Chapter Thirteen

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Chapter Thirteen

• Identity and Philosophical Problems of Symbolic
  Logic

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1. Identity

• We use the identity symbol so we can
  correctly translate statements of identity into
  our logical notation.
• The rule of identity (ID) states that we can
  translate identicals into identicals.
• The rule of Identity Reflexivity (IR) allows
  introduction of the formula (x)(x x) into a
  proof at any time.

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2. Definite Descriptions

• A description that picks out one definite entity
  is a definite description.

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3. Properties of Relations

• Relational properties are used in mathematics and
  the sciences to generate orders between things.

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Properties of Relations, continued

• A relation is symmetrical iff when one thing
  bears that relation to the second the second must
  bear it to the first.
• A relation is asymmetrical iff when one thing
  bears that relation to a second, the second thing
  cannot bear it to the first.

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Properties of Relations, continued

• All two-place relations are transitive,
  intransitive, or nontransitive.

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Properties of Relations, continued

• A relation is transitive iff when one thing bears
  that relation to a second, and the second to a
  third, then the first must bear it to the third.

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Properties of Relations, continued

• A relation is intransitive iff when one thing
  bears that relation to a second, and the second
  to a third, then the first cannot bear it to the
  third.

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Properties of Relations, continued

• All relations that are neither transitive or
  intransitive are nontransitive.

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Properties of Relations, continued

• A situation is totally reflexive iff everything
  must bear that relation to itself.

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Properties of Relations, continued

• A relation is reflexive iff x bears that relation
  to y, then x must bear it to itself.

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Properties of Relations, continued

• A relation is irreflexive iff nothing can bear
  that relation to itself.

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4. Higher-Order Logics

• A predicate logic that forbids sentences that
  ascribe properties to properties themselves and
  restricts quantification to individual variables
  is a first-order predicate logic.

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Higher-Order Logics, continued

• In a higher-order logic, we can have property
  variables as well as individual variables.
• In a higher-order logic properties can have
  properties.

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5. Limitations of Predicate Logic

• There seem to be arguments that, while invalid
  using the notation and proof techniques of
  predicate logic, are valid in some wider (perhaps
  ideal) deductive system, for example, arguments
  using indirect or intensional contexts.

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Limitations of Predicate Logic, continued

• Contexts that use terms that refer to states of
  the mind are called intentional contexts.
• These are a species of intensional contexts.

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Limitations of Predicate Logic, continued

• Arguments with missing premises are called
  enthymemes.

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Limitations of Predicate Logic, continued

• Dispositional properties are powers, potentials,
  or dispositions of objects.

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Limitations of Predicate Logic, continued

• It has been suggested that the correct analysis
  of dispositional sentences is into subjunctive or
  contrary-to-fact conditionals.

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6. Philosophical Problems

• One basic philosophical issue concerns whether
  logic deals with sentences or propositions.

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Philosophical Problems, continued

• Sentential logic is a two-valued truth-functional
  logic. But it has been argued that most natural
  language sentences do not have two truth-values.

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Philosophical Problems, continued

• There are philosophical issues concerning the
  status of sentence connectives in predicate logic.

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Philosophical Problems, continued

• There are philosophical difficulties with
  truth-functional connectives.
• For example, the use of a truth-functional
  conditional has been objected to on the grounds
  that is generates so-called paradoxes of material
  implication.

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Philosophical Problems, continued

• It is not clear what a deductively valid
  argument is, since the terms must and
  impossible that are used to describe such
  arguments are ambiguous.

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7. Logical Paradoxes

• Higher-order logics are plagued by logical
  paradoxes.

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Logical Paradoxes, continued

• If we allow the predication of properties to
  properties, then syntactic paradoxes can be
  generated, such as the impredicable paradox.

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Logical Paradoxes, continued

• A property that can be truly predicated of itself
  is a predicable property, and a property that
  cannot be predicated of itself is an impredicable
  property.

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Logical Paradoxes, continued

• Is the property of being impredicable predicable
  or impredicable?
• One solution to this paradox is the simple theory
  of types.

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Logical Paradoxes, continued

• There are also semantic paradoxes, such as the
  paradox of the liar.
• One way to solve these paradoxes is to
  distinguish between levels of language languages
  used to talk about non-linguistic things and
  languages used to talk about language.

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Key Terms

• Asymmetrical relation
• Contrary-to-fact conditional (counterfactual)
• Definite description
• Dispositional property
• Enthymeme
• First-order predicate logic
• Impredicable paradox
• Indirect context

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Key Terms, continued

• Intensional context
• Intentional context
• Intransitive relation
• Inflexive relation
• Levels of language theory
• Logical paradoxes
• Nonreflexive relation
• Nonsymmetrical relation

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Key Terms, continued

• Nontransitive relation
• Paradoxes of material implication
• Property variable
• Reflexive relation
• Semantic paradox
• Sentence token
• Sentence type
• Simple theory of types

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Key Terms, continued

• Subjunctive conditional
• Symmetrical relation
• Syntactic paradox
• Totally reflexive relation
• Transitive relation