postprocessing step. As these lemmata cannot be generated by resolution, we will employ function introduction rules as presented by Baaz and Leitsch and give a correct and complete criterion for their applicability to the proofs. Applying function introduction rules to resolution proofs enables us to detect and eliminate certain homomorphic subproofs immune to the known redundancy elimination rules. For the cases when the criterion is satisfied we will characterize the transformation of tree resolution proofs to their condensation-reduced functional extensions, which may result in a double exponential reduction of proof height. Moreover, the proofs obtained by this transformation are more structured and hence more intelligible.

On paradox without self-reference

The basic idea behind that account is that we need to distinguish straightforward inconsistency, or self-contradiction, of a set of assumptions, from paradoxicality. Both involve proofs of absurdity (1). But the proofs of absurdity in connection with straightforward contradictions are normalizable, whereas those in connection with paradoxes are not. Proofs are normalizable when they can be brought into normal form by a finite sequence of applications of reduction procedures. These reduction procedures are designed to get rid of unnecessary prolixity. Such prolixity can arise, most importantly, by applying an introduction rule for a logical operator and then immediately applying the corresponding elimination rule. The result is a sentence occurrence within the proof standing as the conclusion of an application of the introduction rule and as the major premiss of an application of the corresponding elimination rule. Reductions get rid of such 'maximal' sentence occurrences, which stand as unwanted 'knuckles' in the proof. The reduction procedures for the logical operators are designed to eliminate such unnecessary detours within proofs.1

Natural deduction for intuitionistic linear logic

The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential!) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL +, is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL +, instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different introduction rule for the exponential, !I +, which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph “Natural Deduction”. It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL +; in particular one can formulate a notion of strong validity (as in Prawitz's “Ideas and Results in Proof Theory”) permitting a proof of strong normalization. The conversion rules for ILL explicitly mentioned in the paper by Benton et al. do not suffice for normal forms with subformula property, but we can show that this can be remedied by addition of a special permutation conversion plus some “satellite” permutation conversions. Some discussion of the categorical models which might correspond to ILL + is given.

Embedding a Second-Order Type System into an Intersection Type System

This paper presents the relationship between a second-order type assignment system T∀ and an intersection type assignment system T∧. First we define a translation tr from intersection types to second-order types. Then we define a system T∧* obtained from T∧ by restricting the use of the intersection type introduction rule, and show that T∧* and T∀ are equivalent in the following senses: (a) if a λ-term M has a type σ in T∧*, then M has the type tr(σ) in T∀; and conversely, (b) if M has a type T in T∀, then M has a type σ in T∧* such that tr(σ) is equivalent to T. These two theorems mean that T∀ is embedded into T∧.

Interaction Systems I: The theory of optimal reductions

We introduce a new class of higher order rewriting systems, called Interaction Systems (IS's). IS's are derived from Lafont's (Intuitionistic) Interaction Nets (Lafont 1990) by dropping the linearity constraint. In particular, we borrow from Interaction Nets the syntactical bipartitions of operators into constructors and destructors and the principle of binary interaction. As a consequence, IS's are a subclass of Klop's Combinatory Reduction Systems (Klop 1980), where the Curry-Howard analogy still ‘makes sense’. Destructors and constructors, respectively, correspond to left and right logical introduction rules: interaction is cut and reduction is cut-elimination.Interaction Systems have been primarily motivated by the necessity of extending the practice of optimal evaluators for λ-calculus (Lamping 1990; Gonthier et al. 1992a) to other computational constructs such as conditionals and recursion. In this paper we focus on the theoretical aspects of optimal reductions. In particular, we generalize the family relation in Lévy (1978; 1980), thus defining the amount of sharing an optimal evaluator is required to perform. We reinforce our notion of family by approaching it in two different ways (generalizing labelling and extraction in Levy (1980)) and proving their coincidence. The reader is referred to Asperti and Laneve (1993c) for the paradigmatic description of optimal evaluators of IS's.

Inductive families

AbstractA general formulation of inductive and recursive definitions in Martin-Löf's type theory is presented. It extends Backhouse's ‘Do-It-Yourself Type Theory’ to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalisation to type theory of Martin-Löf's theory of iterated inductive definitions in predicate logic.Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalised induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion.The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applications by showing several basic examples: predicate logic, generalised induction, and a formalisation of the untyped lambda calculus.

Intuitionistic relevant logic and perfect validity

In [3] Neil Tennant shows that the provable sequents of his system of Classical Relevant Logic (without a conditional) are exactly the substitution instances of classically perfectly valid sequents. A sequent is perfectly valid, relative to a notion of validity, if it is valid but ceases to be so on removal of any sentence that occurs in it as either a premiss or as conclusion. In [4] he conjectures that the analogous property holds of his Intuitionistic Relevant Logic ([4], p. 255). Here I shall refute that conjecture, suggest an amendment that circumvents the refutation, and indicate a further difficulty. In its sequent formulation Intuitionistic Relevant Logic (IR) is like standard intuitionist propositional logic save that the structural rules of Cut and Thinning (Dilution) are forsworn and left v-introduction is slightly strengthened. All sequents are of the form X : Y where X is a possibly empty set of sentences and Y is either empty or contains a single sentence. Below we indicate sets by listing their members, i.e. we omit set braces. The differences between IR in its natural deduction formulation and intuitionist propositional logic are more marked: the introduction and elimination rules, other than v-elimination, are those of intuitionist propositional logic without the rule of absurdity; to make up for that omission the rule of v-elimination is strengthened so as to allow derivation of disjunctive syllogism (modus tollendo ponens); all proofs must be in normal form and there can be no vacuous discharge of assumptions in -introduction and v-elimination. Tennant has shown that the sequent and natural deduction formulations are equivalent. (For further details see [4], pp. 185-200, 253-65.) I shall now derive in IR a sequent that is intuitionistically valid but neither itself intuitionistically perfectly valid nor a substitution instance of an intuitionistically perfectly valid sequent:

An Information System Interpretation of Martin-Löf′s Partial Type Theory with Universes

Martin-Löf′s partial type theory, i.e., type theory with a fixed point operator, is extended with universes and given a domain-theoretic interpretation using Scott′s information systems. The dependent types are interpreted as monotone families (parametrizations) of information systems and elements as relativized approximable mappings. Universes are interpreted as fixed points of the continuous operators (on the class of parametrizations) that are induced by the introduction rules for universes. Partial type theory has a rich type structure. In fact, as is shown elsewhere, Martin-Löf′s inconsistent type theory, Type : Type, is definable in it. However, in this paper we give a direct interpretation of (an extension of) Type : Type as an application of the domain-theoretic method of solving type universe equations introduced here.

Extending the Curry-Howard interpretation to linear, relevant and other resource logics

The so-called Curry-Howard interpretation (Curry [1934], Curry and Feys [1958], Howard [1969], Tait [1965]) is known to provide a rather neat term-functional account of intuitionistic implication. Could one refine the interpretation to obtain an almost as good account of other neighbouring implications, including the so-called ‘resource’ implications (e.g. linear, relevant, etc.)?We answer this question positively by demonstrating that just by working with side conditions on the rule of assertability conditions for the connective representing implication (‘→’) one can characterise those ‘resource’ logics. The idea stems from the realisation that whereas the elimination rule for conditionals (of which implication is a particular case) remains virtually unchanged no matter what kind of conditional one has (i.e. linear, relevant, intuitionistic, classical, etc., all have modus ponens), the corresponding introduction rule carries an element of vagueness which can be explored in the characterisation of several sorts of conditionals. The rule of →-introduction is classified as an ‘improper’ inference rule, to use a terminology from Prawitz [1965]. Now, the so-called improper rules leave room for manoeuvre as to how a particular logic can be obtained just by imposing conditions on the discharge of assumptions that would correspond to the particular logical discipline one is adopting (linear, relevant, ticket entailment, intuitionistic, classical, etc.). The side conditions can be ‘naturally’ imposed, given that a degree of ‘vagueness’ is introduced by the presentation of those improper inference rules, such as the rule of →-introduction:which says: starting from assumption ‘A’, and arriving at ‘B’ via an unspecified number of steps, one can discharge the assumption and conclude that ‘A’ implies ‘B’.

Complexity of resolution proofs and function introduction

The length of resolution proofs is investigated, relative to the model-theoretic measure of Herband complexity. A concept of resolution deduction (R-deduction) is introduced which is somewhat more general than the classical concepts. It is shown that proof complexity is exponential in terms of Herband complexity and that this bound is tight. The concept of R-deduction is extended to FR-deduction, where, besides resolution, a function introduction rule (based on a re-Skolemization technique) is allowed. As an example, consider the clause P(x, y) ∨ Q(x, y): conclude P(x,f(x)) ∨ Q(a, y), where a, f(x) are new function symbols extending Herbrand's universe. (We use the derivability of (∀x)(∃y) P(x, y) ∨ (∃x)(∀) Q(x, y) from (∀x)(∀y)( P(x, y) ∨ Q(x, y)) and Skolemization). By modification of a construction of Statman it is shown that FR-deductions may be nonelementarily shorter than R-deductions (thus giving a nonelementary speedup with respect to Herbrand complexity and search space). Finally it is proved that FR-deduction has weaker effects only (maximally double exponential speedup) if clausal logic is restricted to Horn logic.

A Proof-Theoretic Approach to Logic Programming

We introduce a definitional extension of logic programming by means of an inference schema (PV), which, in a certain sense, is dual to the (\-P) schema of rule application discussed in Part I. In the operational semantics, this schema is dual to the resolution principle. We prove soundness and completeness for the extended system, discuss the computation of substitutions that this new schema gives rise to, and also consider the notion of negation intrinsic to the system and its relation to negation by failure. 1. The definitional reading of logic programs When one writes a logic program, one intends to define somehow the predicates involved, i.e., to give them a meaning via those rules whose heads start with the predicates in question. This is reflected in the terminology used in several PROLOG textbooks and manuals, where the set of rules for a predicate p is called the 'definition' of p. The proof-theoretic approach can be used to obtain an extension of definite Horn clause programming, which makes this definitional reading of logic programs precise. Prooftheoretically, we can look at a program rule X^A as an introduction rule for A or as a clause in an inductive definition. The corresponding analogy is that in ordinary natural deduction, the introduction rules for a logical constant a can be viewed as rules giving meaning to a, whereas the elimination rule for a is uniquely determined by the introduction rules for a in the following sense: the elimination rule for a states that everything that can be derived from the premises of each introduction rule for a can be derived from the conclusion of these introduction rules (which is the same for all introduction rules). For example, in the case of a disjunction Ex v E2, This paper continues 'A Proof-Theoret ic Approach to Logic Programming. I. Clauses as Rules', which appeared in Vol. 1 No. 2 of this journal. In the following this first part is referred to as 'Part I'.

A sound base for decision-making

Community units have suffered from the problems of inadequate funding, lack of support and poor planning. Wendy King lays down the ground rules for the successful introduction of computers into the community setting

Proof systems for satisfiability in Hennessy-Milner Logic with recursion

An extension of Hennessy-Milner Logic with recursion is presented. A recursively specified formula will have two standard interpretations: a minimal one and a maximal one. Minimal interpretations of formulas are useful for expressing liveness properties of processes, whereas maximal interpretations are useful for expressing safety properties. We present sound and complete proof systems for both interpretations for when a process satisfies a (possibly recursive) formula. The rules of the proof systems consist of an introduction rule for each possible structure of a formula and are intended to extend the work of Stirling and Winskel. Moreover the proof systems may be presented directly in PROLOG to yield a decision procedure for verifying when a finite-state process satisfies a specification given as a (possibly recursive) formula.

A Proof-Theoretic Approach to Logic Programming. I. Clauses as Rules

In this paper definite Horn clause programs are investigated within a proof-theoreti c framework; program clauses being considered rules of a formal system. Based on this approach, the soundness and completeness of SLD-resolutio n is established by purely proof-theoretic methods. Extended Horn clauses are defined as rules of higher levels and related to an approach based on implication formulae in the bodies of clauses. In a further extension, which is treated in Part II of this series, program clauses are viewed as clauses in inductive definitions of atoms, justifying an additional inference schema: a reflection principle that roughly corresponds to interpreting the program clauses as introduction rules in the sense of natural deduction. The evaluation procedures for queries with respect to the defined extensions of definite Horn clause programs are shown to be sound and complete. The sequent calculus with the general elimination schema even permits the

A normalization theorem for set theory

In this paper we present a normalization theorem for a natural deduction formulation of Zermelo set theory. Our result gets around M. Crabbe's counterexample to normalizability (Hallnäs [3]) by adding an inference rule of the formand requiring that this rule be used wherever it is applicable. Alternatively, we can regard the result as pertaining to a modified notion of normalization, in which an inferenceis never considered reducible if A is T Є T, even if R is an elimination rule and the major premise of R is the conclusion of an introduction rule. A third alternative is to regard (1) as a derived rule: using the general well-foundedness rulewe can derive (1). If we regard (2) as neutral with respect to the normality of derivations (i.e., (2) counts as neither an introduction nor an elimination rule), then the resulting proofs are normal.

On a theory of weak implications

The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

On a theory of weak implications

The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

Logic and Meaning: The Philosophical Significance of the Sequent Calculus

In his doctoral dissertation of I935,1 Gentzen introduced two new approaches to the theory of proofs: natural deduction and the sequent calculus. He formulated both classical and intuitionist predicate logics using these approaches. For the resulting sequent calculi, he proved his Hauptsatz, or main theorem, which states that the rule of cut may be dispensed with in all proofs. The significance of this 'cut-elimination theorem' has been much debated over the past fifty years, but no clear consensus has arisen. In this paper, I will examine some of the ways in which this theorem has been interpreted. In particular, I will focus on the idea that the introduction rules for a logical constant in a sequent calculus may be taken as defining the constant or 'giving its meaning'; and I will consider the relation of cut-elimination to this thesis. My discussion will be illustrated by a specific example, drawing on recent work on truth and the semantic paradoxes.

Some consistency proofs and a characterization of inconsistency proofs in illative combinatory logic

It is well known that combinatory logic with unrestricted introduction and elimination rules for implication is inconsistent in the strong sense that an arbitrary term Y is provable. The simplest proof of this, now usually called Curry's paradox, involves for an arbitrary term Y, a term X defined by X = Y(CPy).The fact that X = PXY = X ⊃ Y is an essential part of the proof.The paradox can be avoided by placing restrictions on the implication introduction rule or on the axioms from which it can be proved.In this paper we determine the forms that must be taken by inconsistency proofs of systems of propositional calculus based on combinatory logic, with arbitrary restrictions on both the introduction and elimination rules for the connectives. Generally such a proof will involve terms without normal form and cut formulas, i.e. formulas formed by an introduction rule that are immediately removed by an elimination with at most some equality steps intervening. (Such a sequence of steps we call a “cut”.)The above applies not only to the strong form of inconsistency defined above, but also to the weak form of inconsistency defined by: all propositions are provable, if this can be represented in the system.Any inconsistency proof of this kind of system can be reduced to one where the major premise of the elimination rule involved in the cut and its deduction must also appear in the deduction of the minor premise involved in the cut.We can, using this characterization of inconsistency proofs, put appropriate restrictions on certain introduction rules so that the systems, including a full classical propositional one, become provably consistent.