➕Logic and Formal Reasoning Unit 9 – Identity and Definite Descriptions

Key Concepts and Definitions

• Identity is a fundamental concept in logic that refers to the relation each thing bears to itself
• Leibniz's Law, also known as the indiscernibility of identicals, states that if a is identical to b, then any property that a has, b will have as well
• Definite descriptions are phrases that purport to pick out a unique individual, such as "the tallest man in the room" or "the current King of France"
• A proper name is a word or phrase that refers to a specific individual (Aristotle, New York City)
  • Proper names are considered rigid designators, meaning they refer to the same individual in all possible worlds
• Propositional attitudes are mental states, such as beliefs or desires, that are directed towards propositions
  • Examples include "John believes that the morning star is Venus" and "Mary hopes that the evening star is bright tonight"
• The morning star and evening star both refer to the planet Venus, despite having different senses or cognitive significance
• Frege's distinction between sense and reference is central to understanding the puzzles surrounding identity statements

Historical Context and Philosophers

• The problem of identity and definite descriptions has roots in ancient Greek philosophy, with thinkers like Parmenides and Heraclitus grappling with the nature of change and identity
• Gottlob Frege, a German philosopher and logician, made significant contributions to the study of identity in the late 19th century
  • Frege introduced the distinction between sense and reference in his seminal work "Über Sinn und Bedeutung" (On Sense and Reference)
• Bertrand Russell, a British philosopher, developed his theory of descriptions in the early 20th century as a response to Frege's puzzle and other problems surrounding definite descriptions
• Saul Kripke, an American philosopher and logician, introduced the concept of rigid designators in the 1970s, which has important implications for the semantics of proper names and identity statements
• Contemporary philosophers continue to debate and refine theories of identity and definite descriptions, drawing on insights from linguistics, cognitive science, and other fields

Identity in Logic

• Identity is typically represented using the equals sign (=), with statements like $a = b$ asserting that a and b are the same thing
• The reflexive property of identity states that everything is identical to itself: $\forall x (x = x)$
• The symmetric property of identity holds that if $a = b$, then $b = a$
• The transitive property of identity asserts that if $a = b$ and $b = c$, then $a = c$
• Substitutivity of identicals is a principle that allows the replacement of co-referential terms in a sentence without changing the truth value
  • For example, if "the morning star" and "the evening star" refer to the same object (Venus), then the sentences "The morning star is bright" and "The evening star is bright" should have the same truth value
• Identity is closely related to the concepts of necessity and possibility in modal logic
  • If $a = b$ is true, then it is necessarily true in all possible worlds

Definite Descriptions Explained

• Definite descriptions are phrases that refer to a unique individual by specifying a property or set of properties that only that individual possesses
• The logical form of a definite description can be represented as $\exists x (Fx \land \forall y (Fy \to y = x))$, where $Fx$ is the property (or properties) that uniquely identifies the individual
• Definite descriptions can be used in both subject and predicate positions in a sentence
  • Subject position: "The tallest man in the room is wearing a hat"
  • Predicate position: "John is the tallest man in the room"
• Improper or non-denoting descriptions are definite descriptions that fail to refer to any existing individual, such as "the current King of France"
  • The treatment of non-denoting descriptions is a central problem in the philosophy of language and logic
• Definite descriptions can give rise to de re and de dicto ambiguities in propositional attitude reports
  • De re (of the thing) interpretations attribute a property directly to the individual referred to by the description
  • De dicto (of what is said) interpretations attribute a property to the individual under the description used

Frege's Puzzle and Related Problems

• Frege's puzzle arises from the observation that identity statements like "the morning star is the evening star" can be informative, while statements like "the morning star is the morning star" are trivial
  • This suggests that there is more to the meaning of an expression than just its reference
• Frege proposed that expressions have both a sense (Sinn) and a reference (Bedeutung)
  • The sense of an expression is the cognitive significance or mode of presentation, while the reference is the actual object denoted
• The substitution problem concerns the failure of co-referential terms to be substitutable in certain contexts, particularly in propositional attitude reports
  • For example, "John believes that the morning star is Venus" may be true, while "John believes that the evening star is Venus" could be false, even though "the morning star" and "the evening star" refer to the same object
• Opaque contexts are linguistic environments in which the substitution of co-referential terms does not preserve truth-value, such as in propositional attitude reports or modal contexts
• Frege's distinction between sense and reference offers a potential solution to these puzzles, as it allows for expressions to differ in cognitive significance even when they share the same reference

Russell's Theory of Descriptions

• Bertrand Russell developed his theory of descriptions as a response to Frege's puzzle and other problems surrounding definite descriptions
• According to Russell, definite descriptions are not genuine referring expressions but rather disguised existential quantifications
• The logical form of a sentence containing a definite description, such as "The F is G," can be analyzed as $\exists x (Fx \land \forall y (Fy \to y = x) \land Gx)$
  • This captures the uniqueness and existence implications of definite descriptions
• Russell's theory provides a uniform treatment of both proper and improper descriptions
  • For improper descriptions like "the current King of France," the analyzed logical form would be false, as there is no unique individual satisfying the description
• The theory of descriptions allows for the elimination of definite descriptions from a logical language, reducing them to more basic quantificational and predicational structures
• Russell's approach has been influential in the development of formal semantics and the analysis of natural language

Applications in Natural Language

• Definite descriptions are ubiquitous in natural language and play a crucial role in communication and reference
• Anaphoric uses of definite descriptions involve referring back to a previously mentioned or contextually salient individual (John bought a car. The car was red.)
• Bridging uses of definite descriptions rely on associative relationships or inference to identify a unique referent (Mary bought a book. The author was famous.)
• Generic uses of definite descriptions refer to a kind or class of individuals rather than a specific one (The lion is a majestic animal.)
• Definite descriptions can interact with other linguistic phenomena, such as presupposition, implicature, and ellipsis
  • Presupposition: The use of a definite description typically presupposes the existence and uniqueness of the referent
  • Implicature: The choice of a particular definite description can convey additional information beyond its literal meaning
• The interpretation of definite descriptions often depends on contextual factors and pragmatic considerations, requiring a nuanced approach to their analysis in natural language

Criticisms and Alternative Approaches

• P.F. Strawson, a British philosopher, criticized Russell's theory of descriptions for failing to capture the referential uses of definite descriptions in ordinary language
  • Strawson argued that definite descriptions presuppose, rather than assert, the existence and uniqueness of their referents
• Keith Donnellan, an American philosopher, distinguished between the attributive and referential uses of definite descriptions
  • Attributive uses focus on the descriptive content, while referential uses aim to pick out a specific individual, even if the description is not strictly satisfied
• Saul Kripke's causal theory of reference challenges the descriptivist view of proper names and definite descriptions
  • According to Kripke, the reference of a name is determined by a causal-historical chain rather than a set of identifying descriptions
• Discourse representation theory (DRT) and file change semantics (FCS) offer dynamic approaches to the interpretation of definite descriptions, taking into account the evolving context and information states of speakers and hearers
• Game-theoretic and pragmatic approaches to definite descriptions emphasize the role of strategic reasoning and communicative goals in their use and interpretation
• Ongoing research in linguistics, philosophy, and cognitive science continues to refine and extend theories of definite descriptions, incorporating insights from various subfields and methodologies