Ideas from 'Nominalism and Substitutional Quantifiers' by Ruth Barcan Marcus [1978], by Theme Structure

[found in 'Philosophy of Logic: an anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21868-8]].

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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
The nominalist is tied by standard semantics to first-order, denying higher-order abstracta
                        Full Idea: The nominalist finds that standard semantics shackles him to first-order languages if, as nominalists are wont, he is to make do without abstract higher order objects.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.166)
                        A reaction: Aha! Since I am pursuing a generally nominalist strategy in metaphysics, I suddenly see that I must adopt a hostile attitude to higher-order logic! Maybe plural quantification is the way to go, with just first-order objects.
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
Nominalists see proper names as a main vehicle of reference
                        Full Idea: For a nominalist with an ontology of empirically distinguishable objects, proper names are seen as a primary vehicle of reference.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.162)
Anything which refers tends to be called a 'name', even if it isn't a noun
                        Full Idea: The tendency has been to call any expression a 'name', however distant from the grammatical category of nouns, provided it is seen as referring.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.162)
5. Theory of Logic / G. Quantification / 1. Quantification
Nominalists should quantify existentially at first-order, and substitutionally when higher
                        Full Idea: For the nominalist, at level zero, where substituends are referring names, the quantifiers may be read existentially. Beyond level zero, the variables and quantifiers are read sustitutionally (though it is unclear whether this program is feasible).
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.167)
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Quantifiers are needed to refer to infinitely many objects
                        Full Idea: An adequate language for referring to infinitely many objects would seem to require variables and quantifiers in addition to names.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.164)
Substitutional semantics has no domain of objects, but place-markers for substitutions
                        Full Idea: On a substitutional semantics of a first-order language, a domain of objects is not specified. Variables do not range over objects. They are place markers for substituends (..and sentences are true-for-all-names, or true-for-at-least-one-name).
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.165)
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Maybe a substitutional semantics for quantification lends itself to nominalism
                        Full Idea: It has been suggested that a substitutional semantics for quantification theory lends itself to nominalistic aims.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.161)
A true universal sentence might be substitutionally refuted, by an unnamed denumerable object
                        Full Idea: Critics say if there are nondenumerably many objects, then on the substitutional view there might be true universal sentences falsified by an unnamed object, and there must always be some such, for names are denumerable.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.167)
                        A reaction: [See Quine 'Reply to Prof. Marcus' p.183] The problem seems to be that there would be names which are theoretically denumerable, but not nameable, and hence not available for substitution. Marcus rejects this, citing compactness.
Substitutional language has no ontology, and is just a way of speaking
                        Full Idea: Translation into a substitutional language does not force the ontology. It remains, literally, and until the case for reference can be made, a façon de parler. That is the way the nominalist would like to keep it.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.166)
7. Existence / A. Nature of Existence / 3. Being / i. Deflating being
Is being just referent of the verb 'to be'?
                        Full Idea: Being itself has been viewed as referent of the verb 'to be'.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.162)
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
Nominalists say predication is relations between individuals, or deny that it refers
                        Full Idea: Nominalists have the major task of explaining how predicates work. They usually construct it as a relation between individuals, or deny the referential function of predicates.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.163)
9. Objects / A. Existence of Objects / 3. Objects in Thought
If objects are thoughts, aren't we back to psychologism?
                        Full Idea: If objects are thoughts, aren't we back to psychologism?
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.166)
                        A reaction: Personally I don't think that would be the end of the world, but Fregeans go into paroxyms at the mention of 'psychology', because they fear that it destroys objectivity. That may be because they haven't understood thought properly.
9. Objects / F. Identity among Objects / 2. Defining Identity
Substitutivity won't fix identity, because expressions may be substitutable, but not refer at all
                        Full Idea: Substitutivity 'salve veritate' cannot define identity since two expressions may be everywhere intersubstitutable and not refer at all.
                        From: Ruth Barcan Marcus (Nominalism and Substitutional Quantifiers [1978], p.167)