bluuebunny/crossref_metadata_2025_split huggingface dataset:
Tense and temporal logic
Authors: Smith Quentin 
Published: September 2018

A special kind of logic is needed to represent the valid kinds of arguments involving tensed sentences. The first significant presentation of a tense logic appeared in Prior (1957). Sentential tense logic, in its simplest form, adds to classical sentential logic two tense operators, P and F. The basic idea is to analyse past and future tenses in terms of prefixes ‘It was true that’ and ‘It will be true that’, attached to present-tensed sentences. (Present-tensed sentences do not need present tense operators, since ‘It is true that Jane is walking’ is equivalent to ‘Jane is walking’.) Translating the symbols into English is merely a preliminary to a semantics for tense logic; we may translate ‘P’ as ‘it was true that’ but we still have the question of the meaning of ‘it was true that’. There are at least two versions of the tensed theory of time – the minimalist version and the maximalist version – that can be used for the interpretation of the tense logic symbols.
The minimalist version implies that there are no past or future particulars, and thus no things or events that have properties of pastness or futurity. What exists are the things, with their properties and relations, that can be mentioned in certain present-tensed sentences. If ‘Jane is walking’ is true, then there is a thing, Jane, which possesses the property of walking. ‘Socrates was discoursing’, even if true, does not contain a name that refers to a past thing, Socrates, since there are no past things. The ontological commitments of past and future tensed sentences are merely to propositions, which are sentence-like abstract objects that are the meanings or senses of sentences. ‘Socrates was discoursing’ merely commits us to the proposition expressed by the sentence ‘It was true that Socrates is discoursing’.
The maximalist tensed theory of time, by contrast, implies that there are past, present and future things and events; that past items possess the property of pastness, present items possess the property of presentness, and future items possess the property of being future. ‘Socrates was discoursing’ involves a reference to a past thing, Socrates, and implies that the event of Socrates discoursing has the property of being past.
Deontic logic
Authors: Belzer Marvin 
Published: September 2018

Deontic logic is the investigation of the logic of normative concepts, especially obligation (‘ought’, ‘should’, ‘must’), permission (‘may’) and prohibition (‘ought not’, ‘forbidden’). Deontic logic differs from normative legal theory and ethics in that it does not attempt to determine which principles hold, nor what obligations exist, for any given system. Rather it seeks to develop a formal language that can adequately represent the normative expressions of natural languages, and to regiment such expressions in a logical system.
The theorems of deontic logic specify relationships both among normative concepts (for example, whatever is obligatory is permissible) and between normative and non-normative concepts (for example, whatever is obligatory is possible). Contemporary research beginning with von Wright treats deontic logic as a branch of modal logic, in so far as (as was noted already by medieval logicians) the logical relations between the obligatory, permissible and forbidden to some extent parallel those between the necessary, possible and impossible (concepts treated in ‘alethic’ modal logic).
Second- and higher-order logics
Authors: Lavine Shaughan 
Published: September 2018

In first-order predicate logic there are symbols for fixed individuals, relations and functions on a given universe of individuals and there are variables ranging over the individuals, with associated quantifiers. Second-order logic adds variables ranging over relations and functions on the universe of individuals, and associated quantifiers, which are called second-order variables and quantifiers. Sometimes one also adds symbols for fixed higher-order relations and functions among and on the relations, functions and individuals of the original universe. One can add third-order variables ranging over relations and functions among and on the relations, functions and individuals on the universe, with associated quantifiers, and so on, to yield logics of even higher order. It is usual to use proof systems for higher-order logics (that is, logics beyond first-order) that include analogues of the first-order quantifier rules for all quantifiers.
An extensional n-ary relation variable in effect ranges over arbitrary sets of n-tuples of members of the universe. (Functions are omitted here for simplicity: remarks about them parallel those for relations.) If the set of sets of n-tuples of members of a universe is fully determined once the universe itself is given, then the truth-values of sentences involving second-order quantifiers are determined in a structure like the ones used for first-order logic. However, if the notion of the set of all sets of n-tuples of members of a universe is specified in terms of some theory about sets or relations, then the universe of a structure must be supplemented by specifications of the domains of the various higher-order variables. No matter what theory one adopts, there are infinitely many choices for such domains compatible with the theory over any infinite universe. This casts doubt on the apparent clarity of the notion of ‘all n-ary relations on a domain’: since the notion cannot be defined categorically in terms of the domain using any theory whatsoever, how could it be well-determined?
Intensional logics
Authors: Garson James W. 
Published: September 2018

Intensional logics are systems that distinguish an expression’s intension (roughly, its sense or meaning) from its extension (reference, denotation). The purpose of bringing intensions into logic is to explain the logical behaviour of so-called intensional expressions. Intensional expressions create contexts which violate a cluster of standard principles of logic, the most notable of which is the law of substitution of identities – the law that from a = b and P(a) it follows that P(b). For example, ‘obviously’ is intensional because the following instance of the law of substitution is invalid (at least on one reading): Scott = the author of Waverley; obviously Scott = Scott; so, obviously Scott = the author of Waverley. By providing an analysis of meaning, intensional logics attempt to explain the logical behaviour of expressions such as ‘obviously’. On the assumption that it is intensions and not extensions which matter in intensional contexts, the failure of substitution and related anomalies can be understood.
Alonzo Church pioneered intensional logic, basing it on his theory of types. However, the widespread application of intensional logic to linguistics and philosophy began with the work of Richard Montague, who crafted a number of systems designed to capture the expressive power of natural languages. One important feature of Montague’s work was the application of possible worlds semantics to the analysis of intensional logic. The most difficult problems concerning intensional logic concern the treatment of propositional attitude verbs, such as ‘believes’, ‘desires’ and ‘knows’. Such expressions pose difficulties for the possible worlds treatment, and have thus spawned alternative approaches.
Provability logic
Authors: Visser Albert 
Published: September 2018

Central to Gödel’s second incompleteness theorem is his discovery that, in a sense, a formal system can talk about itself. Provability logic is a branch of modal logic specifically directed at exploring this phenomenon. Consider a sufficiently rich formal theory T. By Gödel’s methods we can construct a predicate in the language of T representing the predicate ‘is formally provable in T’. It turns out that T is able to prove statements of the form
- (1) If A is provable in T, then it is provable in T that A is provable in T.
In modal logic, predicates such as ‘it is unavoidable that’ or ‘I know that’ are considered as modal operators, that is, as non-truth-functional propositional connectives. In provability logic, ‘is provable in T’ is similarly treated. We write □A for ‘A is provable in T’. This enables us to rephrase (1) as follows:
- (1′) □A →□□A.
This is a well-known modal principle amenable to study by the methods of modal logic.
Provability logic produces manageable systems of modal logic precisely describing all modal principles for □A that T itself can prove. The language of the modal system will be different from the language of the system T under study. Thus the provability logic of T (that is, the insights T has about its own provability predicate as far as visible in the modal language) is decidable and can be studied by finitistic methods. T, in contrast, is highly undecidable. The advantages of provability logic are: (1) it yields a very perspicuous representation of certain arguments in a formal theory T about provability in T; (2) it gives us a great deal of control of the principles for provability in so far as these can be formulated in the modal language at all; (3) it gives us a direct way to compare notions such as knowledge with the notion of formal provability; and (4) it is a fully worked-out syntactic approach to necessity in the sense of Quine.
Definition
Authors: Antonelli G. Aldo 
Published: September 2018

A definition is a statement, declaration or proposal establishing the meaning of an expression. In virtue of the definition, the expression being defined (the ‘definiendum’) is to acquire the same meaning as the expression in terms of which it is defined (the ‘definiens’). For example, ‘Man is a rational animal’ determines the meaning of the term ‘man’ by making it synonymous with ‘rational animal’. Classical theory maintains that a good definition captures the ‘real nature’ of what is defined: ‘A ‘‘definition’’ is a phrase signifying a thing’s essence’ (Aristotle). Historically, philosophers have come to distinguish these ‘real’ definitions from ‘nominal’ definitions that specify the meaning of a linguistic expression rather than signify the essential nature of an object, ‘making another understand by Words, what Idea, the term defined stands for’ (Locke).
A further distinction can be drawn between contextual or implicit definitions, on the one hand, and explicit definitions, on the other. Often a definition fixes meaning directly and explicitly: for example, the definition of a proper name might well take the form of an explicit identity statement (‘Pegasus = the winged horse’) and a definition of a predicate is usually given (or can be re-cast) in the form of an equivalence (‘For every x: x is a man if and only if x is a rational animal’). But sometimes the meaning of a term is specified in context, by way of the meaning of larger expressions in which the term occurs. A paradigmatic example of this is Bertrand Russell’s analysis of the meaning of the definite article.
Hilbert’s programme and formalism
Authors: Detlefsen Michael 
Published: September 2018

In the first, geometric stage of Hilbert’s formalism, his view was that a system of axioms does not express truths particular to a given subject matter but rather expresses a network of logical relations that can (and, ideally, will) be common to other subject matters.
The formalism of Hilbert’s arithmetical period extended this view by emptying even the logical terms of contentual meaning. They were treated purely as ideal elements whose purpose was to secure a simple and perspicuous logic for arithmetical reasoning – specifically, a logic preserving the classical patterns of logical inference. Hilbert believed, however, that the use of ideal elements should not lead to inconsistencies. He thus undertook to prove the consistency of ideal arithmetic with its contentual or finitary counterpart and to do so by purely finitary means.
In this, ‘Hilbert’s programme’, Hilbert and his followers were unsuccessful. Work published by Kurt Gödel in 1931 suggested that such failure was perhaps inevitable. In his second incompleteness theorem, Gödel showed that for any consistent formal axiomatic system T strong enough to formalize what was traditionally regarded as finitary reasoning, it is possible to define a sentence that expresses the consistency of T, and is not provable in T. From this it has generally been concluded that the consistency of even the ideal arithmetic of the natural numbers is not finitarily provable and that Hilbert’s programme must therefore fail.
Despite problematic elements in this reasoning, post-Gödelian work on Hilbert’s programme has generally accepted it and attempted to minimize its effects by proposing various modifications of Hilbert’s programme. These have generally taken one of three forms: attempts to extend Hilbert’s finitism to stronger constructivist bases capable of proving more than is provable by strictly finitary means; attempts to show that for a significant family of ideal systems there are ways of ‘reducing’ their consistency problems to those of theories possessing more elementary (if not altogether finitary) justifications; and attempts by the so-called ‘reverse mathematics’ school to show that the traditionally identified ideal theories do not need to be as strong as they are in order to serve their mathematical purposes. They can therefore be reduced to weaker theories whose consistency problems are more amenable to constructivist (indeed, finitist) treatment.
Realism in the philosophy of mathematics
Authors: Blanchette Patricia A. 
Published: September 2018

Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. As the realist sees it, mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create. These form the subject matter of mathematical discourse: a mathematical statement is true just in case it accurately describes the mathematical facts.
An important form of mathematical realism is mathematical Platonism, the view that mathematics is about a collection of independently existing mathematical objects. Platonism is to be distinguished from the more general thesis of realism, since the objectivity of mathematical truth does not, at least not obviously, require the existence of distinctively mathematical objects.
Realism is in a fairly clear sense the ‘natural’ position in the philosophy of mathematics, since ordinary mathematical statements make no explicit reference to human activities, beliefs or capacities. Because of the naturalness of mathematical realism, reasons for embracing antirealism typically stem from perceived problems with realism. These potential problems concern our knowledge of mathematical truth, and the connection between mathematical truth and practice. The antirealist argues that the kinds of objective facts posited by the realist would be inaccessible to us, and would bear no clear relation to the procedures we have for determining the truth of mathematical statements. If this is right, then realism implies that mathematical knowledge is inexplicable. The challenge to the realist is to show that the objectivity of mathematical facts does not conflict with our knowledge of them, and to show in particular how our ordinary proof-procedures can inform us about these facts.
Analysis, nonstandard
Authors: Machover Moshe 
Published: September 2018

Nonstandard analysis is an important application of mathematical logic to the rest of mathematics. Invented in 1960, it provided a long-sought-for rigorous justification for the use of infinitely large and infinitely small (infinitesimal) quantities in the differential and integral calculus, and the first sound canon for manipulating such quantities.
Consider the structure ℜ of real numbers, that is, the set of real numbers together with operations and relations on them. We specify a formal language L, which has names for all individual real numbers as well as for the operations and relations of ℜ. We can then obtain an enlargement (a special kind of extension) *ℜ (‘pseudo-ℜ‘) of ℜ such that *ℜ and ℜ have exactly the same formal properties: that is, properties expressible in L. But, far from being merely a copy of ℜ, *ℜ has convenient properties not expressible in L, which are not shared by ℜ. In particular, *ℜ has additional ‘nonstandard’ objects (which have no names in L), some of which behave as infinite or infinitesimal quantities. If, using these novel objects and properties of *ℜ, we prove a proposition about *ℜ that can be expressed in L, then this proposition holds automatically also for ℜ. Such ‘nonstandard’ proofs of results about ℜ are often easier and more intuitive than conventional proofs, which operate wholly within ℜ.
More generally, this method is applied to other structures, in virtually every branch of mathematics – including algebraic number theory, various branches of classical and modern analysis, probability theory and mathematical physics, to yield highly intuitive characterizations of various infinitary concepts, to simplify proofs and to provide new, efficient ways of generating mathematical constructs.
Dedekind, Julius Wilhelm Richard (1831–1916)
Authors: Stein Howard 
Published: September 2018

Dedekind is known chiefly, among philosophers, for contributions to the foundations of the arithmetic of the real and the natural numbers. These made available for the first time a systematic and explicit way, starting from very general notions (which Dedekind himself regarded as belonging to logic), to ground the differential and integral calculus without appeal to geometric ‘intuition’. This work also forms a pioneering contribution to set theory (further advanced in Dedekind’s correspondence with Georg Cantor) and to the general notion of a ‘mathematical structure’.
Dedekind’s foundational work had a close connection with his advancement of substantive mathematical knowledge, particularly in the theories of algebraic numbers and algebraic functions. His achievements in these fields make him one of the greatest mathematicians of the nineteenth century.
Category theory, introduction to
Authors: McLarty Colin 
Published: September 2018

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets.
This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.
Polish logic
Authors: Zygmunt Jan 
Published: September 2018

The term ‘Polish logic’ was coined by McCall to signal the important contributions to modern logic by logicians from Poland between the wars. There were several centres of research, of which the Warsaw school, which grew out of the earlier Lwów–Warsaw philosophical movement, was the most significant. Its development was closely connected with the Warsaw school of mathematics, which gave it its characteristic mathematical bent.
Polish logic took as its point of departure the main trends in logical research of the time and it has influenced both subsequent logical research and subsequent work in the Western analytic tradition of philosophy. Its chief contributions were: (1) an enrichment of existing logical theory (including work on Boolean algebras, the sentential calculus, set theory, the theory of types); (2) new logical theories (for example, Leśniewski’s systems, Łukasiewicz’s many-valued logics, Tarski’s theory of truth, theory of the consequence operation and the calculus of systems); (3) new methods and tools as well as improvements of existing methods (for example, the matrix method of constructing sentential calculi, axiomatizability of logical matrices, algebraic and topological interpretations of deductive systems, permutation models for set theory, the application of quantifier elimination to decidability and definability problems); and (4) the application of formal methods to the study of the history of logic, resulting in a new understanding of the logics of Aristotle, the Stoics and the medievals.
Fictionalism in the philosophy of mathematics
Authors: Colyvan Mark 
Published: September 2018

Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘∏ is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or antirealist) account of mathematics in that it denies the existence of a realm of abstract mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and moreover are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities; and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics).
Although fictionalism holds that mathematical statements implying the existence of mathematical entities are strictly speaking false, there is a sense in which these statements are true - they are true in the story of mathematics. The idea here is borrowed from literary fiction, where statements like ‘Bilbo Baggins is a hobbit’ is strictly speaking false (because there are no hobbits), but true in Tolkien’s fiction The Hobbit. Fictionalism about mathematics shares the virtue of ontological parsimony with other nominalist accounts of mathematics. It also lends itself to a very straightforward epistemology: there is nothing to know beyond the human-authored story of mathematics. And coming to know the various fictional claims requires nothing more than knowledge of the story in question. The most serious problem fictionalism faces is accounting for the applicability of mathematics. Mathematics, unlike Tolkien’s stories, is apparently indispensable to our best scientific theories and this, according to some, suggests that we ought to be realists about mathematical entities.
It is fair to say that there are serious difficulties facing all extant philosophies of mathematics, and fictionalism is no exception. Despite its problems fictionalism remains a popular option in virtue of a number of attractive features. In particular, it endorses a uniform semantics across mathematical and nonmathematical discourse and it provides a neat answer to questions about attaining mathematical knowledge. The major challenge for fictionalism is to provide an adequate account of mathematics in applications.