bluuebunny/crossref_metadata_2025 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.
Descriptions
Authors: Neale Stephen
Published: September 2018

‘Definite descriptions’ are noun phrases of the form ‘the’ + noun complex (for example, ‘the finest Greek poet’, ‘the cube of five’) or of the form possessive + noun complex (for example, ‘Sparta’s defeat of Athens’). As Russell realized, it is important to philosophy to be clear about the semantics of such expressions. In the sentence ‘Aeschylus fought at Marathon’, the function of the subject, ‘Aeschylus’, is to refer to something; it is a referential noun phrase (or ‘singular term’). By contrast, in the sentence ‘Every Athenian remembers Marathon’, the subject noun phrase, ‘every Athenian’, is not referential but quantificational. Definite descriptions appear at first sight to be referential. Frege treated them referentially, but Russell held that they should be treated quantificationally in accordance with his theory of descriptions, and argued that certain philosophical puzzles were thereby solved.
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.
Identity of indiscernibles
Authors: Simons Peter
Published: September 2018

The principle of the identity of indiscernibles states that objects which are alike in all respects are identical. It is sometimes called Leibniz’s Law. This name is also frequently used for the converse principle, the indiscernibility of identicals, that objects which are identical are alike in all respects. Both principles together are sometimes taken to define the concept of identity. Unlike the indiscernibility of identicals, which is widely accepted as a logical truth, the identity of indiscernibles principle has frequently been doubted and rejected. The principle is susceptible of more precise formulation in a number of ways, some more dubitable than others.
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.
Predication
Authors: Mulligan Kevin
Published: September 2018

Some sentences have a very simple structure, consisting only of a part which serves to pick out a particular object and a part which says something about the object picked out. Expressions which can be used to say something about objects picked out are called predicates. Thus ‘smokes’ in ‘Sam smokes’ is a predicate. But ‘predication’ may refer either to the activity of predicating or to what is predicated. To understand either we need to know what predicates are and how they combine with other expressions.
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.
Use/mention distinction and quotation
Authors: Washington Corey
Published: September 2018

Speakers ‘use’ the expressions they utter and ‘mention’ the individuals they talk about. Connected with the roles of used expressions and mentioned individuals is a way of uniting them and a characteristic mistake involving them. Usually the expression used in an utterance will not be the same as the individual mentioned, but the two can be made to converge. The means is quotation. Quotation is a special usage in which an expression is used to mention itself. A failure to distinguish between the roles of used expressions and mentioned individuals can lead to mistakes. Such mistakes are called use/mention confusions. In themselves use/mention confusions are a minor linguistic faux pas, but under unfavourable conditions, they have the potential to cause greater problems.
Vagueness
Authors: Tye Michael
Published: September 2018

It seems obvious that there are vague ways of speaking and vague ways of thinking – saying that the weather is hot, for example. Common sense also has it that there is vagueness in the external world (although this is not the usual view in philosophy). Intuitively, clouds, for example, do not have sharp spatiotemporal boundaries. But the thesis that vagueness is real has spawned a number of deeply perplexing paradoxes and problems. There is no general agreement among philosophers about how to understand vagueness.
Turing reducibility and Turing degrees
Authors: Hodes Harold
Published: September 2018

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.
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.
Turing machines
Authors: Tamburrini Guglielmo
Published: September 2018

Turing machines are abstract computing devices, named after Alan Mathison Turing. A Turing machine operates on a potentially infinite tape uniformly divided into squares, and is capable of entering only a finite number of distinct internal configurations. Each square may contain a symbol from a finite alphabet. The machine can scan one square at a time and perform, depending on the content of the scanned square and its own internal configuration, one of the following operations: print or erase a symbol on the scanned square or move on to scan either one of the immediately adjacent squares. These elementary operations are possibly accompanied by a change of internal configuration. Turing argued that the class of functions calculable by means of an algorithmic procedure (a mechanical, stepwise, deterministic procedure) is to be identified with the class of functions computable by Turing machines. The epistemological significance of Turing machines and related mathematical results hinges upon this identification, which later became known as Turing’s thesis; an equivalent claim, Church’s thesis, had been advanced independently by Alonzo Church. Most crucially, mathematical results stating that certain functions cannot be computed by any Turing machine are interpreted, by Turing’s thesis, as establishing absolute limitations of computing agents.
Tarski’s definition of truth
Authors: Gupta Anil
Published: September 2018

Alfred Tarski’s definition of truth is unlike any that philosophers have given in their long struggle to understand the concept of truth. Tarski’s definition is more clear and precise than any previous definition, but it is also unusual in character and more restricted in scope. Tarski does not provide a general definition of truth. He provides instead a method of constructing, for a range of formalized languages L, definitions of the notions ‘true sentence of L’. A remarkable feature of Tarski’s work on truth is his ‘Criterion T’, which lays down a general condition that any definition of ‘true sentence of L’ must satisfy. Tarski’s ideas have exercised an enormous influence in philosophy. They have played an important role in the formulation and defence of a range of views in logic, semantics and metaphysics.
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.
Cantor’s theorem
Authors: Tiles Mary
Published: September 2018

Cantor’s theorem states that the cardinal number (‘size’) of the set of subsets of any set is greater than the cardinal number of the set itself. So once the existence of one infinite set has been proved, sets of ever increasing infinite cardinality can be generated. The philosophical interest of this result lies (1) in the foundational role it played in Cantor’s work, prior to the axiomatization of set theory, (2) in the similarity between its proof and arguments which lead to the set-theoretic paradoxes, and (3) in controversy between intuitionist and classical mathematicians concerning what exactly its proof proves.
Axiom of choice
Authors: Moore Gregory H.
Published: September 2018

The axiom of choice is a mathematical postulate about sets: for each family of non-empty sets, there exists a function selecting one member from each set in the family. If those sets have no member in common, it postulates that there is a set having exactly one element in common with each set in the family. First formulated in 1904, the axiom of choice was highly controversial among mathematicians and philosophers, because it epitomized ‘non-constructive’ mathematics. Nevertheless, as time passed, it had an increasingly broad range of consequences in many branches of mathematics.
Theory of types
Authors: Cocchiarella Nino B.
Published: September 2018

The theory of types was first described by Bertrand Russell in 1908. He was seeking a logical theory that could serve as a framework for mathematics, and, in particular, a theory that would avoid the so-called ‘vicious-circle’ antinomies, such as his own paradox of the property of those properties that are not properties of themselves – or, similarly, of the class of those classes that are not members of themselves. Such paradoxes can be thought of as resulting when logical distinctions are not made between different types of entities, and, in particular, between different types of properties and relations that might be predicated of entities, such as the distinction between concrete objects and their properties, and the properties of those properties, and so on. In ‘ramified’ type theory, the hierarchy of properties and relations is, as it were, two-dimensional, where properties and relations are distinguished first by their order, and then by their level within each order. In ‘simple’ type theory properties and relations are distinguished only by their orders.
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 in the 17th and 18th centuries
Authors: Capozzi Mirella
Published: September 2018

Logic in the seventeenth century was characterized by attempts to reconcile older viewpoints, such as those of Ramus and Melanchthon, and by criticism of the nature and scope of traditional logic. F. Bacon indicated induction, rather than deduction, as the object of logic, thus opening the way for a logic of the empirical sciences. Descartes proposed to replace the complicated precepts of old logic by simple rules of method. However, even the authors of the Port-Royal Logic, who were influenced by Descartes, could not follow him all the way and continued to teach traditional doctrines, albeit with a new attention to the doctrine of ideas. Other logicians, following Locke, tried to modernize logic by concentrating on an analysis of human cognitive faculties, of the idea–word relation and of other than certain knowledge, thus broadening the scope of logic so as to account for probability. Another suggestion for the improvement of logic came from those who thought that logic should assume mathematics as an example either for its axiomatic-deductive method or for the inventive techniques of algebra. The last of these suggestions prompted research in the area of logical calculi. But this kind of research benefited from the doctrines devised by non-mathematically oriented authors who thus provided the logical framework in which algebraic techniques would be tried. This general background accounts not only for the exceptional logic of Leibniz, but also for some logical calculi worked out in the eighteenth century.
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.
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.
Paraconsistent logic
Authors: Priest Graham
Published: September 2018

A logic is paraconsistent if it does not validate the principle that from a pair of contradictory sentences, A and ∼A, everything follows, as most orthodox logics do. If a theory has a paraconsistent underlying logic, it may be inconsistent without being trivial (that is, entailing everything). Sustained work in formal paraconsistent logics started in the early 1960s. A major motivating thought was that there are important naturally occurring inconsistent but non-trivial theories. Some logicians have gone further and claimed that some of these theories may be true. By the mid-1970s, details of the semantics and proof-theories of many paraconsistent logics were well understood. More recent research has focused on the applications of these logics and on their philosophical underpinnings and implications.
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.