Bases for first-order theories and subtheories

1960 ◽  
Vol 25 (2) ◽  
pp. 97-142 ◽  
Author(s):  
William Craig
Keyword(s):  
Axiom System ◽  
Logical Theory ◽  
Theoretical Terms ◽  
First Order ◽  
Right Turns

The extent to which we can grasp the content of a (non-logical) theory, expressing it economically by means of an axiom system or basis, varies greatly. In this paper we shall investigate what degree of economy, or at least regularity, can be achieved for all recursively axiomatizable first-order theories. A useful approach, also of interest in its own right, turns out to be the study of bases for subtheories, where a subtheory of a given theory consists of those theorems from which certain predicate symbols are absent. These predicate symbols might be thought of as the formal counterparts of the “purely theoretical” terms employed by a science, the theory corresponding to the science itself and the subtheory to its “observational consequences”. Roughly speaking, the types of operations involving such predicate symbols will be reduced to a minimum, so that their syntactical role in deductions will emerge more clearly.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

scholarly journals A Certain Kind of Formal Theories

1965 ◽  
Vol 25 ◽  
pp. 59-86 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Classical Logic ◽  
Axiom System ◽  
Logical System ◽  
First Order ◽  
Formal Theories

A common feature of formal theories is that each theory has its own system of axioms described in terms of some symbols for its primitive notions together with logical symbols. Each of these theories is developed by deduction from its axiom system in a certain logical system which is usually the classical logic of the first order.

On generalized quantifiers in arithmetic

1982 ◽  
Vol 47 (1) ◽  
pp. 187-190 ◽  
Author(s):  
Carl Morgenstern
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
First Order ◽  
Order Language ◽  
Truth Definition ◽  
Order Arithmetic ◽  
Infinite Set

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

A relative consistency proof

1954 ◽  
Vol 19 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Joseph R. Shoenfield
Keyword(s):  
Set Theory ◽  
Functional Calculus ◽  
Direct Method ◽  
Axiom System ◽  
Similar Process ◽  
Consistency Proof ◽  
First Order ◽  
Definition Of ◽  
A Direct Method ◽  
Exact Definition

LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. (An exact definition ofC′ will be given later.) Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. (The converse is obvious.) Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. We could, of course, obtain such a contradiction by proving the theorems ofCone by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem inCfrom its proof inC′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.The advantage of the proof may be stated more specifically by arithmetizing the syntax ofCandC′.

A Logical Theory of Truth-Makers and Falsity-Makers

2018 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Logical Consequence ◽  
Suitable Method ◽  
Elimination Rule ◽  
Logical Theory ◽  
Relative Truth ◽  
Logical Operator ◽  
First Order ◽  
Introduction Rule ◽  
Infinite Case ◽  
Ways Of Being

We explicate the different ways that a first-order sentence can be true (resp., false) in a model M, as formal objects, called (M-relative) truth-makers (resp., falsity-makers). M-relative truth-makers and falsity-makers are co-inductively definable, by appeal to the “atomic facts” in M, and to certain rules of verification and of falsification, collectively called rules of evaluation. Each logical operator has a rule of verification, much like an introduction rule; and a rule of falsification, much like an elimination rule. Applications of the rules (∀) and (∃) involve infinite furcation when the domain of M is infinite. But even in the infinite case, truth-makers and falsity-makers are tree-like objects whose branches are at most finitely long. A sentence φ is true (resp., false) in a model M (in the sense of Tarski) if and only if there existsπ such that π is an M-relative truth-maker (resp., falsity-maker) for φ. With “ways of being true” explicated as these logical truthmakers, one can re-conceive logical consequence between given premises and a conclusion. It obtains just in case there is a suitable method for transforming M-relative truthmakers for the premises into an M-relative truthmaker for the conclusion, whatever the model M may be.

scholarly journals Proving Semantic Properties as First-Order Satisfiability (Extended Abstract)

2020 ◽  
Author(s):  
Salvador Lucas
Keyword(s):  
Logic Programming ◽  
Programming Languages ◽  
Answer Set Programming ◽  
Deductive Databases ◽  
Query Answering ◽  
Logical Theory ◽  
Data Bases ◽  
First Order ◽  
Computational Systems ◽  
Semantic Properties

The semantics of computational systems (e.g., relational and knowledge data bases, query-answering systems, programming languages, etc.) can often be expressed as (the specification of) a logical theory Th. Queries, goals, and claims about the behavior or features of the system can be expressed as formulas φ which should be checked with respect to the intended model of Th, which is often huge or even incomputable. In this paper we show how to prove such semantic properties φ of Th by just finding a model A of Th∪{φ}∪Zφ, where Zφ is an appropriate (possibly empty) theory depending on φ only. Applications to relational and deductive databases, rewriting-based systems, logic programming, and answer set programming are discussed.

Gödel’s Proof Based on ω-Consistency

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Predicate Symbol ◽  
Order Theory ◽  
Modus Ponens ◽  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Underlying Assumption ◽  
First Order ◽  
First Order Theory

The proof that we have just given of the incompleteness of Peano Arithmetic was based on the underlying assumption that Peano Arithmetic is correct—i.e., that every sentence provable in P.A. is a true sentence. Gödel’s original incompleteness proof involved a much weaker assumption—that of ω-consistency to which we now turn. We consider an arbitrary axiom system S whose formulas are those of Peano Arithmetic, whose axioms include all those of Groups I and II (or alternatively, any set of axioms for first-order logic with identity such that all logically valid formulas are provable from them), and whose inference rules are modus ponens and generalization. (It is also possible to axiomatize first-order logic in such a way that modus ponens is the only inference rule—cf. Quine [1940].) In place of the axioms of Groups III and IV, however, we can take a completely arbitrary set of axioms. Such a system S is an example of what is termed a first-order theory, and we will consider several such theories other than Peano Arithmetic. (For the more general notion of a first-order theory, the key difference is that we do not necessarily start with + and × as the undefined function symbols, nor do we necessarily take ≤ as the undefined predicate symbol. Arbitrary function symbols and predicate symbols can be taken, however, as the undefined function and predicate symbols—cf. Tarski [1953] for details. However, the only theories (or “systems”, as we will call them) that we will have occasion to consider are those whose formulas are those of P.A.) S is called simply consistent (or just “consistent” for short) if no sentence is both provable and refutable in S.

The Incompleteness of Peano Arithmetic with Exponentiation

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
First Order ◽  
Axiom Scheme

We shall now turn to a formal axiom system which we call Peano Arithmetic with Exponentiation and which we abbreviate “P.E.”. We take certain correct formulas which we call axioms and provide two inference rules that enable us to prove new correct formulas from correct formulas already proved. The axioms will be infinite in number, but each axiom will be of one of nineteen easily recognizable forms; these forms are called axiom schemes. It will be convenient to classify these nineteen axiom schemes into four groups (cf. discussion that follows the display of the schemes). The axioms of Groups I and II are the so-called logical axioms and constitute a neat formalization of first-order logic with identity due to Kalish and Montague [1965], which is based on an earlier system due to Tarski [1965]. The axioms of Groups III and IV are the so-called arithmetic axioms. In displaying these axiom schemes, F, G and H are any formulas, vi and vj are any variables, and t is any term. For example, the first scheme L1 means that for any formulas F and G, the formula (F ⊃ (G ⊃ F)) is to be taken as an axiom; axiom scheme L4 means that for any variable Vi and any formulas F and G, the formula . . . (∀vi (F ⊃ G) ⊃ (∀vi (F ⊃ ∀vi G) . . . is to be taken as an axiom.

The completeness of a predicate-functor logic

1985 ◽  
Vol 50 (4) ◽  
pp. 903-926 ◽  
Author(s):  
John Bacon
Keyword(s):  
Predicate Logic ◽  
Axiom System ◽  
Combinatory Logic ◽  
Completeness Proof ◽  
First Order ◽  
Full Theory ◽  
History Of ◽  
Quantificational Logic ◽  
Term Logic ◽  
Individual Variables

Predicate-functor logic, as founded by W. V. Quine ([1960], [1971], [1976], [1981]), is first-order predicate logic without individual variables. Instead, adverbs or predicate functors make explicit the permutations and replications of argument-places familiarly indicated by shifting variables about. For the history of this approach, see Quine [1971, 309ff.]. With the evaporation of variables, individual constants naturally assimilate to singleton predicates or adverbs, leaving no logical subjects whatever of type 0. The orphaned “predicates” may then be taken simply as terms in the sense of traditional logic: class and relational terms on model-theoretic semantics, schematic terms on Quine's denotational or truth-of semantics. Predicate-functor logic thus stands forth as the pre-eminent first-order term logic, as distinct from propositional-quantificational logic. By the same token, it might with some justification qualify as “first-order combinatory logic”, with allowance for some categorization of the sort eschewed in general combinatory logic, the ultimate term logic.Over the years, Quine has put forward various choices of primitive predicate functors for first-order logic with or without the full theory of identity. Moreover, he has provided translations of quantificational into predicate-functor notation and vice versa ([1971, 312f.], [1981, 651]). Such a translation does not of itself establish semantic completeness, however, in the absence of a proof that it preserves deducibility.An axiomatization of predicate-functor logic was first published by Kuhn [1980], using primitives rather like Quine's. As Kuhn noted, “The axioms and rules have been chosen to facilitate the completeness proof” [1980, 153]. While this expedient simplifies the proof, however, it limits the depth of analysis afforded by the axioms and rules. Mindful of this problem, Kuhn ([1981] and [1983]) boils his axiom system down considerably, correcting certain minor slips in the original paper.

The Logic of Constructivism

Disputatio ◽  
2002 ◽  
Vol 1 (11) ◽  
pp. 37-42 ◽  
Author(s):  
Gustavo Fernández Díez
Keyword(s):  
Predicate Logic ◽  
Philosophy Of Mathematics ◽  
Axiom System ◽  
Current View ◽  
Logical System ◽  
First Order ◽  
Common Basis ◽  
The Status ◽  
Logical Principle ◽  
The Common

Abstract In this paper I dispute the current view that intuitionistic logic is the common basis for the three main trends of constructivism in the philosophy of mathematics: intuitionism, Russian constructivism and Bishop’s constructivism. The point is that the so-called ‘Markov’s principle’, which is accepted by Russian constructivists and rejected by the other two, is expressible in intuitionistic first-order logic, and so it appears to have the status of a logical principle. The result of appending this principle to a complete intuitionistic axiom system for first-order predicate logic constitutes a new logic, which could well be called ‘Markov’s logic’, and which should be regarded as the true logical system underlying Russian constructivism.

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