Abstract
According to truthmaker maximalism, each truth has a truthmaker. Peter Milne has attempted to refute truthmaker maximalism on mere logical grounds via the construction of a self-referential truthmaker sentence M “saying” of itself that it doesn’t have a truthmaker. Milne argues that M (similar to a Gödel sentence and different from a paradox-generating Liar sentence) turns out to be a true sentence without a truthmaker and thus provides a counterexample to truthmaker maximalism. In this paper, I show that Milne’s refutation of truthmaker maximalism does not succeed. In particular, I argue that the notion of truthmaker meets two structural principles which, if added to a formal language of a theory (that allows for diagonalization), are already sufficient to produce a provable contradiction—a contradiction that gives rise to a socalled “Truthmaker paradox”. I also address the question of how to possibly resolve the Truthmaker paradox. I thereby show that the Truthmaker paradox, just as the strengthened Liar paradox, yields a “revenge problem” for paracomplete theories and might lead to triviality for Priest’s dialetheist account LP if the notion of truthmaker is defined as a certain semantic predicate within LP. But regardless of how one tries to cope with the Truthmaker paradox, this paradox is surely interesting in its own right. However, its significance is completely orthogonal to the question of whether truthmaker maximalism is a philosophically sound view.
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Notes
Gonzalo Rodriguez-Pereyra contends that accepting the existence of truthmakers just means to endorse the idea that truth is “grounded in” or “determined by reality”. He, however, emphasizes that a commitment to the existence of truthmakers does not necessarily involve a commitment to a substantive form of realism: “For idealists can accept that truth is determined by reality—they will simply add that this reality is not mind-independent or language-independent” (Rodriguez-Pereyra 2005, p. 21).
For the rest of this paper and with an eye to Milne’s argument, I will only consider sentences as truthbearers. So in this paper, truthmaker necessitarianism will be formulated as an account of truthmaker targeted at sentences. But, of course, there are also versions of truthmaker necessitarianism which apply to other truthbearers, as, for example, propositions, statements, beliefs etc.
The numbers in brackets after the line numbers indicate the line numbers of the premises and assumptions from which the sentence in the line depends.
The free ∃-elimination rule is as follows: from a given sentence ∃xA and a derivation of a formula B from A(a/x) & ∃y(y = a) (where A(a/x) is the result of replacing all free occurrences of x in A with a, and where a is new and does not occur in either A or B), one can eliminate A(a/x) and ∃y(y = a) and infer B from ∃xA (see, for example, Tennant 1978, p. 168). Let A be the sentence □(∃y(y = x) → M) and B the sentence M. In the above proof, we have derived M from A(a/x), i.e., □(∃y(y = a) → M) and ∃y(y = a). Using free ∃-elimination, we can thus eliminate □(∃y(y = a) → M) and ∃y(y = a) and derive M from ∃xA, i.e., ∃x□(∃y(y = x) → M).
According to diagonalization (see, for example, the diagonal lemma in Boolos/Jeffrey 31989, p. 173), there is for any formula B(y), containing just the variable y free, a sentence σ such that: ⊢Q+ σ ≡ B(⎾σ⏋), where ⎾σ⏋ is the numeral of the Gödel number of σ, and “B(⎾σ⏋)” results from “B(y)” by replacing all free occurrences of “y” in “B(y)” with ⎾σ⏋.
If a theory includes some false axioms but has truth-preserving rules, the theory’s false theorems are not “made true” by the theory’s inference rules, although one could, of course, say that the false theorems could have been true, had the axioms been true. Furthermore, if a true sentence σ is derived from false axioms via truth-preserving rules, σ cannot be said to be “made true” by this derivation.
For similar views of proofs as truthmakers for mathematical sentences, see also Read 2010, p. 53. It is also remarkable to note that Milne himself seems to accept Gödel numbers of proofs of sentences as truthmakers of those sentences when claiming with respect to the Gödel sentence, that “a number encoding a proof of the Gödel sentence would ground or establish it from the perspective of arithmetic.” (Milne 2005, p. 222).
Barrio and Rodriguez-Pereyra (2015) also point out that there are essential dissimilarities between M and a Gödel sentence and similarities between M and a Liar sentence. Milne attempts to justify the analogy between M and a Gödel sentence in claiming that his proof is merely on the object level since no semantic terms explicitly occur in the proof. However, Barrio and Rodriguez-Pereyra argue that Milne’s proof against TMM only succeeds if the notion of truthmaker were spelled out in terms of truth. A truthmaker of a sentence in a necessitarian account is an entity whose mere existence necessitates the truth of that sentence. Thus, Barrio and Rodriguez-Pereyra claim that “a reference to truth seems ineliminable from any sentence that speaks about truthmakers.” (Barrio and Rodriguez-Pereyra 2015, p. 6) Furthermore, they contend that the conclusion in Milne’s proof would only be a counterexample to TMM if a principle of semantic ascent is implicitly assumed, i.e., if it is assumed that if M then M is true. Although I think that these are serious objections, the parallel between a Liar sentence and the sentence M in this paper is shown via applying Montague’s theorem. The inconsistency proof using Montague’s theorem does not presuppose the philosophically controversial necessitarian account of truthmakers. It only assumes some basic structural principles that govern the use of a truthmaker predicate within a formal system Q +*. It is not assumed that Q +* also contains a truth predicate. When speaking of the “truth” of M or the “truth” of a Gödel sentence, this notion of truth is a certain meta-theoretical notion.
Barrio and Rodriguez-Pereyra also discuss Milne’s sentence M in paracomplete approaches. They, however, reject this approach since Milne’s proof uses weak reductio which is not valid for sentences which are neither true nor false (see Barrio and Rodriguez-Pereyra 2015, p. 5).
Since “not having a truthmaker” and “being false only” are coextensive in Priest’s dialetheist semantics, the self-referential sentence M “saying” of itself that it does not have truthmaker is extensionally equivalent to the self-referential sentence “saying” of itself that it is false only.
For a similar reply with regard to the Extended Liar paradox, a paradox derived in LP via a sentence “saying” of itself that it is false only, see Priest 22010, p. 288.
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Acknowledgements
I am very grateful for helpful comments and suggestions from Filippo Ferrari. I would also like to thank Gerhard Schurz and the participants of his research colloquium for discussing an earlier version of this paper. I am also grateful for helpful comments of three anonymous reviewers.
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Brendel, E. Truthmaker maximalism and the truthmaker paradox. Synthese 197, 1647–1660 (2020). https://doi.org/10.1007/s11229-018-01980-2
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DOI: https://doi.org/10.1007/s11229-018-01980-2