bluuebunny/crossref_metadata_2025_split huggingface dataset:
Linear logic
Authors: Bierman G.M. 
Published: September 2018

Linear logic was introduced by Jean-Yves Girard in 1987. Like classical logic it satisfies the law of the excluded middle and the principle of double negation, but, unlike classical logic, it has non-degenerate models. Models of logics are often given only at the level of provability, in that they provide denotations of formulas. However, we are also interested in models which provide denotations of deductions, or proofs. Given such a model two proofs are said to be equivalent if their denotations are equal. A model is said to be ‘degenerate’ if there are no formulas for which there exist at least two non-equivalent proofs. It is easy to see that models of classical logic are essentially degenerate because any formula is either true or false and so all proofs of a formula are considered equivalent. The intuitionist approach to this problem involves altering the meaning of the logical connectives but linear logic attacks the very connectives themselves, replacing them with more refined ones. Despite this there are simple translations between classical and linear logic.
One can see the need for such a refinement in another way. Both classical and intuitionistic logics could be said to deal with static truths; both validate the rule of modus ponens: if A→B and A, then B; but both also validate the rule if A→B and A, then A∧B. In mathematics this is correct since a proposition, once verified, remains true – it persists. Many situations do not reflect such persistence but rather have an additional notion of causality. An implication A→B should reflect that a state B is accessible from a state A and, moreover, that state A is no longer available once the transition has been made. An example of this phenomenon is in chemistry where an implication A→B represents a reaction of components A to yield B. Thus if two hydrogen and one oxygen atoms bond to form a water molecule, they are consumed in the process and are no longer part of the current state. Linear logic provides logical connectives to describe such refined interpretations.
Essentialism
Authors: Yablo Stephen 
Published: September 2018

Essentialists maintain that an object’s properties are not all on an equal footing: some are ‘essential’ to it and the rest only ‘accidental’. The hard part is to explain what ‘essential’ means.
The essential properties of a thing are the ones it needs to possess to be the thing it is. But this can be taken in several ways. Traditionally it was held that F is essential to x if and only if to be F is part of ‘what x is’, as elucidated in the definition of x. Since the 1950s, however, this definitional conception of essence has been losing ground to the modal conception: x is essentially F if and only if necessarily whatever is x has the property F; equivalently, x must be F to exist at all. A further approach conceives the essential properties of x as those which underlie and account for the bulk of its other properties. This entry emphasizes the modal conception of essentiality.
Acceptance of some form of the essential/accidental distinction appears to be implicit in the very practice of metaphysics. For what interests the metaphysician is not just any old feature of a thing, but the properties that make it the thing it is. The essential/accidental distinction helps in other words to demarcate the subject matter of metaphysics. But it also constitutes a part of that subject matter. If objects have certain of their properties in a specially fundamental way, then this is a phenomenon of great metaphysical significance.
Indicative conditionals
Authors: Jackson Frank 
Published: September 2018

Examples of indicative conditionals are ‘If it rained, then the match was cancelled’ and ‘If Alex plays, Carlton will win’. The contrast is with subjunctive or counterfactual conditionals, such as ‘If it had rained, then the match would have been cancelled’, and categoricals, such as ‘It will rain’.
Despite the ease with which we use and understand indicative conditionals, the correct account of them has proved to be very difficult. Some say that ‘If it rained, the match was cancelled’ is equivalent to ‘Either it did not rain, or the match was cancelled’. Some say that the sentence asserts that the result of ‘adding’ the supposition that it rained to the actual situation is to give a situation in which the match was cancelled. Some say that to assert that if it rained then the match was cancelled is to make a commitment to inferring that the match was cancelled should one learn that it rained. This last view is often combined with the view that indicative conditionals are not, strictly speaking, true or false; rather, they are more or less assertible or acceptable.
Intensional entities
Authors: Bealer George 
Published: September 2018

Intensional entities are such things as concepts, propositions and properties. What makes them ‘intensional’ is that they violate the principle of extensionality; the principle that equivalence implies identity. For example, the concept of being a (well-formed) creature with a kidney and the concept of being a (well-formed) creature with a heart are equivalent in so far as they apply to the same things, but they are different concepts. Likewise, although the proposition that creatures with kidneys have kidneys and the proposition that creatures with hearts have kidneys are equivalent (both are true), they are not identical. Intensional entities are contrasted with extensional entities such as sets, which do satisfy the principle of extensionality. For example, the set of creatures with kidneys and the set of creatures with hearts are equivalent in so far as they have the same members and, accordingly, are identical. By this standard criterion, each of the following philosophically important types of entity is intensional: qualities, attributes, properties, relations, conditions, states, concepts, ideas, notions, propositions and thoughts.
All (or most) of these intensional entities have been classified at one time or another as kinds of universals. Accordingly, standard traditional views about the ontological status of universals carry over to intensional entities. Nominalists hold that they do not really exist. Conceptualists accept their existence but deem it to be mind-dependent. Realists hold that they are mind-independent. Ante rem realists hold that they exist independently of being true of anything; in re realists require that they be true of something.
Logical constants
Authors: McCarthy Timothy 
Published: September 2018

A fundamental problem in the philosophy of logic is to characterize the concepts of ‘logical consequence’ and ‘logical truth’ in such a way as to explain what is semantically, metaphysically or epistemologically distinctive about them. One traditionally says that a sentence p is a logical consequence of a set S of sentences in a language L if and only if (1) the truth of the sentences of S in L guarantees the truth of p and (2) this guarantee is due to the ‘logical form’ of the sentences of S and the sentence p. A sentence is said to be logically true if its truth is guaranteed by its logical form (for example, ‘2 is even or 2 is not even’). There are three problems presented by this picture: to explicate the notion of logical form or structure; to explain how the logical forms of sentences give rise to the fact that the truth of certain sentences guarantees the truth of others; and to explain what such a guarantee consists in.
The logical form of a sentence may be exhibited by replacing nonlogical expressions with a schematic letter. Two sentences have the same logical form when they can be mapped onto the same schema using this procedure (‘2 is even or 2 is not even’ and ‘3 is prime or 3 is not prime’ have the same logical form: ‘p or not-p’). If a sentence is logically true then each sentence sharing its logical form is true. Any characterization of logical consequence, then, presupposes a conception of logical form, which in turn assumes a prior demarcation of the logical constants. Such a demarcation yields an answer to the first problem above; the goal is to generate the demarcation in such a way as to enable a solution of the remaining two.
Approaches to the characterization of logical constants and logical consequence are affected by developments in mathematical logic. One way of viewing logical constanthood is as a semantic property; a property that an expression possesses by virtue of the sort of contribution it makes to determining the truth conditions of sentences containing it. Another way is proof-theoretical: appealing to aspects of cognitive or operational role as the defining characteristics of logical expressions. Broadly, proof-theoretic accounts go naturally with the conception of logic as a theory of formal deductive inference; model-theoretic accounts complement a conception of logic as an instrument for the characterization of structure.
Logical form
Authors: Menzel Christopher 
Published: September 2018

Consider the following argument: All men are mortal; Socrates is a man; therefore, Socrates is mortal. Intuitively, what makes this a valid argument has nothing to do with Socrates, men or mortality. Rather, each sentence in the argument exhibits a certain ‘logical form’ and those forms, taken together, constitute a pattern that guarantees the truth of the conclusion given the truth of the premises. More generally, the logical form of a sentence of natural language is what determines both its logical properties and its logical relations to other sentences.
The logical form of a sentence of natural language is typically represented in a theory of logical form by a well-formed formula in a ‘logically pure’ language whose only meaningful symbols are expressions with fixed, distinctly logical meanings (for example, quantifiers). Thus, the logical forms of the sentences in the above argument would be represented in a theory based on pure predicate logic by the formulas ‘∀x(Fx → Gx)’, ‘Fy’ and ‘Gy’, respectively, where ‘F’ and ‘G’ are free predicate variables and ‘y’ a free individual variable. The argument’s intuitive validity is then explained by the fact that the logical forms of the premises formally entail the logical form of the conclusion. The primary goal of a theory of logical form is to explain as broad a range of such intuitive logical phenomena as possible in terms of the logical forms that it assigns to sentences of natural language.
Meaning and verification
Authors: Hart W.D. 
Published: September 2018

The verifiability theory of meaning says that meaning is evidence. It is anticipated in, for example, Hume’s empiricist doctrine of impressions and ideas, but it emerges into full notoriety in twentieth-century logical positivism. The positivists used the theory in a critique of metaphysics to show that the problems of philosophy, such as the problem of the external world and the problem of other minds, are not real problems at all but only pseudoproblems. Their publicists used the doctrine to argue that religion, ethics and fiction are meaningless, which is how verificationism became notorious among the general public.
Seminal criticism of verification from around 1950 argues that no division between sense and nonsense coincides tidily with a division between science and metaphysics, as the positivists had claimed. Quine later developed verificationism into a sort of semantic holism in which metaphysics is continuous with science. In contrast, Dummett argues from a reading of Wittgenstein’s claim that meaning is use to a rejection of any sort of truth surpassing the possibility of knowledge, and thence to a defence of intuitionistic logic. But the claim that all truths can be known yields in an otherwise innocuous setting the preposterous consequence that all truths actually are known. There are ways to tinker with the setting so as to avoid this consequence, but it is best to conclude by reductio that some truths cannot be known and that verificationism is false. That in turn seems to show that the prospects for an empiricist theory of meaning are dim, which might well shake a complacent confidence in meaning.
Ontological commitment
Authors: Jubien Michael 
Published: September 2018

A person may believe in the existence of God, or numbers or ghosts. Such beliefs may be asserted, perhaps in a theory. Assertions of the existence of specific entities or kinds of entities are the intuitive source of the notion of ontological commitment, for it is natural to think of a person who makes such an assertion as being ‘committed’ to an ‘ontology’ that includes such entities. So ontological commitment appears to be a relation that holds between persons or existence assertions (including theories), on the one hand, and specific entities or kinds of entities (or ontologies), on the other.
Ontological commitment is thus a very rich notion – one in which logical, metaphysical, linguistic and epistemic elements are intermingled. The main philosophical problem concerning commitment is whether there is a precise criterion for detecting commitments in accordance with intuition. It once seemed extremely important to find a criterion, for it promised to serve as a vital tool in the comparative assessment of theories. Many different criteria have been proposed and a variety of problems have beset these efforts. W.V. Quine has been the central figure in the discussion and we will consider two of his formulations below.
Many important philosophical topics are closely connected with ontological commitment. These include: the nature of theories and their interpretation; interpretations of quantification; the nature of kinds; the question of the existence of merely possible entities; extensionality and intensionality; the general question of the nature of modality; and the significance of Occam’s razor.
Quantifiers, substitutional and objectual
Authors: Richard Mark 
Published: September 2018

Understood substitutionally, ‘Something is F’ is true provided one of its substitution instances (a sentence of the form ‘a is F’) is true. This contrasts with the objectual understanding, on which it is true provided ‘is F’ is true of some object in the domain of the quantifier. Substitutional quantifications have quite different truth-conditions from objectual ones. For instance, ‘Something is a mythological animal’ is true if understood substitutionally, since the substitution instance ‘Pegasus is a mythological animal’ is true. But understood objectually, the sentence is not true, since there are no mythological creatures to make up a domain for the quantifier.
Since substitutional quantifiers do not need domains over which they range, it is easy to introduce substitutional quantifiers which bind predicate or sentential variables, even variables within quotation marks. One reason for interest in substitutional quantification is the hope that it may provide a way to understand discourse which appears to be about numbers, properties, propositions and other ‘troublesome’ sorts of entities as being free of exceptional ontological commitments. Whether natural language quantification is sometimes plausibly construed as substitutional is not, however, clear.
Computability and information
Authors: Calude Cristian S. 
Published: September 2018

The standard definition of randomness as considered in probability theory and used, for example, in quantum mechanics, allows one to speak of a process (such as tossing a coin, or measuring the diagonal polarization of a horizontally polarized photon) as being random. It does not allow one to call a particular outcome (or string of outcomes, or sequence of outcomes) ‘random’, except in an intuitive, heuristic sense. Information-theoretic complexity makes this possible. An algorithmically random string is one which cannot be produced by a description significantly shorter than itself; an algorithmically random sequence is one whose initial finite segments are almost random strings.
Gödel’s incompleteness theorem states that every axiomatizable theory which is sufficiently rich and sound is incomplete. Chaitin’s information-theoretic version of Gödel’s theorem goes a step further, revealing the reason for incompleteness: a set of axioms of complexity N cannot yield a theorem that asserts that a specific object is of complexity substantially greater than N. This suggests that incompleteness is not only natural, but pervasive; it can no longer be ignored by everyday mathematics. It also provides a theoretical support for a quasi-empirical and pragmatic attitude to the foundations of mathematics.
Information-theoretic complexity is also relevant to physics and biology. For physics it is convenient to reformulate it as the size of the shortest message specifying a microstate, uniquely up to the assumed resolution. In this way we get a rigorous, entropy-like measure of disorder of an individual, microscopic, definite state of a physical system. The regulatory genes of a developing embryo can be ultimately conceived as a program for constructing an organism. The information contained by this biochemical computer program can be measured by information-theoretic complexity.
Numbers
Authors: Priest Graham 
Published: September 2018

Numbers are, in general, mathematical entities whose function is to express the size, order or magnitude of something or other. Historically, starting from the most basic kind of number, the positive integers (1, 2, 3,…), which appear in the earliest written records, the notion of number has been generalized and extended in several different directions – often in the face of considerable opposition.
Other than the positive integers, the most venerable are the rational numbers (fractions), which were known to the Egyptians and Mesopotamians. The discovery, by Pythagorean mathematicians, that there are lengths that cannot be expressed as fractions occasioned the introduction of irrational numbers, such as the square root of 2, though the Greeks managed only a geometric understanding of these. The number zero was recognized, first in Indian mathematics, by the seventh century; the use of negative numbers evolved after this time; and complex numbers, such as the square root of −1, appeared first at the end of the Middle Ages. Infinitesimal numbers were developed by the founders of the calculus, Newton and Leibniz, in the seventeenth century (and were later to disappear from mathematics – for a time); and infinite numbers (ordinals and cardinals) were introduced by the founder of modern set theory, Cantor, in the nineteenth century.
The introductions of three of these kinds of number, in particular, occasioned crises in the foundations of mathematics. The first (concerning irrational numbers) was finally resolved in the nineteenth century by the work of Cauchy and Weierstrass. The second (concerning infinitesimals) was also resolved then, by the work of Weierstrass and Dedekind. The third (concerning infinite numbers), which involves paradoxes such as Russell’s, still awaits a convincing solution.
It is seemingly impossible to give a rigorous definition of what it is to be a number. The closest one can get is a family-resemblance notion, with very ill-defined boundaries.
Logic, ancient
Authors: Thom Paul 
Published: September 2018

Western antiquity produced two great bodies of logical theory – those of Aristotle and the Stoics. Both aim to explain what distinguishes good arguments from bad. Both see that the best arguments are valid and that an argument’s validity depends on its form. For both, therefore, logic’s business is to identify the valid argument forms. Both theories do this by laying down a small number of basic argument forms – Aristotle’s ‘perfect syllogisms’, the Stoics’ ‘indemonstrables’ – and rigorously deriving other valid forms from them. Both theories also try – though in a less systematic manner – to classify the ways in which an argument can go wrong.
Here the similarities between these two logics end. Their most significant differences can be illustrated by comparing basic argument forms from each. The argument ‘Every swan is an animal and every animal is moving, so every swan is moving’ has the same form as the argument ‘Every musician is human and every human is a substance, so every musician is a substance’. The Aristotelian expression of this form is ‘A belongs to all B and B belongs to all C, so A belongs to all C’. In this form the letters ‘A’, ‘B’ and ‘C’ stand for any terms whatever, and ‘A belongs to all B’ replaces ‘Every B is an A’. This represents the Aristotelian approach. Compare it with the following. The argument ‘If it is day then it is light, it is day, so it is light’ has the same form as the argument ‘If Dion walks then Dion moves, Dion walks, so Dion moves’. This form is expressed by the Stoics as ‘If the first then the second, the first, so the second’. Here the expressions ‘the first’ and ‘the second’ stand for any declarative sentences whatever.
In both cases, the validity of the argument form is tantamount to the validity of all arguments having that form (though the Stoics, unlike Aristotle, require that the precise words used in an argument should recur in its form). But the Aristotelian argument form is different in kind from the Stoic one: while it abstracts from terms, the Stoic form abstracts from sentences. Aristotelian logic is a term logic, Stoic logic a sentential one.
Logic machines and diagrams
Authors: Dipert Randall R. 
Published: September 2018

By ‘logical diagrams’ we generally mean any two-dimensional representations of logical relationships, such as of class inclusion or consequence. One usually also means representations using non-typographical symbols or geometric figures. Such diagrams were first used in the seventeenth and eighteenth centuries, but gained wide currency only in the nineteenth; the best known are the Euler and Venn diagrams. It is an open question whether logical diagrams are useful only as elementary pedagogical devices, or have implications for advanced logical research.
The conception of an organism, the mind, or of the universe as ‘machine’ was not really attractive and useful until machines were widespread, complex and able to perform interesting tasks. This occurred first in the late Renaissance, and initiated ways of thinking that dominated the seventeenth and eighteenth centuries. It was also becoming evident that machines could be used to perform some complex, repetitive or difficult tasks more reliably or faster than human beings.
The very idea of machines that can perform ‘symbolic’ tasks, such as mathematical, logical or, eventually, linguistic ones required first a symbolism. For this reason, the idea of computers for mathematical or logical tasks, and systems of mathematical and logical notation, are strongly intertwined: one must have efficient ways of feeding information into a machine, and interpreting the results.