Abstract
I attempt an explication of what it means for an operation across domains to be the same on all domains, an issue that (Feferman, S.: Logic, logics and logicism. Notre Dame J. Form. Log. 40, 31–54 (1999)) took to be central for a successful delimitation of the logical operations. Some properties that seem strongly related to sameness are examined, notably isomorphism invariance, and sameness under extensions of the domain. The conclusion is that although no precise criterion can satisfy all intuitions about sameness, combining the two properties just mentioned yields a reasonably robust and useful explication of sameness across domains.
*Sol Feferman was an inspiration to me for most of my adult life and, for almost as long, a friend. I’m delighted to have been asked to contribute to this volume in his honor. As to the paper, thanks to Lauri Hella and Jouko Väänänen for helpful conversations on the model-theoretic parts, and to Stanley Peters who saved me from a rather embarrassing mistake (remaining ones are mine, of course). I also want to thank the two extremely encouraging and patient editors of this volume.
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Notes
- 1.
- 2.
Thereby avoiding to treat as logical, for example, an operator which behaves as the universal quantifier on universes containing the number 0, and as the existential quantifier on other universes. Also, Sher restricted attention to operations of type level at most 2, and Feferman in general follows this restriction.
- 3.
Stipulating that the cartesian product of the empty sequence () of sets is \(\{\emptyset \}\), we have \(M_{()} = \mathcal {P}(\{\emptyset \}) = \{\emptyset \{\emptyset \}\} = \{0,1\}\), so () corresponds to the type t in TFT.
- 4.
If \(\pi _m\) is the TFT type symbol \(\langle e^m, t\rangle \) of m-ary relations between individuals (and \(\pi _0=t\)), type \(\langle k_1,\ldots ,k_n\rangle \) corresponds to \(\langle \pi _{k_1}\ldots \pi _{k_n}, t\rangle \). Similarly in RT, it corresponds to \(((e^{k_1}),\ldots ,(e^{k_n}))\).
- 5.
I am using the same notation for the conjunction symbol and its interpretation (its corresponding quantifier), and similarly later on for other quantifier symbols; this should not generate confusion.
- 6.
- 7.
Call a TFT-type symbol \(\tau \) truth-functional if it does not contain e, and extended truth-functional if it is either truth-functional or of the form
-
\(\langle \sigma _{11}\ldots \sigma _{1k_1},\ldots ,\langle \sigma _{n1}\ldots \sigma _{nk_n},e\rangle \ldots \rangle \)
where each \(\sigma _{ij}\) is truth-functional. [24] shows that the monotone types in TFT are exactly the extended truth-functional ones, and that all other TFT-types \(\tau \) are disjoint in the sense that \(M\ne M'\) implies \(\mathcal {M}_\tau \cap \mathcal {M}'_\tau = \emptyset \).
One way to make TFT-types monotone is to allow partial functions rather than just total ones. But this raises other issues for a formulation of Ext; the situation is further discussed in [24].
-
- 8.
E.g. in type \(\langle 1,2\rangle \), \(\lnot \mathbf {0}_M(A,R)\) for all \(A\subseteq M\) and \(R\subseteq M^2\), and \(\mathbf {1}_M(A,R)\) for all \(A\subseteq M\) and \(R\subseteq M^2\).
- 9.
In the literature, the label ‘Rescher quantifier’ is often used for a stronger quantifier of type \(\langle 1,1\rangle \), let us call it more, where \(\textit{more}_M(A,B) \Leftrightarrow |A|>|B|\) (which, incidentally, is Ext). (For example, the survey article [10] uses ‘\(Q^R\)’ for more.) My usage here is historically more accurate, since precisely the type \(\langle 1\rangle \) quantifier \(Q^R\) was introduced in the abstract [13]. Rescher also mentioned the relativization of \(Q^R\), which is the quantifier most, defined by \(\textit{most}_M(A,B) \Leftrightarrow |A\cap B|>|A-B|\); most is strictly stronger than \(Q^R\), but strictly weaker than more; see, for example, [11], Ch. 14.3.
- 10.
More precisely, it means that determiner denotations are relativizations of type \(\langle 1\rangle \) quantifiers (see Sect. 5), which, for type \(\langle 1,1\rangle \) quantifiers, can easily be shown to be equivalent to saying that they satisfy Ext and conservativity:
(Conserv) For all M and all \(A,B\subseteq M\), \(Q_M(A,B) \Leftrightarrow Q_M(A,A\cap B)\).
- 11.
These denotations are \(\forall \), \(Q^R\), and \((\exists _{=5})\lnot \), respectively. On the other hand, something, or at least three things, also contain thing, but have Ext denotations.
- 12.
We assume that L extends FOL and is closed under substitution of formulas for predicate letters.
- 13.
If \(L = L(Q_1,Q_2,\ldots )\), then L relativizes if each \(Q_i^{rel}\) is definable in L. Similarly for other logics; for example, FOL, \(L_{\omega _1\omega }\), and \(L_{\omega _1\omega _1}\) relativize, but not e.g. \(L_{\omega \omega _1}\).
- 14.
\(Q^H\) essentially involves the whole universe, which is reflected in the fact that \((Q^H)^{uni} = Q^H\).
- 15.
I have not attempted a general statement and proof of this fact here, but intuitively it should be fairly clear. Consider, for example, \(\textit{ WO}^{\,uni}\), where \(\textit{WO}_M(R)\) says that R is a well-ordering of its field. Let \(\eta \) be the order type of the rationals, and let \(\mathcal {M}= (M,A,R)\) be a model where R is a linear order of M with the order type \(\eta + \omega + \eta \) and A is the subset corresponding to \(\omega \), and let \(\mathcal {M}' = (M',A',R')\) be a similar model, but where \(R'\) has the order type \(\eta + \omega + \omega ^* + \omega + \eta \), and \(A'\) corresponds to \(\omega + \omega ^* + \omega \). Then \((\textit{ WO}^{\,uni})^{rel}_M(A,R)\) holds and \((\textit{ WO}^{\,uni})^{rel}_{M'}(A',R')\) fails, but \(\mathcal {M}\) and \(\mathcal {M}'\) cannot be distinguished in \(L(\textit{WO}^{\,uni})\). Indeed, over \(\mathcal {M}'\), \(L(\textit{WO}^{\,uni})\) is equivalent to FOL, since there are no definable well-orders of the universe in \(\mathcal {M}'\).
- 16.
Of course, in this case, this is immediate from a slight adjustment of the order axioms; the point here was to bring out the general situation.
- 17.
However, Jouko Väänänen pointed out an exception (p.c.): the quantifier (not mentioned in [10]) \((Q^{card})_M(R) \Leftrightarrow R\) is a \( |\textit{field}(R)|\)-like linear ordering of its field, that is, each proper initial segment has cardinality \(< |\textit{field}(R)|\). Then the sentence saying that R is discrete with a least element, and that for all a, \((Q^{card})_M(R\!\upharpoonright \!\{b\!: bRa\}))\), characterizes \((\omega ,<)\), so \(L(Q^{card})\) is not compact. On the other hand, it can be shown with techniques that go back to Vaught and Fuhrken that \(L((Q^{card})^{uni})\) is (countably) compact. So my claim is indeed tentative and does not hold across the board; it may be still of some interest to find sufficient conditions for when it does.
- 18.
In other respects, \(L(Q^R)\) and \(L(Q^C)\) are quite different: \(L(Q^C)\) reduces to FOL on finite models, whereas [1] showed that FOL \(< L(Q^R) < L(\textit{most})\) even on finite models (illustrating the need for type \(\langle 1,1\rangle \) quantifiers in natural language semantics).
- 19.
[1] uses thing as a logical constant in its formal language.
- 20.
Provided Q is Conserv and Ext (see [11], Ch. 6.1).
- 21.
As we noted, the proposal that Same = Perm + Ext is equivalent. We can also formulate the proposal more succinctly: say that an n-ary operation O across domains of relational type is closed under injections if for all M, all relations \(R_i\) over M of suitable type, and all injections \(\pi \) from M to \(M'\),
- (Inj):
-
\(O_M(R_1,\ldots ,R_n) \Leftrightarrow O_{M'}(\pi (R_1),\ldots ,\pi (R_n))\)
It is easy to check that Inj \(\equiv \) Isom + Ext. However, the ideas behind Isom and Ext are quite different, so they might as well be kept separate.
- 22.
This is a special case of the more general truth conditions for possessives discussed at length in [12].
- 23.
We may note that, although test2 and \(\forall \)even appear to be very different quantifiers, and intuitively the latter seems clearly non-Same, they have essentially the same ‘jumping behavior’.
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Westerståhl, D. (2017). Sameness. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_16
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