Chapter Thirteen

1. Chapter Thirteen Identity and Philosophical Problems of Symbolic Logic

2. 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.

3. 2. Definite Descriptions A description that picks out one definite entity is a definite description.

4. 3. Properties of Relations Relational properties are used in mathematics and the sciences to generate orders between things.

5. 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.

6. Properties of Relations, continued All two-place relations are transitive, intransitive, or nontransitive.

7. 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.

8. 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.

9. Properties of Relations, continued All relations that are neither transitive or intransitive are nontransitive.

10. Properties of Relations, continued A situation is totally reflexive iff everything must bear that relation to itself.

11. Properties of Relations, continued A relation is reflexive iff x bears that relation to y, then x must bear it to itself.

12. Properties of Relations, continued A relation is irreflexive iff nothing can bear that relation to itself.

13. 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.

14. 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.

15. 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.

16. 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.

17. Limitations of Predicate Logic, continued Arguments with missing premises are called enthymemes.

18. Limitations of Predicate Logic, continued Dispositional properties are powers, potentials, or dispositions of objects.

19. Limitations of Predicate Logic, continued It has been suggested that the correct analysis of dispositional sentences is into subjunctive or contrary-to-fact conditionals.

20. 6. Philosophical Problems One basic philosophical issue concerns whether logic deals with sentences or propositions.

21. 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.

22. Philosophical Problems, continued There are philosophical issues concerning the status of sentence connectives in predicate logic.

23. 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.

24. 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.

25. 7. Logical Paradoxes Higher-order logics are plagued by logical paradoxes.

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

27. 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.

28. Logical Paradoxes, continued Is the property of being impredicable predicable or impredicable? One solution to this paradox is the simple theory of types.

29. 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.

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

31. Key Terms, continued • Intensional context • Intentional context • Intransitive relation • Inflexive relation • Levels of language theory • Logical paradoxes • Nonreflexive relation • Nonsymmetrical relation

32. Key Terms, continued • Nontransitive relation • Paradoxes of material implication • Property variable • Reflexive relation • Semantic paradox • Sentence token • Sentence type • Simple theory of types

33. Key Terms, continued • Subjunctive conditional • Symmetrical relation • Syntactic paradox • Totally reflexive relation • Transitive relation