Download Identity and Philosophical Problems of Symbolic Logic

Chapter Thirteen
Identity and Philosophical
Problems of Symbolic Logic
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.
2. Definite Descriptions
A description that picks out one definite entity is a definite
description.
3. Properties of Relations
Relational properties are used in mathematics and the
sciences to generate orders between things.
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.
Properties of Relations, continued
All two-place relations are transitive, intransitive, or
nontransitive.
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.
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.
Properties of Relations, continued
All relations that are neither transitive or intransitive are
nontransitive.
Properties of Relations, continued
A situation is totally reflexive iff everything must
bear that relation to itself.
Properties of Relations, continued
A relation is reflexive iff x bears that relation to y,
then x must bear it to itself.
Properties of Relations, continued
A relation is irreflexive iff nothing can bear that
relation to itself.
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.
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.
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.
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.
Limitations of Predicate Logic,
continued
Arguments with missing premises are called
enthymemes.
Limitations of Predicate Logic,
continued
Dispositional properties are powers, potentials, or
dispositions of objects.
Limitations of Predicate Logic,
continued
It has been suggested that the correct analysis of
dispositional sentences is into subjunctive or
contrary-to-fact conditionals.
6. Philosophical Problems
One basic philosophical issue concerns whether
logic deals with sentences or propositions.
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.
Philosophical Problems, continued
There are philosophical issues concerning the status
of sentence connectives in predicate logic.
Philosophical Problems, continued
There are philosophical difficulties with truthfunctional 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.
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.
7. Logical Paradoxes
Higher-order logics are plagued by logical
paradoxes.
Logical Paradoxes, continued
If we allow the predication of properties to
properties, then syntactic paradoxes can be
generated, such as the impredicable paradox.
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.
Logical Paradoxes, continued
Is the property of being impredicable predicable or
impredicable?
One solution to this paradox is the simple theory of
types.
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.
Key Terms
• Asymmetrical relation
• Contrary-to-fact conditional (counterfactual)
• Definite description
• Dispositional property
• Enthymeme
• First-order predicate logic
• Impredicable paradox
• Indirect context
Key Terms, continued
• Intensional context
• Intentional context
• Intransitive relation
• Inflexive relation
• Levels of language theory
• Logical paradoxes
• Nonreflexive relation
• Nonsymmetrical relation
Key Terms, continued
• Nontransitive relation
• Paradoxes of material implication
• Property variable
• Reflexive relation
• Semantic paradox
• Sentence token
• Sentence type
• Simple theory of types
Key Terms, continued
• Subjunctive conditional
• Symmetrical relation
• Syntactic paradox
• Totally reflexive relation
• Transitive relation