bluuebunny/crossref_metadata_2025_split huggingface dataset:
Proper names
Authors: Forbes Graeme 
Published: September 2018

The Roman general Julius Caesar was assassinated on 14 March 44 bc by conspirators led by Brutus and Cassius. It is a remarkable fact that, in so informing or reminding the reader, the proper names ‘Julius Caesar‘, ‘Brutus’ and ‘Cassius’ are used to refer to three people each of whom has been dead for about two thousand years. Our eyes could not be used to see any of them, nor our voices to talk to them, yet we can refer to them with our words.
The central philosophical issue about proper names is how this sort of thing is possible: what exactly is the mechanism by which the user of a name succeeds in referring with the name to its bearer? As the example indicates, whatever the mechanism is, it must be something that can relate the use of a name to its bearer even after the bearer has ceased to exist.
In modern philosophy of language there are two main views about the nature of the mechanism. On one account, which originated with Frege, a use of a name expresses a conception or way of thinking of an object, and the name refers to whatever object fits, or best fits, that conception or way of thinking. Thus with ‘Cassius‘, for example, I may associate the conception ‘the conspirator whom Caesar suspected because of his size’ (recalling a famous speech in Shakespeare’s Julius Caesar). Conception theories are usually called ‘sense’ theories, after Frege’s term ‘Sinn‘. The other account is the ‘historical chain’ theory, due to Kripke and Geach. In Geach’s words, ’ for the use of a word as a proper name there must in the first instance e someone acquainted with the object named…. But…the use of a given name for a given object…can be handed on from one generation to another…Plato knew Socrates, and Aristotle knew Plato, and Theophrastus knew Aristotle, and so on in apostolic succession down to our own times. That is why we can legitimately use "Socrates" as a name the way we do’ (1969–70: 288–9).
Quantifiers, generalized
Authors: Westerstahl Dag 
Published: September 2018

Generalized quantifiers are logical tools with a wide range of uses. As the term indicates, they generalize the ordinary universal and existential quantifiers from first-order logic, ‘∀x’ and ‘∃x’, which apply to a formula A(x), binding its free occurrences of x. ∀xA(x) says that A(x) holds for all objects in the universe and ∃xA(x) says that A(x) holds for some objects in the universe, that is, in each case, that a certain condition on A(x) is satisfied. It is natural then to consider other conditions, such as ‘for at least five’, ‘at most ten’, ‘infinitely many’ and ‘most’. So a quantifier Q stands for a condition on A(x), or, more precisely, for a property of the set denoted by that formula, such as the property of being non-empty, being infinite, or containing more than half of the elements of the universe. The addition of such quantifiers to a logical language may increase its expressive power.
A further generalization allows Q to apply to more than one formula, so that, for example, Qx(A(x),B(x)) states that a relation holds between the sets denoted by A(x) and B(x), say, the relation of having the same number of elements, or of having a non-empty intersection. One also considers quantifiers binding more than one variable in a formula. Qxy,zu(R(x,y),S(z,u)) could express, for example, that the relation (denoted by) R(x,y) contains twice as many pairs as S(z,u), or that R(x,y) and S(z,u) are isomorphic graphs.
In general, then, a quantifier (the attribute ‘generalized’ is often dropped) is syntactically a variable-binding operator, which stands semantically for a relation between relations (on individuals), that is, a second-order relation. Quantifiers are studied in mathematical logic, and have also been applied in other areas, notably in the semantics of natural languages. This entry first presents some of the main logical facts about generalized quantifiers, and then explains their application to semantics.
Computability theory
Authors: Mundici Daniele , Sieg Wilfried 
Published: September 2018

The effective calculability of number-theoretic functions such as addition and multiplication has always been recognized, and for that judgment a rigorous notion of ‘computable function’ is not required. A sharp mathematical concept was defined only in the twentieth century, when issues including the decision problem for predicate logic required a precise delimitation of functions that can be viewed as effectively calculable. Predicate logic emerged from Frege’s fundamental ‘Begriffsschrift’ (1879) as an expressive formal language and was described with mathematical precision by Hilbert in lectures given during the winter of 1917–18. The logical calculus Frege had also developed allowed proofs to proceed as computations in accordance with a fixed set of rules; in principle, according to Gödel, the rules could be applied ‘by someone who knew nothing about mathematics, or by a machine’.
Hilbert grasped the potential of this mechanical aspect and formulated the decision problem for predicate logic as follows: ‘The Entscheidungsproblem [decision problem] is solved if one knows a procedure that permits the decision concerning the validity, respectively, satisfiability of a given logical expression by a finite number of operations.’ Some, for example, von Neumann (1927), believed that the inherent freedom of mathematical thought provided a sufficient reason to expect a negative solution to the problem. But how could a proof of undecidability be given? The unsolvability results of other mathematical problems had always been established relative to a determinate class of admissible operations, for example, the impossibility of doubling the cube relative to ruler and compass constructions. A negative solution to the decision problem obviously required the characterization of ‘effectively calculable functions’.
For two other important issues a characterization of that informal notion was also needed, namely, the general formulation of the incompleteness theorems and the effective unsolvability of mathematical problems (for example, of Hilbert’s tenth problem). The first task of computability theory was thus to answer the question ‘What is a precise notion of "effectively calculable function"?’. Many different answers invariably characterized the same class of number-theoretic functions: the partial recursive ones. Today recursiveness or, equivalently, Turing computability is considered to be the precise mathematical counterpart to ‘effective calculability’. Relative to these notions undecidability results have been established, in particular, the undecidability of the decision problem for predicate logic. The notions are idealized in the sense that no time or space limitations are imposed on the calculations; the concept of ‘feasibility’ is crucial in computer science when trying to capture the subclass of recursive functions whose values can actually be determined.
Church’s theorem and the decision problem
Authors: Parikh Rohit 
Published: September 2018

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. Church’s paper exhibited an undecidable combinatorial problem P and showed that P was representable in first-order logic. If first-order logic were decidable, P would also be decidable. Since P is undecidable, first-order logic must also be undecidable.
Church’s theorem is a negative solution to the decision problem (Entscheidungsproblem), the problem of finding a method for deciding whether a given formula of first-order logic is valid, or satisfiable, or neither. The great contribution of Church (and, independently, Turing) was not merely to prove that there is no method but also to propose a mathematical definition of the notion of ‘effectively solvable problem’, that is, a problem solvable by means of a method or algorithm.
Complexity, computational
Authors: Urquhart Alasdair 
Published: September 2018

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible.
The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.
Model theory
Authors: Hodges Wilfrid 
Published: September 2018

Model theory studies the relations between sentences of a formal language and the interpretations (or ‘structures’) which make these sentences true or false. It offers precise definitions of truth, logical truth and consequence, meanings and modalities. These definitions and their consequences have revolutionized the teaching of elementary logic.
Model theory also forms a branch of mathematics concerned with the ways in which mathematical structures can be classified. This technical work has led to philosophically interesting results in at least two areas: it has thrown light on the nature of the set-theoretic universe, and in nonstandard analysis it has suggested new forms of argument (where we prove something different from what we intended, but then use a general model-theoretic argument to change the result into what we wanted).
The word ‘model’ has many other uses. For example, model theory is not about scientific theories as models of the world. It is also a controversial question – not considered here – how model theory is connected with the ‘mental models’ which appear in the psychology of reasoning.
Semantic paradoxes and theories of truth
Authors: McGee Vann 
Published: September 2018

The Cretan philosopher Epimenides said that Cretans always lie. Assuming, for the sake of argument, the mendacity of all other statements by Cretans, we get a paradox: if what Epimenides said was true, it must have been a lie, whereas if what he said was a lie, it would have made his statement true. The citizens of Crete have long since forgiven the insult, but semantics has never recovered.
Alfred Tarski perceived the consequences of Epimenides’ paradox with particular clarity. Our common-sense intuitions about truth follow the paradigm: ‘Snow is white’ is true if and only if snow is white. As Tarski rigorously shows, if the language we are describing (the object language) is the same as the language in which we are formulating our theory (the metalanguage), this paradigm will be inconsistent with the rudimentary laws of syntax. The conclusion Tarski drew was that, if we are to develop a satisfactory theory of truth, our metalanguage must be essentially richer in expressive power than the object language. Since there is no human language essentially richer than English (or any other natural language), there can be no satisfactory theory of truth for English.
One earnestly hopes that this is not the end of the matter. Tarski’s analysis leaves open the prospect that we can develop a fully satisfactory theory of truth for a substantial fragment of English; also the prospect that we can develop a theory of truth for English as a whole which, while not fully satisfying our intuitions, is none the less useful and illuminating.
Both prospects have been substantially advanced by Saul Kripke’s ‘Outline of a Theory of Truth’, which exploits the idea that there are truth-value gaps, statements which are neither true nor false, and that Epimenides’ insult was one of them.
Invocation of truth-value gaps does not resolve the paradox in any straightforward way. If we let the phrase ‘the simple liar sentence’ refer to the sentence ‘The simple liar sentence is false’, we see that we can readily account for the paradoxical features of the sentence by declaring the sentence neither true nor false; but if we let the strengthened liar sentence be ‘The strengthened liar sentence is not true’, we get a sentence we cannot dispose of so tidily. If the strengthened liar is neither true nor false, then it is not true; but that it is not true is precisely what the sentence says.
Truth-value gaps have not vanquished the liar paradox. Nor have any of the alternatives, the most prominent of which are a contextualist account, which sees the English word ‘true’ as radically ambiguous, and so-called ‘revision theory’, which investigates the cyclic reasoning that occurs when we try to evaluate the simple liar sentence: if the sentence is true, then it must be false; but if, then, it is false, it must be true; and so on. While these approaches have not eliminated the paradox, they have opened new approaches that have greatly improved our prospects for finding a comfortable way to live with it.
Beth’s theorem and Craig’s theorem
Authors: Swijtink Zeno 
Published: September 2018

Beth’s theorem is a central result about definability of non-logical symbols in classical first-order theories. It states that a symbol P is implicitly defined by a theory T if and only if an explicit definition of P in terms of some other expressions of the theory T can be deduced from the theory T. Intuitively, the symbol P is implicitly defined by T if, given the extension of these other symbols, T fixes the extension of the symbol P uniquely. In a precise statement of Beth’s theorem this will be replaced by a condition on the models of T. An explicit definition of a predicate symbol states necessary and sufficient conditions: for example, if P is a one-place predicate symbol, an explicit definition is a sentence of the form (x) (Px ≡φ(x)), where φ(x) is a formula with free variable x in which P does not occur. Thus, Beth’s theorem says something about the expressive power of first-order logic: there is a balance between the syntax (the deducibility of an explicit definition) and the semantics (across models of T the extension of P is uniquely determined by the extension of other symbols).
Beth’s definability theorem follows immediately from Craig’s interpolation theorem. For first-order logic with identity, Craig’s theorem says that if φ is deducible from ψ, there is an interpolant θ, a sentence whose non-logical symbols are common to φ and ψ, such that θ is deducible from ψ, while φ is deducible from θ. Craig’s theorem and Beth’s theorem also hold for a number of non-classical logics, such as intuitionistic first-order logic and classical second-order logic, but fail for other logics, such as logics with expressions of infinite length.
Paradoxes of set and property
Authors: Moore Gregory H. 
Published: September 2018

Emerging around 1900, the paradoxes of set and property have greatly influenced logic and generated a vast literature. A distinction due to Ramsey in 1926 separates them into two categories: the logical paradoxes and the semantic paradoxes. The logical paradoxes use notions such as set or cardinal number, while the semantic paradoxes employ semantic concepts such as truth or definability. Both often involve self-reference.
The best known logical paradox is Russell’s paradox concerning the set S of all sets x such that x is not a member of x. Russell’s paradox asks: is S a member of itself? A moment’s reflection shows that S is a member of itself if and only if S is not a member of itself – a contradiction.
Russell found this paradox by analysing the paradox of the largest cardinal. The set U of all sets has the largest cardinal number, since every set is a subset of U. But there is a cardinal number greater than that of any given set M, namely the cardinal of the power set, or set of all subsets, of M. Thus the cardinal of the power set of U is greater than that of U, a contradiction. (The paradox of the largest ordinal, discussed below, is similar in structure.)
Among the semantic paradoxes, the best known is the liar paradox, found by the ancient Greeks. A man says that he is lying. Is what he says true or false? Again, either conclusion leads to its opposite. Although this paradox was debated in medieval Europe, its modern interest stems from Russell, who placed it in the context of a whole series of paradoxes, including his own.
Set theory, different systems of
Authors: Potter Michael 
Published: September 2018

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory.
Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals).
Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception.
Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.
Property theory
Authors: Cocchiarella Nino B. 
Published: September 2018

Traditionally, a property theory is a theory of abstract entities that can be predicated of things. A theory of properties in this sense is a theory of predication – just as a theory of classes or sets is a theory of membership. In a formal theory of predication, properties are taken to correspond to some (or all) one-place predicate expressions. In addition to properties, it is usually assumed that there are n-ary relations that correspond to some (or all) n-place predicate expressions (for n≥ 2). A theory of properties is then also a theory of relations.
In this entry we shall use the traditional labels ‘realism’ and ‘conceptualism’ as a convenient way to classify theories. In natural realism, where properties and relations are the physical, or natural, causal structures involved in the laws of nature, properties and relations correspond to only some predicate expressions, whereas in logical realism properties and relations are generally assumed to correspond to all predicate expressions.
Not all theories of predication take properties and relations to be the universals that predicates stand for in their role as predicates. The universals of conceptualism, for example, are unsaturated concepts in the sense of cognitive capacities that are exercised (saturated) in thought and speech. Properties and relations in the sense of intensional Platonic objects may still correspond to predicate expressions, as they do in conceptual intensional realism, but only indirectly as the intensional contents of the concepts that predicates stand for in their role as predicates. In that case, instead of properties and relations being what predicates stand for directly, they are what nominalized predicates denote as abstract singular terms. It is in this way that concepts – such as those that the predicate phrases ‘is wise’, ‘is triangular’ and ‘is identical with’ stand for – are distinguished from the properties and relations that are their intensional contents – such as those that are denoted by the abstract singular terms ‘wisdom’, ‘triangularity’ and ‘identity’, respectively. Once properties are represented by abstract singular terms, concepts can be predicated of them, and, in particular, a concept can be predicated of the property that is its intensional content. For example, the concept represented by ‘is a property’ can be predicated of the property denoted by the abstract noun phrase ‘being a property’, so that ‘Being a property is a property’ (or, ‘The property of being a property is a property’) becomes well-formed. In this way, however, we are confronted with Russell’s paradox of (the property of) being a non-self-predicable property, which is the intensional content of the concept represented by ‘is a non-self-predicable property’. That is, the property of being a non-self-predicable property both falls and does not fall under the concept of being a non-self-predicable property (and therefore both falls and does not fall under the concept of being self-predicable).