Lecture 2: The “Traditional” Concerns (And A Little More)

1 Marcus Rossberg, Arché Centre1, University of St Andrews, mr30@st-andrews.ac.uk

Second-Order Logic
Socrates Teaching Mobility Intensive Seminar, University of Helsinki, 16-19 May, 2005

Lecture 2: The “Traditional” Concerns (and a little more)
The Worries 1. 2. 3. 4. Quine I: Quantification over Attributes Quine II: “Staggering Ontological Commitment” to Sets Incompleteness MathematicalContent

Quine I: Quantification over Attributes In On What There Is Quine lets his fictional character McX (read: Armstrong) who believes in the existence of universals or attributes say: There are red houses, red roses, red sunsets; this much is prephilosophical common sense in which we must all agree. These houses, roses, and sunsets, then, have something in common; and this which they havein common is all I mean by the attribute of redness. Quine responds: The words ‘houses’, ‘roses’, and ‘sunsets’ are true of sundry individual entities which are houses, roses, and sunsets, and the word ‘red’ or ‘red object’ is true of each of sundry individual entities which are red houses, red roses, and red sunsets; but there is not, in addition, any entity whatever, individual or otherwise,which is named by the word ‘redness’, nor, for that matter, by the word ‘househood’, ‘rosehood’, ‘sunsethood’. (Quine 1948/1953, p. 10) Since we do not have a good criterion of identity for attributes, such obscure entities have to be disqualified from serious intellectual enterprises: No entity without identity. (Quine 1958/1969, p. 23; see also Quine 1941/1966, §6)

2 At least at some pointQuine also seemed to be willing to suggest that there is something incoherent about quantifying into predicate position. He proposes the following “derivation” of a contradiction analogous to Russell’s Paradox of the class of non-selfmembered classes (Quine 1947, p. 78; I simplified Quine’s “proof” slightly here): (1) GH ≡ GH (2) ∀H (GH ≡ GH) (3) ∃F ∀H (FH ≡ GH) (4) ∃F ∀H (FH ≡ ~HH) (5) ∃F (FF ≡~FF) Exercise: What is wrong with this? tautology (1), universal generalisation (2), existential generalisation (3), uniform substitution [~HH/GH] (4)

Quine II: “Staggering Ontological Commitment” to Sets Classes, down the years I have grudgingly admitted; attributes not. I have felt that if I must come to terms with Platonism, the least I can do is keep it extensional. (Quine 1975, p. 100). Inthe first edition of his Philosophy of Logic (1970), Quine speaks of the staggering ontological commitment of second-order logic to the set theoretic hierarchy; in the second edition, he more carefully phrases: “a fair bit of set theory has slipped in unheralded” (Quine 1986, p. 68). How so? [E]ntities of a given sort are assumed by a theory if and only if some of them must be counted among thevalues of the variables in order that the statements affirmed in the theory be true. (Quine 1953, p. 103) The universe of entities is the range of values of variables. To be is to be the value of a variable. (Quine 1939, p. 708)

Incompleteness Curiously enough, this charge is not raised by Quine against second-order logic in any of his major publications on the issue, although he complainsabout incompleteness regarding logics with branching quantifiers, such as IF logic (see Quine 1986, p. 90). Others, though, see completeness as a necessary ingredient of a “proper” logic (e.g. Tharp, Wolenski; cf. also Shapiro 1991, chapter 5, and the discussion in Kreisel).

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Second-order consequence is taken to be intractable due to the lack of a complete proof procedure. (For a rebuttal ofother intractability allegations see Shapiro 1999.) • Boolos asks: Why the emphasis on completeness, and not decidability? Monadic second-order logic is decidable (cf. Löwenheim; it remains decidable even if we add two successor functions, cf. Rabin); polyadic first-order logic is not. Wagner demands semi-decidability – but semi-decidability of what? Theoremhood? Validity? The former holds for...