Towards a Model Theory of Venn Diagrams

1996 ◽  
Author(s):  
Eric Hammer ◽  
Norman Danner
Keyword(s):  
Deductive Reasoning ◽  
Logical Consequence ◽  
Formal System ◽  
Semantic Content ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Venn Diagrams ◽  
Linguistic Representations ◽  
Finite Set

One of the goals of logical analysis is to construct mathematical models of various practices of deductive inference. Traditionally, this is done by means of giving semantics and rules of inference for carefully specified formal languages. While this has proved to be an extremely fruitful line of analysis, some facets of actual inference are not accurately modeled by these techniques. The example we have in mind concerns the diversity of types of external representations employed in actual deductive reasoning. Besides language, these include diagrams, charts, tables, graphs, and so on. When the semantic content of such non-linguistic representations is made clear, they can be used in perfectly rigorous proofs. A simple example of this is the use of Venn diagrams in deductive reasoning. If used correctly, valid inferences can be made with these diagrams, and if used incorrectly, they can be the source of invalid inferences; there are standards for their correct use. To analyze such standards, one might construct a formal system of Venn diagrams where the syntax, rules of inference, and notion of logical consequence have all been made precise and explicit, as is done in the case of first-order logic. In this chapter, we will study such a system of Venn diagrams, a variation of Shin’s system VENN formulated and studied in Shin [1991] and Shin [1991a] (see Chapter IV of this book). Shin proves a soundness theorem and a finite completeness theorem (if ∆ is a finite set of diagrams, D is a diagram, and D is a logical consequence of ∆ , then D is provable from ∆ ). We extend Shin’s completeness theorem to the general case: if ∆ is any set of diagrams, D is a, diagram, and D is a logical consequence of ∆. then D is provable from ∆. We hope that the fairly simple diagrammatic system discussed here will help motivate closer study of the use of more complicated diagrams in actual inference.

A weak completeness theorem for infinite valued first-order logic

1963 ◽  
Vol 28 (1) ◽  
pp. 43-50 ◽  
Author(s):  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Unit Interval ◽  
Order Logic ◽  
First Order Logic ◽  
Order System ◽  
First Order ◽  
Recursively Enumerable ◽  
Weak Completeness ◽  
Truth Value

This paper contains some results concerning the completeness of a first-order system of infinite valued logicThere are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

A Weak Completeness Theorem for Infinite Valued First-Order Logic.

1971 ◽  
Vol 36 (2) ◽  
pp. 332
Author(s):  
Bruno Scarpellini ◽  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Weak Completeness

Proof theory

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Completeness Theorem ◽  
Formal Proof ◽  
Order Logic ◽  
Truth Table ◽  
First Order Logic ◽  
Formal Proofs ◽  
Systematic Method ◽  
First Order

As with any logic, the semantics of first-order logic yield rules for deducing the truth of one sentence from that of another. In this chapter, we develop both formal proofs and resolution for first-order logic. As in propositional logic, each of these provides a systematic method for proving that one sentence is a consequence of another. Recall the Consequence problem for propositional logic. Given formulas F and G, the problemis to decide whether or not G is a consequence of F. From Chapter 1, we have three approaches to this problem: • We could compute the truth table for the formula F → G. If the truth values are all 1s then we conclude that F → G is a tautology and G is a consequence of F. Otherwise, G is not a consequence of F. • Using Tables 1.5 and 1.6, we could try to formally derive G from {F}. By the Completeness Theorem for propositional logic, G is a consequence of F if and only if {F} ├ G. • We could use resolution. By Theorem1.76, G is a consequence of F if and only if ∅ ∈ Res(H) where H is a formula in CNF equivalent to (F ∧￢G). Using these methods not only can we determine whether one formula is a consequence of another, but also we can determine whether a given formula is a tautology or a contradiction. A formula F is a tautology if and only if F is a consequence of (A∨￢A) if and only if ￢F is a contradiction. In this chapter, we consider the analogous problems for first-order logic. Given formulas φ and ψ, how can we determine whether ψ is a consequence of φ? Equivalently, how can we determine whether a given formula is a tautology or a contradiction? We present three methods for answering these questions. • In Section 3.1, we define a notion of formal proof for first-order logic by extending Table 1.5. • In Section 3.3, we “reduce” formulas of first-order logic to sets of formulas of propositional logic where we use resolution as defined in Chapter 1.

Prerequisites

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Numbers ◽  
Logical Validity ◽  
First Order ◽  
Gödel’S Completeness Theorem ◽  
Logical Connectives ◽  
Incompleteness Theorems ◽  
Individual Variables

As we remarked in the preface, although this volume is a sequel to our earlier volume G.I.T. (Gödel’s Incompleteness Theorems), it can be read independently by those readers familiar with at least one proof of Gödel’s first incompleteness theorem. In this chapter we give the notation, terminology and main results of G.I.T. that are needed for this volume. Readers familiar with G.I.T. can skip this chapter or perhaps glance through it briefly as a refresher. §0. Preliminaries. we assume the reader to be familiar with the basic notions of first-order logic—the logical connectives, quantifiers, terms, formulas, free and bound occurrences of variables, the notion of interpretations (or models), truth under an interpretation, logical validity (truth under all interpretations), provability (in some complete system of first-order logic with identity) and its equivalence to logical validity (Gödel’s completeness theorem). we let S be a system (theory) couched in the language of first-order logic with identity and with predicate and/or function symbols and with names for the natural numbers. A system S is usually presented by taking some standard axiomatization of first-order logic with identity and adding other axioms called the non-logical axioms of S.we associate with each natural number n an expression n̅ of S called the numeral designating n (or the name of n).we could, for example, take 0̅,1̅,2̅, . . . ,to be the expressions 0,0', 0",..., as we did in G.I.T. we have our individual variables arranged in some fixed infinite sequence v1, v2,..., vn , . . . . By F(v1, ..., vn) we mean any formula whose free variables are all among v1,... ,vn, and for any (natural) numbers k1,...,kn by F(к̅1 ,... к̅n), we mean the result of substituting the numerals к̅1 ,... к̅n, for all free occurrences of v1,... ,vn in F respectively.

Ranked partial structures

2003 ◽  
Vol 68 (4) ◽  
pp. 1109-1144
Author(s):  
Timothy J. Carlson
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
Basic Theory ◽  
First Order Logic ◽  
Theoretic Method ◽  
First Order ◽  
Partial Structures ◽  
Skolem Theorem

AbstractThe theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.

scholarly journals A proof of completeness for continuous first-order logic

2010 ◽  
Vol 75 (1) ◽  
pp. 168-190 ◽  
Author(s):  
Itaï Ben Yaacov ◽  
Arthur Paul Pedersen
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Strong Completeness ◽  
First Order ◽  
Completeness Result ◽  
Metric Structures ◽  
Natural Classes

AbstractContinuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies anapproximatedform of strong completeness, whereby Σ⊧φ(if and) only if Σ⊢φ∸2−nfor alln < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth ofφ.Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theoryTisdecidableif for every sentenceφ, the valueφTis a recursive real, and moreover, uniformly computable fromφ. IfTis incomplete, we say it is decidable if for every sentenceφthe real numberφTois uniformly recursive fromφ, whereφTois the maximal value ofφconsistent withT. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

A Logic for Frege’s Theorem

2020 ◽  
pp. 24-54
Author(s):  
Richard Kimberly Heck
Keyword(s):  
Formal System ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Second Order Logic ◽  
Special Case

It has been known for a few years that no more than Π‎1 1 ∆1 3 comprehension is needed for the proof of “Frege’s Theorem.” One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, Delta-3-1 comprehension axioms are not logical truths. What I suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called “ancestral logic” is developed, and it is then extended to yield what I call “Arché logic.”

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