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| A variable inside the scope of a quantifier is bound whereas a variable not bound by a quantifier is free. |
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| A sentence with more than one free variables is open whereas a sentence with no free variables is closed. |
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| These are individual constants (‘a’, ‘b’) which denote a specific individual. |
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To say that a three legged dog exists, according to frege, is to say that the concept three legged dog is not empty, the existential quantifier is a concept which applies to concepts, a second level function.
Fregian quantifiers are concepts which may apply to other concepts. |
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| Objectual versus substitutional interpretation |
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Definition
Objectual interpretation (Quine, Davidson) appeals to the values of the variables, the objects over which the variables range:
‘(x)Fx’ is interpreted as ‘For all objects, x, in the domain, D, Fx’
Substitutional interpretation (Mates, Marcus) appeals not to values but to substituends for the variables:
‘(x)Fx’ is interpretaed as ‘All substitution instances of ‘F…’ are true’ |
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| Quine on quantification and ontology |
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Definition
Ontology is part of metaphysics which concerns the question, what kinds of thing there are. Quine gives two key ideas:
To be is to be the value of a variable (criterion of ontological commitment)
No entity without identity (standards of ontological admissibility)
Intentional (meaning) notions are hopelessly unclear, therefore entities which can only be individuated by appeal to meaning—properties or propositions, for example-are not, but his standards, admissible. |
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| Criterion of ontological commitment |
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Definition
Quine Entities of a given sort are assumed by a theory if and only if some of them must be counted among the values of the variables, in order that the statements affirmed by the tory be true.
One tells what a theory says there is by putting it in predicate calculus notation, and asking what kinds of thing are required as values of its variables if theorems with existential quantiiers are to be true. So a theory which makes an existential claim “there exists an x such that x is prime and x is greater than 1000000’ is a theorem committed to the existence of prime numbers greater than a million, and a fortiori to the existence of prime numbers and to the existence of numbers.
Quine’s criteria is a trest of what a theory says there is, not what there is; what there is is what a true theory says there is.
Quine’s refusal to admit intensional entities acts as a filter: theories which say there are intensional entities are not really intelligible so a fortiori they are not true. |
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| Russell’s theorty of descriptions (elimination of singular terms)+Tarski’s theory of truth |
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| (objectual interpretation)→Quine’s ontological criterion (to be is to be the value of a variable)→rejection of second order quantification |
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| Quine’s ontological standard (no entity without identity)+attack on intensional |
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| Quine’s ontological standard (no entity without identity)+attack on intensional concepts→objection to propositions, properties, etc.-->rejection of second order quantification |
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| Quine’s rejection of singular terms |
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| \Quine rejects singular terms in two steps: first singlar terms are replaced by definite descriptions (replace singular terms like ‘a’ with predicates), second definite descriptions are eliminated in favor of quantifiers and variables (Russell). |
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| i. In the case of some proper names, one can supply a definite description which denotes the same thing—for instance “The teacher of Plato” is the definite descriptio of “Socrates”. Quine creates artificial predicates which are true of just the individual denoted by the name. |
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| Quine + Russell's Theory of Definite D |
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| ii. Russell’s theory of descriptions eliminates definite descriptions which noew repace singular terms. This eliminates definite descriptions in favour of quantifies, vriabls and identity. |
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iii. Thus sentences containing names such as “Socrates took poison” can be
a. Replaced by sentences containing descriptions (The x which socratises took poison)
b. And then by sentences containing only quantifiers and variables (There is just one x which socratises and whatever socratises took poison) |
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| Objection to Quine's rejection of singular terms |
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Definition
| An objection might be that Quine’s syetm of replacing singular terms with quantifiers and variables does not really show that singular terms are meaningless, it just asserts the same thing in a different way. However… |
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Quine’s insistence on the objectual intepretation of quantifiers is very important for Quine’s ontology: though the same theory could be expressed using singular terms as well as quantifiers, or combinatory operators instead of quantifiers, its quantificational form reveals its ontological commitments most transparently. (Quine believes that any theory in which a sentence containing a quantifier is true a fortiori presumes the existence of the object which satisfies the quantified predicate. So a theory in which “There is an x such that x is a prime greater than 1,000,000” is true asserts a fortiori the existence of numbers, prime numbers, and a prime niumber greater than 1,000,000).
So according to Quine, a sentence containing an existential quantifier says “there is something which…”
To insist on the correctness of the criterion, is indeed to say that there is no distinction between the ‘there are’ of ‘there are universals’, ‘there are unicorns’ and ‘there are hippopotami’ |
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On the objectual iterretation, ‘(3x)Fx’ means that there is an object x in the domain D which is F. If D is the universe than ‘(3x)Fx’ means that there is an (existant, real) object which is F.
If ‘(3x)Fx’ means that ‘There is an extant object x which is F’ than if ‘(3x)Fx’ is a theorem in a theory that theory is committed to there being Fs.
Haack: Quine’s new slogan is “to be said to be is to be the value of a variable bound by an objectual quantifier” |
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| Ostensibly existential assertions |
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Definition
There needs to be work to decide which ostensibly existential assertions of a theory need remain in primitive notation, and which are eliminabe by suitable paraphrase:
Morton White’s proposal—reduce “There is a possibility that James will come” to “That James will come is not certainly false” |
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| Substitutional quantification and ontology |
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Definition
Substitutional interpretation postpones answers to Quine’s ontological questions:
‘(3x)Fx’ means “Some substitution instance of “F…” is true”
Questions of existence now depend upon the conditions for the truth of the substitution instances.
If “Fa” is true only if ‘a’ is a singular term which denotes an existent object, then there will have to be an object which is F if ‘(3x)Fx’ is to come out true; but it is not inevitable that the truth conditions for the appropriate substitution instances will bring an ontological commitment.
Substitutional interpretation can avoid emberassing ontological commitments in logic if one believes that it ought not be a matter of logic whether anything exists. |
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| substitutional/objectual interpretation of non-existence claims |
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Definition
On the objectual interpretation, ‘(3x)Fx or -Fx’ asserts that there is at least one object which is either F or not F.
On a substitutional interpretation, ‘(3x)Fx or -Fx’ defers ontological commitment—if non-denoting terms are allowed as substituends then ontological commitment may be avoided. A substitutional interpretation allows for quantifiers to be treated metalinguistically (second level quantifiers) and to avoid objectual interpretations and thus avoid ontological commitment—however Quine says that this is evasion of metaphysical responsibility. |
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| Substitutional quantifiers and truth |
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Definition
If interpreted substitutionally, then the truth of quantified formulae can be defined directly in terms of the truth of atomic formulae
‘(3x)Fx’ is true iff some substitution instance of ‘F…’ is true |
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| material adequacy condition |
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| Tarski proposes material adequacy condition—any acceptable definition of truth must have as consequence all instances of the T-schema: ‘S’ is true iff ‘p’ where ‘S’ names the sentence ‘p’. |
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| Substitutional quantifiers and truth |
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Definition
If interpreted substitutionally, then the truth of quantified formulae can be defined directly in terms of the truth of atomic formulae
‘(3x)Fx’ is true iff some substitution instance of ‘F…’ is true
Tarski proposes material adequacy condition—any acceptable definition of truth must have as consequence all instances of the T-schema: ‘S’ is true iff ‘p’ where ‘S’ names the sentence ‘p’.
First order predicate calculus:
Substitutional interpretation will obviously make existentially quantified wffs true only if suitable substituends are available, so
‘(3x)(Fx or –Fx)’
will not, on its own, come out true since there are no singular terms which maye serve as substituends.
Second order quantification
On the objectual interpretation, ‘(3x)…’ says that there is an object such that… so it is expected that the appropriate substituends for bound variables should be expressions whose role is to denote objects, that is to say, singular terms. Quine sometimes defines singular terms as an expression which can take the position of the bound variable.
On substitutional interpretation, quantification is related directly not to objects but substituends so there is no need to insist that only expressions of the category of singular terms may be boud by quantifiers. |
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Objectual interpretation: quantifiers→objects
Substitutional interpretation: quantifiers→substituends→expressions of other syntactic categories OR singular terms→non-denoting OR denoting→objects |
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Nominalists admit the existence only of particulars whereas Paltonists allow for the reality of universals.
If only names are substitutable for bound variables, than one is forced to choose between nominalism (only first order quantification with variables replaced by names of particulars, Quine) |
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Nominalists admit the existence only of particulars whereas Paltonists allow for the reality of universals.
If only names are substitutable for bound variables, than one is forced to choose between nominalism (only first order quantification with variables replaced by names of particulars, Quine)
Or nominalistic Platonism (allow second-order quantification, with variables replaceable by names of abstract objects, properties or propositions: Church) |
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