Abstract
Wittgensteinian predicate logic (W-logic) is characterized by the requirement that the objects mentioned within the scope of a quantifier be excluded from the range of the associated bound variable. I present a sound and complete tableaux calculus for this logic and discuss issues of translatability between Wittgensteinian and standard predicate logic in languages with and without individual constants. A metalinguistic co-denotation predicate, akin to Frege’s triple bar of the Begriffsschrift, is introduced and used to bestow the full expressive power of first-order logic with identity on W-logic in the presence of constants.
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Notes
See also Hintikka’s examples in Hintikka (1956, p. 225).
It is less clear whether the mutual interpretability can serve as a positive argument for the elimination of identity. Since the translation goes both ways, friends of identity might argue that FOL= simply makes a commitment to identity explicit that W-logic hides in its quantifier semantics.
We are here, for the most part, following Smullyan (1968, chapter 4). Note that Smullyan’s parameters are essentially typographically distinguished free variables, as used e.g. in the Gentzen tradition (see for example Schütte [1977]). The use of such parameters is convenient for tableaux calculi, but by no means indispensible. W-logic can just as well be done with just one sort of individual variable; we will indicate the adjustments necessary to the translations between W-logic and FOL= in the notes. Similarly, the use of elements of a domain as constituents of formulas is simply a matter of convenience that allows one to suppress mention of variable assignments in the definition of truth in a structure; the influential textbook by Shoenfield (1967) has made this technique popular.
The language \({\mathcal{L}}^=\) of FOL= results from \({\mathcal{L}}\) by the addition of a distinguished binary predicate symbol =, the equality symbol; semantically, it is interpreted as true identity.
As promised, we briefly address languages without a set of designated free variables (parameters). The translation ψ is actually simpler when working in such a language. The clauses for atomic formulae and Boolean connectives are essentially the same; however, in the universal quantifier case, ψ(∀xF) is literally \({\forall}x \left(\bigwedge_{i=1}^n x\not= y_i \to \psi(F)\right),\) without needing to introduce ψ(F) as an abbreviation (where y 1,..., y n are the variables free in \(\forall xF).\)
Again, here are the relevant translation clauses with respect to first-order languages that don’t have typographically distinguished free variables (parameters):
(1) φ(x = x) is Px→ Px.
(2) φ(x = y) is \(\neg (Rxy \to Rxy),\) where x and y are distinct individual variables.
(3) φ(∀xF) is
$$ {\forall}x\varphi(F)\wedge \bigwedge_{i=1}^n \varphi(F^x_{x_i}), $$where it is assumed that, before substituting x i for x in F, bound variables in F have been renamed so as to avoid unintentional capturing of x i in the result of the substitution (and similarly for the existential case).
Note how, in the quantifier case, our original version with parameters obviates the need for renaming bound variables in the second conjunct. On the other hand, languages with parameters are clumsier with respect to the first conjunct, where one must first replace the bound variable x with a fresh parameter a, then effect the translation, and then re-substitute x for a in the result.
These tableaux rules are the obvious analogues of the rules for the Gentzen sequent calculus introduced in Wehmeier (2004). Soundness and completeness for the tableaux calculus may thus be inferred from the corresponding properties of that sequent calculus. I wish to note, however, that the soundness and completeness of the sequent calculus were established only indirectly in the mentioned paper, that is, by reference to a sound and complete calculus for FOL=. The proofs outlined here connect the W-procedure directly with the semantics for W-logic and should thus be of independent interest. For the record, I would like to note that soundness and completeness proofs for what is essentially the W-procedure were first given in an unpublished term paper by Sam Hillier, written under my supervision.
Let us note here that (W-UI) must insist on the instantiating parameter already occurring on the branch (except in the case where there are no parameters on the branch whatsoever). In tableaux calculi for FOL, it is strategically a good idea to use such instantiating parameters, but certainly not required for the soundness of the calculus. The W-procedure, however, actually becomes unsound if arbitrary instantiating parameters are allowed (even if these are restricted to those not ocurring in the quantified sentence itself). This is due to the fact that it is impossible to properly extend σ in a 1−1 way if the range of σ already exhausts U. Here’s an example:
$$ \frac{(1)\quad {\forall}x{\forall}y (Rxy \wedge \neg Rxy)} {(2)\quad {\forall}y (Ray \wedge\neg Ray)}\hbox{UI from (1)} $$At this point, the tree cannot be continued in accordance with the W-procedure: Instantiating the sentence in line (2) to the parameter a is not permissible, because a occurs in that very sentence. Instantiating to another parameter is not allowed, because there are already parameters on the branch, viz. a. And this is as it should be, because the sentence at the origin is W-valid in every domain of cardinality 1. If we allowed instantiation to arbitrary parameters not occurring in the quantified sentence under consideration, we’d be able to continue the above tree (obtaining line (3) from line (2) by an illegitimate application of universal instantiation) as follows
$$ \frac{(3)\quad Rab \wedge\neg Rab} {\frac{(4)\quad Rab}{(5)\quad \neg Rab} \wedge\hbox{-rule from} (3)} \wedge\hbox{-rule from} (3) $$and end up with a closed tree, incorrectly indicating that the sentence in line (1) should be W-unsatisfiable.
Beginning with a single-node tree T 0 and successively applying the rules of the W-procedure, one obtains a sequence T 0,T 1,T 2,… of trees. If every T i is extendable by a rule application, this sequence may be continued indefinitely. In this case, we will also refer to its limit \(\bigcup_{i=0}^\infty T_i\) as a tree constructed according to the W-procedure.
It is perhaps interesting to note that, by imposing restrictions on the quantifier rules of the Burgess–Boolos calculus for FOL, one obtains a calculus sound and complete for W-logic, which we have shown to be stronger than FOL, viz. equivalent to FOL=. Hintikka makes a similar observation with respect to his calculus in Hintikka (1956, p. 237).
Indeed, it will produce a finite model of minimal cardinality. Proof: let the model \({\mathcal{U}}\) of the argument just given be of minimal cardinality. This was observed, for the Burgess–Boolos calculus, by Perry Smith (1985).
Parameters are not individual constants in the sense considered here, but rather typographically distinguished free variables. Constants, as explained below, are interpreted as fixed objects in a structure, whereas parameters may assume, in the same model, arbitrary values over the domain of discourse.
Wittgenstein seems to have had a similar usage in mind for the equality symbol; cf. Tractatus 4.241 (Wittgenstein 1922).
See also Charles Caton’s discussion in Caton (1976, pp. 174–175).
Mendelsohn (2005, pp. 60–61) also points out that neither the autonymous use of individual constants in ≡-contexts nor the simultaneous autonymous and non-autonymous use of constants in separate contexts constitutes a logical error. It is only the simultaneous quantification into a position reserved for autonymous terms and a position reserved for non-autonymous terms that arguably creates gibberish.
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Acknowledgements
Thanks to Sam Hillier for various discussions concerning W-logic, and to Peter Schroeder-Heister for the suggestion to provide tableaux rules for W-logic. I also wish to acknowledge my gratitude to Ulrich Pardey for numerous illuminating discussions regarding identity. Finally, I am grateful to two anonymous referees for this journal for asking a number of technical and philosophical questions that helped improve this paper.
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Wehmeier, K.F. Wittgensteinian Tableaux, Identity, and Co-Denotation. Erkenn 69, 363–376 (2008). https://doi.org/10.1007/s10670-008-9118-x
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DOI: https://doi.org/10.1007/s10670-008-9118-x