Infinitary logic and admissible sets

1969 ◽  
Vol 34 (2) ◽  
pp. 226-252 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Predicate Calculus ◽  
Generalized Quantifiers ◽  
Infinitary Logic ◽  
First Order ◽  
Admissible Sets ◽  
Classical Language ◽  
Second Order Logic ◽  
Order Predicate Calculus

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

Predicate calculus

2018 ◽  
Author(s):  
Timothy Smiley
Keyword(s):  
Propositional Calculus ◽  
Second Order ◽  
Order Logic ◽  
Predicate Calculus ◽  
First Order Logic ◽  
Complex Predicates ◽  
Modern Logic ◽  
Form Complex ◽  
First Order ◽  
Second Order Logic

The predicate calculus is the dominant system of modern logic, having displaced the traditional Aristotelian syllogistic logic that had been the previous paradigm. Like Aristotle’s, it is a logic of quantifiers – words like ‘every’, ‘some’ and ‘no’ that are used to express that a predicate applies universally or with some other distinctive kind of generality, for example ‘everyone is mortal’, ‘someone is mortal’, ‘no one is mortal’. The weakness of syllogistic logic was its inability to represent the structure of complex predicates. Thus it could not cope with argument patterns like ‘everything Fs and Gs, so everything Fs’. Nor could it cope with relations, because a logic of relations must be able to analyse cases where a quantifier is applied to a predicate that already contains one, as in ‘someone loves everyone’. Remedying the weakness required two major innovations. One was a logic of connectives – words like ‘and’, ‘or’ and ‘if’ that form complex sentences out of simpler ones. It is often studied as a distinct system: the propositional calculus. A proposition here is a true-or-false sentence and the guiding principle of propositional calculus is truth-functionality, meaning that the truth-value (truth or falsity) of a compound proposition is uniquely determined by the truth-values of its components. Its principal connectives are negation, conjunction, disjunction and a ‘material’ (that is, truth-functional) conditional. Truth-functionality makes it possible to compute the truth-values of propositions of arbitrary complexity in terms of their basic propositional constituents, and so develop the logic of tautology and tautological consequence (logical truth and consequence in virtue of the connectives). The other invention was the quantifier-variable notation. Variables are letters used to indicate things in an unspecific way; thus ‘x is mortal’ is read as predicating of an unspecified thing x what ‘Socrates is mortal’ predicates of Socrates. The connectives can now be used to form complex predicates as well as propositions, for example ‘x is human and x is mortal’; while different variables can be used in different places to express relational predicates, for example ‘x loves y’. The quantifier goes in front of the predicate it governs, with the relevant variable repeated beside it to indicate which positions are being generalized. These radical departures from the idiom of quantification in natural languages are needed to solve the further problem of ambiguity of scope. Compare, for example, the ambiguity of ‘someone loves everyone’ with the unambiguous alternative renderings ‘there is an x such that for every y, x loves y’ and ‘for every y, there is an x such that x loves y’. The result is a pattern of formal language based on a non-logical vocabulary of names of things and primitive predicates expressing properties and relations of things. The logical constants are the truth-functional connectives and the universal and existential quantifiers, plus a stock of variables construed as ranging over things. This is ‘the’ predicate calculus. A common option is to add the identity sign as a further logical constant, producing the predicate calculus with identity. The first modern logic of quantification, Frege’s of 1879, was designed to express generalizations not only about individual things but also about properties of individuals. It would nowadays be classified as a second-order logic, to distinguish it from the first-order logic described above. Second-order logic is much richer in expressive power than first-order logic, but at a price: first-order logic can be axiomatized, second-order logic cannot.

Categoricity and the sets

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
The Status ◽  
Second Order Logic ◽  
Philosophy Of Set Theory ◽  
Order Set

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order set theory. As in the previous chapter, we must ask who is entitled to invoke full second-order logic. That question is as subtle as before, and raises the same problem for moderate modelists. However, the quasi-categorical nature of Zermelo's Theorem gives rise to some specific questions concerning the aims of axiomatic set theories. Given the status of Zermelo's Theorem in the philosophy of set theory, we include a stand-alone proof of this theorem. We also prove a similar quasi-categoricity for Scott-Potter set theory, a theory which axiomatises the idea of an arbitrary stage of the iterative hierarchy.

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

scholarly journals Definability theorem for the intuitionistic predicate logic with equality

1982 ◽  
Vol 87 ◽  
pp. 1-15
Author(s):  
Chiharu Mizutani
Keyword(s):  
Predicate Logic ◽  
Second Order ◽  
Order Logic ◽  
Infinitary Logic ◽  
Second Order Logic

Svenonius’ definability theorem and its generalizations to the infinitary logic Lω1ω or to a second order logic with countable conjunctions and disjunctions have been studied by Kochen [1], Motohashi [2], [3] or Harnik and Makkai [4], independently. In this paper, we consider a (Svenonius-type) definability theorem for the intuitionistic predicate logic IL with equality.

scholarly journals First order quantifiers in monadic second order logic

2004 ◽  
Vol 69 (1) ◽  
pp. 118-136 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wafik Boulos Lotfallah
Keyword(s):  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Reachability Problem ◽  
Open Problems ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Quantifier Alternation

AbstractThis paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn (S) on properties S that says “there are n components having S”. We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn ⊈ FO(Σn), Σn ⊈ FO(∆n). and ∆n+1 ⊈ FOB(Σn), solving some open problems raised in [Mat98].

scholarly journals Strong Logics of First and Second Order

2010 ◽  
Vol 16 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Peter Koellner
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Close Correspondence ◽  
First Order ◽  
Second Order Logic

AbstractIn this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

A complete infinitary logic

1976 ◽  
Vol 41 (4) ◽  
pp. 730-746
Author(s):  
Kenneth Slonneger
Keyword(s):  
Natural Extension ◽  
Predicate Calculus ◽  
Cut Elimination ◽  
Infinitary Logic ◽  
Logical System ◽  
Natural Numbers ◽  
First Order ◽  
Ordinal Numbers ◽  
Power Set ◽  
Order Predicate Calculus

This paper is concerned with the proof theoretic development of certain infinite languages. These languages contain the usual infinite conjunctions and disjunctions, but in addition to homogeneous quantifiers such as ∀x0x1x2 … and ∃y0y1y2 …, we shall investigate particular subclasses of the dependent quantifiers described by Henkin [1]. The dependent quantifiers of Henkin can be thought of as partially ordered quantifiers defined by a function from one set to the power set of another set. This function assigns to each existentially bound variable, the set of universally bound variables on which it depends.The natural extension of Gentzen's first order predicate calculus to an infinite language with homogeneous quantifiers results in a system that is both valid and complete, and in which a cut elimination theorem can be proved [2]. The problem then arises of devising, if possible, a logical system dealing with general dependent quantifiers that is valid and complete. In this paper a system is presented that is valid and complete for an infinite language with homogeneous quantifiers and dependent quantifiers that are anti-well-ordered, for example, … ∀x2∃y2∀x1∃y1∀x0∃y0.Certain notational conventions will be employed in this paper. Greek letters will be used for ordinal numbers. The ordinal ω is the set of all natural numbers, and 2ω is the set of all ω -sequences of elements of 2 = {0,1}. The power set of S is denoted by P(S). μα[A(α)] stands for the smallest ordinal α such that A (α) holds.

scholarly journals Where First-Order and Monadic Second-Order Logic Coincide

2016 ◽  
Vol 17 (4) ◽  
pp. 1-18 ◽  
Author(s):  
Michael Elberfeld ◽  
Martin Grohe ◽  
Till Tantau
Keyword(s):  
Second Order ◽  
Order Logic ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic

scholarly journals Symbolic WS1S

10.29007/t28j ◽  
2018 ◽  
Author(s):  
Loris D'Antoni ◽  
Margus Veanes
Keyword(s):  
Blow Up ◽  
Finite Automata ◽  
Decision Procedures ◽  
Second Order ◽  
Order Logic ◽  
Symbolic Logic ◽  
First Order ◽  
Symbolic Automata ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We extend weak monadic second-order logic of one successor (WS1S) to symbolic alphabets byallowing character predicates to range over decidable first order theories and not just finite alphabets.We call this extension symbolic WS1S (s-WS1S). We then propose two decision procedures for such alogic: 1) we use symbolic automata to extend the classic reduction from WS1S to finite automata toour symbolic logic setting; 2) we show that every s-WS1S formula can be reduced to a WS1S formulathat preserves satisfiability, at the price of an exponential blow-up.

Sign in / Sign up

Export Citation Format

Share Document