bluuebunny/crossref_metadata_2025_split huggingface dataset:
Modal logic, philosophical issues in
Authors: McKay Thomas J. 
Published: September 2018

In reasoning we often use words such as ‘necessarily’, ‘possibly’, ‘can’, ‘could’, ‘must’ and so on. For example, if we know that an argument is valid, then we know that it is necessarily true that if the premises are true, then the conclusion is true. Modal logic starts with such modal words and the inferences involving them. The exploration of these inferences has led to a variety of formal systems, and their interpretation is now most often built on the concept of a possible world.
Standard non-modal logic shows us how to understand logical words such as ‘not’, ‘and’ and ‘or’, which are truth-functional. The modal concepts are not truth-functional: knowing that p is true (and what ‘necessarily’ means) does not automatically enable one to determine whether ‘Necessarily p’ is true. (‘It is necessary that all people have been people’ is true, but ‘It is necessary that no English monarch was born in Montana’ is false, even though the simpler constituents – ‘All people have been people’ and ‘No English monarch was born in Montana’– are both true.)
The study of modal logic has helped in the understanding of many other contexts for sentences that are not truth-functional, such as ‘ought’ (‘It ought to be the case that p’) and ‘believes’ (‘Alice believes that p’); and also in the consideration of the interaction between quantifiers and non-truth-functional contexts. In fact, much work in modern semantics has benefited from the extension of modal semantics introduced by Richard Montague in beginning the development of a systematic semantics for natural language.
The framework of possible worlds developed for modal logic has been fruitful in the analysis of many concepts. For example, by introducing the concept of relative possibility, Kripke showed how to model a variety of modal systems: a proposition is necessarily true at a possible world w if and only if it is true at every world that is possible relative to w. To achieve a better analysis of statements of ability, Mark Brown adapted the framework by modelling actions with sets of possible outcomes. John has the ability to hit the bull’s-eye reliably if there is some action of John’s such that every possible outcome of that action includes John’s hitting the bull’s-eye.
Modal logic and its semantics also raise many puzzles. What makes a modal claim true? How do we tell what is possible and what is necessary? Are there any possible things that do not exist (and what could that mean anyway)? Does the use of modal logic involve a commitment to essentialism? How can an individual exist in many different possible worlds?
Adverbs
Authors: Higginbotham James 
Published: September 2018

Adverbs are so named from their role in modifying verbs and other non-nominal expressions. For example, in ‘John ran slowly’, the adverb ‘slowly’ modifies ‘ran’ by characterizing the manner of John’s running. The debate on the semantic contribution of adverbs centres on two approaches. On the first approach, adverbs are understood as predicate operators: for example, in ‘John ran slowly’, ‘ran’ would be taken to be a predicate and ‘slowly’ an operator affecting its meaning. Working this out in detail requires the resources of higher-order logic. On the second approach, adverbs are understood as predicates of ‘objects’ such as events and states, reference to which is revealed in logical form. For example, ‘John ran slowly’ would be construed along the lines of ‘there was a running by John and it was slow’, in which the adverb ‘slowly’ has become a predicate ‘slow’ applied to the event that was John’s running.
Since adverbs are exclusively modifiers, they are classed among the syncategorematic words of terminist logic, the investigation of which carried the subject forward from Aristotle in the thirteenth century. (The contrasting ‘categoremata’ – grammatical subjects and predicates – are those words which have meaning independently.) They are of contemporary interest for philosophical logic and semantic theory, because particular accounts of them carry implications for the nature of combinatorial semantics and language understanding, and for ontology.
De re/de dicto
Authors: Gallois Andre 
Published: September 2018

‘De re’ and ‘de dicto’ have been used to label a host of different, albeit interrelated, distinctions. ‘De dicto’ means ‘of, or concerning, a dictum’, that is, something having representative content, such as a sentence, statement or proposition. ‘De re’ means ‘of, or concerning, a thing’. For example, a de dicto belief is a belief that a bearer of representative content is true, while a de re belief is a belief concerning some thing, that it has a particular characteristic.
Consider the following example:
- John believes his next-door neighbour is a Buddhist.
This statement is ambiguous. Construed de dicto, it is true in the following circumstance. John has never had any contact with his next-door neighbour. Nevertheless, John believes that his next-door neighbour is bound to be a Buddhist. Construed in this de dicto fashion, the statement does not attribute to John a belief that is distinctively about a particular individual. In contrast, construed de re, it does attribute to John a belief that is about a particular individual. For example, construed de re, the statement is true in the following circumstance. John encounters his next-door neighbour, Fred, at a party without realizing that Fred is his next-door neighbour. On the basis of his conversation with Fred, John forms a belief about the individual who is in fact his next-door neighbour to the effect that he is a Buddhist.
Indirect discourse
Authors: Segal Gabriel 
Published: September 2018

Indirect discourse is a mode of speech-reporting whereby a speaker conveys the content of someone’s utterance without quoting the actual words. Thus, if Pierre says, ‘Paris est belle’, an English speaker might truly say,
- (1) Pierre said that Paris is beautiful.
In English, sentences of indirect discourse often have the form ‘A said that s’, where ‘A’ refers to a person and ‘s’ is often called the ‘content sentence’ of the report.
Sentences of indirect discourse have been classed with attributions of belief (and other psychological states) in view of an apparent conflict with the ‘principle of the intersubstitutability of coreferring terms’, which states that the truth-value of a sentence does not alter if one term in a sentence is replaced with another referring to the same thing. If (1) is true and ‘Paris’ and ‘the City of Light’ refer to the same thing, (2) may still be false:
- (2) Pierre said that the City of Light is beautiful.
Logical laws
Authors: Restall Greg 
Published: September 2018

There are at least three different kinds of answer to the question ‘What is a logical law?’ One establishes what it means for something to be a logical law. This answers the semantic question: What is the meaning of ‘logical law’? The second explains what makes something a logical law. This answers the metaphysical question: What is the ground of logical law? The third tells you what the logical laws are. This answers the question: What is the extension of ‘logical law’?
Even though logic is often seen as a complete science, the answers to all three questions are disputed. For example, there are at least three different conceptions of what it means for something to be a law of logic. Different conceptions account for logic in terms of necessity, truth in all models, and proof.
There are also different answers to the metaphysical question. If truth-preservation is central to logic, then the ground of logic depends on the metaphysics of truth. If logic is a matter of the meanings of terms, then the metaphysics of meaning is important for logic. Unfortunately, there is no widespread agreement on the metaphysics of meaning or truth.
Finally, there is no widespread agreement as to what the logical laws are. There are two general disputes here. First, it is not clear what notions count as logical. Does logic contain laws about identity, second-order quantification or modality? Second, given agreement on the scope of logic, there are still questions about the logical laws in that area. Intuitionists, quantum logicians, relevance and paraconsistent logicians each reject things taken as laws by others, even in the language of ‘and’, ‘or’ and ‘not’.
Mass terms
Authors: Pelletier Jeffry 
Published: September 2018

Mass terms are words and phrases such as ‘water’, ‘wood’ and ‘white wallpaper’. They are contrasted with count terms such as ‘woman’, ‘word’ and ‘wild wildebeest’. Intuitively, mass terms refer to ‘stuff’; count terms refer to ‘objects’. Mass terms allow for measurement (‘three kilos of wood’, ‘much water’); count terms allow for counting, quantifying and individuating (‘three women’, ‘each word’, ‘that wildebeest over there’).
Philosophical problems associated with mass terms include (1) distinguishing mass from count terms, (2) describing the semantics of sentences employing mass terms, and (3) explicating the ontology presupposed by our use of mass versus count terms. Associated with these philosophical issues – especially the third – are the meta-philosophical issues concerning the extent to which any investigation into the linguistic practices of speakers of a language can be used as evidence for how those speakers view ‘reality’.
Modal operators
Authors: Schweizer Paul 
Published: September 2018

Modal logic is principally concerned with the alethic modalities of necessity and possibility, although this branch of logic is applied to a wide range of linguistic and conceptual phenomena, including natural language semantics, proof theory, theoretical computer science and the formal characterization of knowledge and belief. This wide range of application stems from the basic form of modal assertions, such as ‘it is necessarily the case that φ’, where an entire statement φ is embedded within a context possessing rich logical structure.
When constructing a formal representation of these embedding contexts, there are several choices concerning their specific symbolic form. The most standard approach symbolizes modal contexts as operators, which combine directly with formulas of the object language to yield new formulas. The primary alternative to this approach is to treat modal contexts as predicates, which attach not to formulas directly, but to names of formulas, and thereby attribute a metalinguistic property to a syntactic object. A variation on the operator approach, which assumes the interpretive framework of possible worlds semantics, is to treat modal contexts as quantifications over possible worlds. Finally, a variation on the predicate approach is to analyse modal contexts as predicates of propositions rather than as predicates of syntactic objects.
Anaphora
Authors: Asher Nicholas 
Published: September 2018

Anaphora describes a dependence of the interpretation of one natural language expression on the interpretation of another natural language expression. For example, the pronoun ‘her’ in (1) below is anaphorically dependent for its interpretation on the interpretation of the noun phrase ‘Sally’ because ‘her’ refers to the same person ‘Sally’ refers to.
- (1) Sally likes her car.
As (2) below illustrates, anaphoric dependencies also occur across sentences, making anaphora a ‘discourse phenomenon’:
- (2) A farmer owned a donkey. He beat it.
The analysis of anaphoric dependence has been the focus of a great deal of study in linguistics and philosophy. Anaphoric dependencies are difficult to accommodate within the traditional conception of compositional semantics of Tarski and Montague precisely because the meaning of anaphoric elements is dependent on other elements of the discourse.
Many expressions can be used anaphorically. For instance, anaphoric dependencies hold between the expression ‘one’ and the indefinite noun phrase ‘a labrador’ in (3) below; between the verb phrase ‘loves his mother’ and a ‘null’ anaphor (or verbal auxiliary) in (4); between the prepositional phrase ‘to Paris’ and the lexical item ‘there’ in (5); and between a segment of text and the pronoun ‘it’ in (6).
- (3) Susan has a labrador. I want one too.
- (4) John loves his mother. Fred does too.
- (5) I didn’t go to Paris last year. I don’t go there very often.
- (6) One plaintiff was passed over for promotion. Another didn’t get a pay increase for five years. A third received a lower wage than men doing the same work. But the jury didn’t believe any of it.
Some philosophers and linguists have also argued that verb tenses generate anaphoric dependencies.
Propositions, sentences and statements
Authors: Engel Pascal 
Published: September 2018

A sentence is a string of words formed according to the syntactic rules of a language. But a sentence has semantic as well as syntactic properties: the words and the whole sentence have meaning. Philosophers have tended to focus on the semantic properties of indicative sentences, in particular on their being true or false. They have called the meanings of such sentences ‘propositions’, and have tied the notion of proposition to the truth-conditions of the associated sentence.
The term ‘proposition’ is sometimes assimilated to the sentence itself; sometimes to the linguistic meaning of a sentence; sometimes to ‘what is said’; sometimes to the contents of beliefs or other ‘propositional’ attitudes. But however propositions are defined, they must have two features: the capacity to be true or false; and compositional structure (being composed of elements which determine their semantic properties).
Sense and reference
Authors: Marti Genoveva 
Published: September 2018

The ‘reference’ of an expression is the entity the expression designates or applies to. The ‘sense’ of an expression is the way in which the expression presents that reference. For example, the ancients used ‘the morning star’ and ‘the evening star’ to designate what turned out to be the same heavenly body, the planet Venus. These two expressions have the same reference, but they clearly differ in that each presents that reference in a different way. So, although coreferential, each expression is associated with a different ‘sense’. The distinction between sense and reference helps explain the cognitive puzzle posed by identity statements. ‘The morning star is the evening star’ and ‘The morning star is the morning star’ are both true, yet the sentences differ in cognitive significance, since the former may be informative, whereas the latter definitely is not. That difference in cognitive significance cannot be explained just by appeal to the references of the terms, for those are the same. It can, however, be naturally accounted for by appeal to a difference in sense. The terms ‘the morning star’ and ‘the evening star’ used in the first sentence, having different senses, present the referent in different ways, whereas no such difference occurs in the second sentence.
The distinction between sense and reference applies to all well-formed expressions of a language. It is part of a general theory of meaning that postulates an intermediate level of sense between linguistic terms and the entities the terms stand for. Senses give significance to expressions, which in and of themselves are just noises or marks on a surface, and connect them to the world. It is because linguistic terms have a sense that they can be used to express judgments, to transmit information and to talk about reality.
Imperative logic
Authors: Green Mitchell 
Published: September 2018

Imperatives lie at the heart of both practical and moral reasoning, yet they have been overshadowed by propositions and relegated by many philosophers to the status of exclamations. One reason for this is that a sentence’s having literal meaning seems to require its having truth-conditions and ‘Keep your promises!’ appears to lack such conditions, just as ‘Ouch!’ does. One reductionist attempt to develop a logic of imperatives translates them into declaratives and construes inferential relations among the former in terms of inferential relations among the latter. Since no such reduction seems fully to capture the meaning of imperatives, others have expanded our notion of inference to include not just truth – but also satisfaction – preservation, according to which an imperative is satisfied just in case what it enjoins is brought about.
A logic capturing what is distinctive about imperatives may shed light on the question whether an ‘ought’ is derivable from an ‘is’; and may elucidate the claim that morality is, or comprises, a system of hypothetical imperatives. Furthermore, instructions, which are often formulated as imperatives (‘Take two tablets on an empty stomach!’), are crucial to the construction of plans of action. A proper understanding of imperatives and their inferential properties may thus also illuminate practical reasoning.
Plural logic
Authors: Oliver Alex , Smiley Timothy 
Published: September 2018

Plural phenomena are significant and inescapable. Granted, the plural idiom is sometimes reducible to the singular, e.g. ‘2 and 3 are prime is equivalent to ‘2 is prime and 3 is prime’. ‘Are prime’, however, belongs to the special class of predicates known as distributives. No such reductions are possible for the general case of collective (nondistributive) predicates, and they are to be found everywhere, from the everyday (‘Whitehead and Russell wrote Principia Mathematica’) to the heart of logic itself (‘The axioms are consistent’, ‘Those premises imply this conclusion’). It is no good dismissing grammatical number as a logically irrelevant complication like person or gender, since plural expressions are crucially involved in valid patterns of argument. To take an elementary example, ‘The Brontë sisters supported one another; the Brontë sisters were Anne, Charlotte and Emily; so Anne, Charlotte and Emily supported one another’. There can be no warrant for ignoring such patterns while attending to their singular counterparts. And some arguments do not even have a singular counterpart. For example, ‘Some numbers are prime. So some numbers are such that they are prime and a number is prime only if it is one of them.’ Logicians wedded to the singular logic of the predicate calculus typically try to dodge the issue of plurals by invoking sets, but we shall see that this is untenable.
Socrates exploited the difference between distributive and collective predicates in Hippias Major, but little of interest happened subsequently until Russell put plurals at the centre of his project for providing a foundation for mathematics, through his idea of the ‘class as many’ in The Principles of Mathematics. After another fallow period, the subject revived in the 1970s and 80s with the work of Black, Morton, Sharvy, Simons and Boolos. It would be premature to attempt a comprehensive survey. This entry offers a nontechnical outline of plural predicate logic, including the major differences between it and singular logic and some matters still to be resolved.
Church’s thesis
Authors: Shapiro Stewart 
Published: September 2018

An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A eventually produces fn as output. A function is ‘computable’ if there is an algorithm that computes it. A set S is ‘decidable’ if there is an algorithm that decides membership in S: if, given any appropriate n as input, the algorithm would output ‘yes’ if n∈S, and ‘no’ if n∉S. The notions of ‘algorithm’, ‘computable’ and ‘decidable’ are informal (or pre-formal) in that they have meaning independently of, and prior to, attempts at rigorous formulation.
Church’s thesis, first proposed by Alonzo Church in a paper published in 1936, is the assertion that a function is computable if and only if it is recursive: ‘We now define the notion…of an effectively calculable function…by identifying it with the notion of a recursive function….’ Independently, Alan Turing argued that a function is computable if and only if there is a Turing machine that computes it; and he showed that a function is Turing-computable if and only if it is recursive.
Church’s thesis is widely accepted today. Since an algorithm can be ‘read off’ a recursive derivation, every recursive function is computable. Three types of ‘evidence’ have been cited for the converse. First, every algorithm that has been examined has been shown to compute a recursive function. Second, Turing, Church and others provided analyses of the moves available to a person following a mechanical procedure, arguing that everything can be simulated by a Turing machine, a recursive derivation, and so on. The third consideration is ‘confluence’. Several different characterizations, developed more or less independently, have been shown to be coextensive, suggesting that all of them are on target. The list includes recursiveness, Turing computability, Herbrand–Gödel derivability, λ-definability and Markov algorithm computability.
Ordinal logics
Authors: Feferman Solomon 
Published: September 2018

By an ordinal logic is meant any uniform effective means of associating a logic (that is, an effectively generated formal system) with each effective ordinal representation. This notion was first introduced and studied by Alan Turing in 1939 as a means to overcome the incompleteness of sufficiently strong consistent formal systems, established by Kurt Gödel in 1931.
The first ordinal logic to consider, in view of Gödel’s results, would be that obtained by iterating into the constructive transfinite the process of adjoining to each system the formal statement expressing its consistency. For that ordinal logic, Turing obtained a completeness result for the class of true statements of the form that all natural numbers have a given effectively decidable property. However, he also showed that any ordinal logic (such as this) which is strictly increasing with increasing ordinal representation cannot have the property of invariance: in general, different representations of the same ordinal will have different sets of theorems attached to them. This makes the choice of representation a crucial one, and without a clear rationale as to how that is to be made, the notion of ordinal logic becomes problematic for its intended use.
Research on ordinal logics lapsed until the late 1950s, when it was taken up again for more systematic development. Besides leading to improvements of Turing’s results in various respects (both positive and negative), the newer research turned to restrictions of ordinal logics by an autonomy (or ‘boot-strap’) condition which limits the choice of ordinal representations admitted, by requiring their recognition as such in advance.