Operator arguments revisited

Abstract

Certain passages in Kaplan’s ‘Demonstratives’ are often taken to show that non-vacuous sentential operators associated with a certain parameter of sentential truth require a corresponding relativism concerning assertoric contents: namely, their truth values also must vary with that parameter. Thus, for example, the non-vacuity of a temporal sentential operator ‘always’ would require some of its operands to have contents that have different truth values at different times. While making no claims about Kaplan’s intentions, we provide several reconstructions of how such an argument might go, focusing on the case of time and temporal operators as an illustration. What we regard as the most plausible reconstruction of the argument establishes a conclusion similar enough to that attributed to Kaplan. However, the argument overgenerates, leading to absurd consequences. We conclude that we must distinguish assertoric contents from compositional semantic values, and argue that once they are distinguished, the argument fails to establish any substantial conclusions. We also briefly discuss a related argument commonly attributed to Lewis, and a recent variant due to Weber.

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Notes

  1. 1.

    See, e.g., Zimmerman (2007) and Cappelen and Hawthorne (2009).

  2. 2.

    In the case of a language as simple as the one described, there is no need to draw a distinction between context and index. For more interesting languages, such as Kaplan’s LD (Kaplan (1989 [1977]), truth at a context is defined by first defining truth at a pair of a context and an index; and \(\varphi \) is defined as true at c just in case \(\varphi \) is true at \(\langle c,i_c\rangle \), where \(i_c\) is the index of c.

  3. 3.

    We will be suggesting later that the literature may have misconstrued the intentions of one or both of these authors.

  4. 4.

    The terms ‘compositional’ and ‘compositionality’ do not occur in Kaplan (1989 [1977]). See also the discussion of ‘propositions’ and ‘contents’ in Kaplan (1989 [1977], p. 546).

  5. 5.

    Of course, others have also argued for the conclusion that these theoretical roles are occupied by different entities: Dummett (1973, 1991), Humberstone (1976), Evans (1979), Davies and Humberstone (1980), Lewis (1980), and Stanley (1997, 2002), to mention a few. But we find these arguments less persuasive than the recent ones by Rabern and Yli-Vakkuri, which concern the semantics of variable-binding operators.

  6. 6.

    We say ‘in the spirit of’ because no quote marks occur in the principles.

  7. 7.

    Although Kaplan’s semantics does not have contents with location-relative truth values, Kaplan (1989 [1977], p. 504) affirms that the presence of locational operators in a language would require the truth values of the contents expressed by its sentences to vary with location.

  8. 8.

    Thanks to Hans Kamp for suggesting this phrase in this connection.

  9. 9.

    \(\Omega \) being vacuous may also be understood as the claim that the truth value of Op at a given time is the same for all choices of p:

    \(({\mathrm {V'_r}})\) :

    \(\forall p\forall q\forall t(T_r(Op,t)\leftrightarrow T_r(Oq,t))\)

    As with \(({\mathrm {V_r}})\), we see no way of constructing an argument from \(({\mathrm {E_r}})\) to \(({\mathrm {V'_r}})\).

    The referee suggests that Kaplan’s intention was not to argue that content eternalism entails that all temporal operators are vacuous in any sense, but rather that only the operators ‘sometimes’ and ‘always’ are vacuous in some sense. We take no position on what Kaplan intended to say as opposed to what he did say.

  10. 10.

    We understand temporal operators to be operators sensitive only to temporal features of their operands; they might be so trivially. There are other intuitive notions of a temporal operator that are not captured by \(({\mathrm {O_r}})\). E.g., on some intuitive notion, ‘someone believes at 3 p.m. that’ is a temporal operator, but it is not a temporal operator on the notion captured by \(({\mathrm {O_r}})\). Here we focus on that notion. This narrow focus is not a problem for our reconstruction of Kaplan insofar as Kaplan’s argument is meant to apply also to operators that are temporal by the criterion \(({\mathrm {O_r}})\), which we assume it is.

  11. 11.

    In the terminology of modal logic, this shows that adding timestamps extends smoothly from a relational or Kripke semantics to a neighborhood or Scott-Montague semantics; see Chellas (1980) for a textbook treatment of the latter under the label ‘minimal models’. Similarly, it is straightforward to adapt the construction to operators taking more than one argument, as well as to so-called general frames, in which contents may be restricted to be only certain sets of worlds; see Blackburn et al. (2001) for a textbook treating such structures.

  12. 12.

    The timestamp semantics indicates that Mark Richard’s discussion of Kaplan’s argument in Richard (1982) is too concessionary. In that paper he argues that insofar as temporal operators operate on propositions (assertoric contents), Kaplan’s view that there is no semantics compatible with eternalism which ‘provides an even minimally acceptable representation of the logical features of tensed English’ is ‘fair’ (p. 342). Many other philosophers have voiced similar thoughts, concerning temporal and other operators: see, e.g., Salmon (1989, p. 373), Fitch (1999), King (2003), Stanley (2008, n. 43), Glanzberg (2007, p. 2), Glanzberg (2009, p. 285, n. 2), Glanzberg (2011), Ninan (2012), Kölbel (2009, p. 384), MacFarlane (2009, section 2), Brogaard (2012, chapter 6).

    As Richard reconstructs Kaplan, a key premise is that ‘if eternalism is true, then it is eternalist intensions which are the formal representatives of propositions’ (p. 341). Here Richard is thinking of those intensions as something like sets of possible worlds. What he and others have not noticed is that this particular choice of formal representatives of eternalist propositions makes all the difference. As the time-stamped semantics illustrates, other choices of formal representatives do much better at providing an eternalist-friendly story about the logical features of tensed English that is ‘minimally acceptable’. Of course, many of these authors eschew sets of worlds as formal representatives in favor of structured propositions, but they have assumed that adding structure in this way is neither here nor there as far as Kaplan’s operator argument is concerned. But if the added structure is our time-stamped structure, that will make all the difference. Of course, our time-stamped structures do not look anything like the structures familiar from the literature on structured propositions, but insofar as those theories are consistent, it’s not hard to dress them up by adding timestamps.

  13. 13.

    Note that Kaplan himself often works with a higher-order language of this kind: see, e.g., Kaplan (1970, 1995).

  14. 14.

    This notation is adapted from Cresswell (1990). These connectives are investigated systematically in hybrid logic, where \([t]\varphi \) is often written as \(t:\varphi \) or \(@_t\varphi \); see, e.g., Braüner (2014).

  15. 15.

    See Schwarz (2013) and Whittle (2017) for discussion of such issues.

  16. 16.

    The same caveats apply here as in footnote 10.

  17. 17.

    Note that this is quite a weak notion of truth-functionality: it is satisfied, for example, by ‘actually’.

  18. 18.

    See the treatment of ‘here’ in section XVIII of Kaplan (1989 [1977]).

  19. 19.

    See Williamson’s (2003) discussion of supervaluationism and epistemicism.

  20. 20.

    Andrew Bacon (2018) develops a sophisticated theory of vagueness in which the truth values of assertoric contents are precisification-relative. We are not aware of any comparably sophisticated theory of location, although Prior (1968) is suggestive of one.

  21. 21.

    In fact, the above argument shows that the operator argument overgenerates even without any particular premises about assertoric contents. For \(({\mathrm {VerQ_m}})\) need not be derived from \(({\mathrm {EQ_m}})\): since x does not occur freely in p, \(({\mathrm {VerQ_m}})\) can simply be derived using classical laws of quantification. Independently of how one thinks about assertoric contents, the argument shows that quantifiers binding variables in sentence position do not range over compositional semantic values. For the purposes of assessing the operator argument, this is the crucial conclusion. Although we will assume for simplicity in the following that quantifiers binding variables in sentence position range over assertoric contents, much of the following does not rely on that assumption—simply read our talk of assertoric contents as concerning whatever quantifiers binding variables in sentence position range over.

  22. 22.

    See Heim and Kratzer (1998, p. 242). Note that we are not theorizing in complete abstraction from standard theoretical frameworks for natural language semantics. We can imagine alternative frameworks in which one assigns semantic values not to expressions in contexts but to occurrences of expressions in contexts in such a way that different occurrences of the same expression in the same context can have different semantic values, as in, e.g., the framework sketched in Salmon (2006), following the precedent of Frege’s doctrine of ‘indirect reference’. And we can imagine theoretical frameworks in which contexts are conceived in such a way that sentences containing free occurrences of variables do not have assertoric contents at all. No doubt the lessons of this paper could be adapted to many such eccentric frameworks, but we think it’s not profitable to try to express things in a way that is completely neutral between the standard framework and these alternatives.

  23. 23.

    Giving a general characterization of the compositional semantic values of sentences, including ones containing free occurrences of variables, is a delicate matter. According to one proposal (Yli-Vakkuri 2013, pp. 554-555) they are something like Kaplanian characters. According to another (Yli-Vakkuri (2013, p. 562, n. 49); Yli-Vakkuri (2016); Yli-Vakkuri and Litland (2016)) they are three-dimensional objects that encode both metasemantic and semantic information. An alternative approach proposed by Fine (2003, 2007) recommends that we dispense entirely with the project of assigning semantic values piecemeal to sentences and theorize instead using relations between occurrences of expressions in sentences.

  24. 24.

    See the Appendix for a statement of this restriction.

  25. 25.

    There are unrelated reasons, such as problems having to do with fictional names and names that fail to refer to anything, for rejecting the validity of universal instantiation. These are standardly used to motivate free logics, i.e., logics that do no validate existential generalization (EG—for the case of propositional quantification: \(\varphi [\psi /p] \rightarrow \exists p\varphi \)). EG is equivalent to UI by contraposition and the duality of the universal and existential quantifiers, so free logics, provided they validate contraposition and treat the quantifiers as duals—as all standard free logics do—also fail to validate UI. Note, then, that the logic of vagueness discussed by Williamson (2003) is a free logic.

  26. 26.

    The referee suggests that there are completely pedestrian counterexamples to the validity of UI/EG, which involve definite descriptions: for example,

    \((\exists \mathrm {Spy})\) :

    \(\exists x({\text {necessarily, }}\,x\,{\text {is\,a\,spy}})\)

    does not follow from
    \((\mathrm {Spy})\) :

    Necessarily, the tallest spy is a spy.

    This is a mistake. First, note that UI/EG only concerns singular terms. On ‘Russellian’ views, on which definite descriptions are not singular terms, they are obviously not counterexamples. Nor are they counterexamples on the most plausible extant versions of ‘Fregean’ views, on which definite descriptions are singular terms: in order to capture the various readings for which Russellians account by scope distinctions, contemporary Fregeans posit free world (and, when tense is at issue, time) variables in definite descriptions, which are bound in \((\mathrm {Spy})\) on its true reading. (See Elbourne (2005, §3.3) for a development of such a Fregean view.) This means that, according to that kind of semantics, \((\exists \mathrm {Spy})\) does not follow from \((\mathrm {Spy})\) by UI/EG: these principles only allow us to instantiate bound occurrences of variables by free occurrences of singular terms, whereas ‘the tallest spy’ does not occur free in \((\exists \mathrm {Spy})\) on the semantics in question. But suppose, contrary to fact, that there is some workable Fregean semantics for definite descriptions that can account for all the readings without positing hidden time and world variables. Even if we adopted some such view and restricted \(\mathrm {UI}\) accordingly, nothing much of substance in our paper would be affected by this. It would still be a further interesting question whether \(\mathrm {UI}\), restricted to singular terms that are not definite descriptions, is valid. Those who think that some such Fregean view is correct can replace our ‘\(\mathrm {UI}\)’ with ‘\(\mathrm {UI}\) restricted to singular terms that are not definite descriptions’ without any loss of essential content.

  27. 27.

    In fact, in Williamson’s semantics, as in supervaluationist semantics, a valuation is an assignment of truth values rather than of contents to the atomic sentences; there is no need for valuations that assign more fine-grained entities to expressions when one is dealing with a language whose only non-truth-functional operator is the definiteness operator. However, when metaphysical modal operators are also present, valuations assign either contents or Kaplanian characters to the language’s atomic sentences: see the reconstructions of epistemicism and different forms of supervaluationism in Yli-Vakkuri (2016) and Yli-Vakkuri and Litland (2016).

  28. 28.

    See Williamson (2003, p. 703–704) for discussion.

  29. 29.

    This is a standard way of treating modality and tense, one found in, e.g., Kaplan (1978, 1989 [1977]) and Fine (2005 [1977]). Montague (1973) gives a different treatment, as do Dorr and Goodman (forthcoming).

  30. 30.

    The referee points out that, here as well as in his (1989), Salmon only explicitly draws a distinction between (in addition to Kaplanian character) two kinds of ‘semantic value’ and says that temporal operators ‘operate on’ one while modal operators ‘operate on’ the other, and that neither is explicitly designated as the compositional semantic value. But what matters for our purposes is not so much which of the layers of ‘semantic value’ Salmon posits obey the principle of compositionality as the fact that Salmon thinks that content eternalists must draw a distinction between at least two kinds of ‘semantic value’, one of which is assertoric content (‘what is said’), and the other of which is what temporal operators ‘operate on’. We wish to emphasize here that we are not arguing against views like Salmon’s, although, for reasons set out in §3, we find the motivation Salmon (1989, p. 373) offers for adding a new layer of ‘semantic value’ to be insufficient.

  31. 31.

    In this connection it is worth noting that the transparency schema

    \(({\mathrm {T_m^s}})\) :

    \(A(\varphi \leftrightarrow T_m\varphi )\)

    should look attractive even if we cannot validly derive each its instances from its universal generalization by \(\mathrm {UI}\). Although \(({\mathrm {T_m^s}})\) plays no role in the formal reconstruction of the operator argument in Sect. 4, an appeal to an analogue of \(({\mathrm {T_m^s}})\) for the relational truth predicate was implicit in our informal discussion of transparency at the beginning of that section. \(({\mathrm {T_m^s}})\) introduces some interesting complications for the eternalist: if they endorse \(({\mathrm {T_m^s}})\) and hold that A operates on features of compositional semantic values not reflected in assertoric contents, then they had better hold that the truth predicate is sensitive to compositional semantic values of its arguments in a coordinated way.

  32. 32.

    Lewis offer an escape route to the view that we must distinguish compositional semantic value from assertoric content. The key idea of the ‘Schmentencite strategy’ is that the sentence that we use to assert that it is raining (‘It is raining’) never combines with temporal operators. When it looks as if we are using the result of applying a temporal operator to that sentence, we are in fact applying a temporal operator to a homonymous expression that is not a sentence and has a different compositional semantic value. As with the other deviant theoretical frameworks we gestured at in note 20, we will not be exploring the Schmentencite strategy here.

  33. 33.

    Note that our definitions of truth at a time for sentences and truth at a time for compositional semantic values induce a kind of disharmony in that sentential truth is time-relative while truth for the compositional semantic values of sentences is not. This kind of disharmony should not be especially discomforting. After all, eternalist orhodoxy has long learned to live with the idea that sentential truth is time-relative while content truth is not. The disharmony here ought to be even less disturbing since it is not disharmony between truth for sentences and their assertoric contents but between truth for sentences and their compositional semantic values. Arguably we are far less entitled to have direct intuitions about the latter.

  34. 34.

    Here we assume the standard treatment of operators like ‘on the 22nd of August 2010 at 2:36 p.m.’ in a stamp-free semantics for tense logic, in which the compositional semantic value of ‘On the 22nd of August 2010 at 2:36 p.m., \(\varphi \)’ is the set of all times if the 22nd of August 2010 at 2:36 p.m. is a member of the compositional semantic value of \(\varphi \) and is the empty set otherwise. When we transform this into a time-stamped semantics following Sect. 3, we get the desired result.

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Acknowledgements

We owe special thanks to Hans Kamp, who provided us with extensive written and oral comments on a draft of this paper, and to Cian Dorr and Timothy Williamson, who did likewise for a draft of a precursor of this paper written by Juhani Yli-Vakkuri. We would also like to thank Berit Brogaard, Vera Flocke, Michael Glanzberg, Jeremy Goodman, Jeff King, Max Kölbel, Øystein Linnebo, Reinhard Muskens, Brian Rabern, Margot Strohminger, Clas Weber, an anonymous referee for Philosophical Studies, and audiences at CSMN at the University of Oslo, the LOGOS Research Group in Analytic Philosophy at the University of Barcelona, the University of Oxford, and Rutgers University for helpful comments and discussions. Juhani Yli-Vakkuri’s work on this paper was supported by the Alexander von Humboldt Foundation and by the University of Tartu ASTRA Project PER ASPERA (European Regional Development Fund).

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Appendix

Appendix

In Sects. 4 and 5, several formulations of the operator argument were given in formal languages using quantifiers binding variables in sentential position. This Appendix shows how to derive the conclusions of these arguments from their premises in natural axiomatic systems, similar to the axiomatic systems for propositional quantifiers already explored in Kaplan (1970).

The formalized principles involved in these arguments use sentential variables \(p,q,\dots \) which are bound by a universal quantifier \(\forall \), and a schematic temporal connective O. The reconstruction in Sect. 4 also involves variables \(t,t',\dots \) ranging over times which are bound by a universal quantifier \(\forall \), a binary truth connective \(T_r(\cdot ,\cdot )\) taking a formula and a time variable as arguments, and a connective \([\cdot ]\) which was used to construct a formula \([t]\varphi \) from any time variable t and formula \(\varphi \); let \(L_r\) be the set of formulas which can be constructed using these resources. In contrast, the arguments formalized in Sect. 5 involve—besides propositional quantifiers and O—two unary sentential operators A and \(T_m\); let \(L_m\) be the set of formulas which can be constructed using these resources.

In both languages, define as usual the notion of a free occurrence of a variable in a formula, and of replacing every free occurrence of a sentential variable p in a formula \(\varphi \) by a formula \(\psi \), written \(\varphi [\psi /p]\). Define a formula \(\psi \) to be free for p in \(\varphi \) if no free occurrence of a variable in \(\psi \) is bound in \(\varphi [\psi /p]\).

We define two axiomatic calculi in the two languages. Both will be based on a common core governing Boolean connectives and propositional quantifiers, consisting of the following axiom schemas and rules:

\(\mathrm {TAUT}\)::

All tautologies

\(\mathrm {MP}\)::

\(\varphi ,\varphi \rightarrow \psi /\psi \)

\(\mathrm {UD}\)::

\(\forall p(\varphi \rightarrow \psi )\rightarrow (\forall p\varphi \rightarrow \forall p\psi )\)

\(\mathrm {UI}\)::

\(\forall p\varphi \rightarrow \varphi [\psi /p]\) if \(\psi \) is free for p in \(\varphi \)

\(\mathrm {UV}\)::

\(\varphi \rightarrow \forall p\varphi \) if p is not free in \(\varphi \)

\(\mathrm {UG}\)::

\(\varphi /\forall p\varphi \)

The axiom system for \(L_r\) contains the following axiom schemas and rules in addition to the common core:
\(\mathrm {TD}\)::

\(\forall t(\varphi \rightarrow \psi )\rightarrow (\forall t\varphi \rightarrow \forall t\psi )\)

\(\mathrm {TI}\)::

\(\forall t\varphi \rightarrow \varphi [t'/t]\) if \(t'\) is free for t in \(\varphi \)

\(\mathrm {TV}\)::

\(\varphi \rightarrow \forall t\varphi \) if t is not free in \(\varphi \)

\(\mathrm {TG}\)::

\(\varphi /\forall t\varphi \)

\(\mathrm {PD}\)::

\([t](\varphi \rightarrow \psi )\rightarrow ([t]\varphi \rightarrow [t]\psi )\)

\(\mathrm {PC}\)::

\([t](\varphi \leftrightarrow \psi )\leftrightarrow ([t]\varphi \leftrightarrow [t]\psi )\)

\(\mathrm {PG}\)::

\(\varphi /[t]\varphi \)

A formula \(\varphi \in L_r\) being derivable in this system will be written \(\vdash _r\varphi \). As usual, this is extended to a finitary consequence relation by letting \(\varphi _1,\dots ,\varphi _n\vdash _r\psi \) abbreviate \(\vdash _r(\varphi _1\wedge \dots \wedge \varphi _n)\rightarrow \psi \).

The axiom system for \(L_m\) contains the following axiom schemas and rules in addition to the common core:

\(\mathrm {AD}\)::

\(A(\varphi \rightarrow \psi )\rightarrow (A\varphi \rightarrow A\psi )\)

\(\mathrm {AT}\)::

\(A\varphi \rightarrow \varphi \)

\(\mathrm {AG}\)::

\(\varphi /A\varphi \)

Analogous to the first system, derivability and consequence in this system will be notated using the symbol \(\vdash _m\).

A few remarks on these systems: First, although we question \(\mathrm {UI}\) in Sect. 6, the rule is unproblematic in the present context, since we are here concerned with formalizing reconstructions of the operator argument based on the assumption that assertoric contents are compositional semantic values. As noted in Sect. 6, \(\mathrm {UI}\) is well-motivated on such an assumption—as is an implausibly strong extension of it, which is not derivable in the present systems.

Second, the axiom systems are in no way intended to be complete, except for being sufficiently strong to carry out the deductions below. It is also unimportant whether derivability entails logical truth, on some understanding of the latter notion; all that is relevant is that derivable closed formulas are true (on the intended interpretation, given any choice of temporal operator O).

Third, we noted in Sect. 4 that one may restrict the arguments of the truth connectives to sentences which themselves do not contain this connective. For simplicity, this is not enforced in the languages used here; nothing in the derivations below would need to be changed if one wanted to impose such restrictions, as the only instances of the truth connectives that are required take formulas of the form p or Op as arguments.

Fourth, the axiom system may allow us to derive false statements if it is not an eternal matter what propositions there are. Again, it would be routine to restrict the systems—along familiar ways involving quantification principles from free logic—without affecting the derivability of the conclusions reached below. For simplicity, such restrictions are not imposed here.

Finally, since the common core of the two axiomatic systems contains \(\mathrm {TAUT}\) and \(\mathrm {MP}\), we can reason from premises using classical propositional logic; we will not note such appeals explicitly. A number of useful rules governing quantifiers are also easily derived in the common core:

\(\mathrm {US}\)::

\(\varphi /\varphi [\psi /p]\) if \(\psi \) is free for p in \(\varphi \)

\(\mathrm {UGC}\)::

\(\varphi \rightarrow \psi /\varphi \rightarrow \forall p\psi \) if p is not free in \(\varphi \)

Similarly, it is routine to derive the following in \(\vdash _r\):
\(\mathrm {TGC}\)::

\(\varphi \rightarrow \psi /\varphi \rightarrow \forall t\psi \) if t not free in \(\varphi \)

\(\mathrm {TPD}\)::

\(\forall t[t](\varphi \rightarrow \psi )\rightarrow (\forall t[t]\varphi \rightarrow \forall t[t]\psi )\)

\(\mathrm {TPG}\)::

\(\varphi /\forall t[t]\varphi \)

We are now ready to substantiate the claim of Sect. 4 that \(({\mathrm {O_r}})\) follows from \(({\mathrm {T_r}})\), \(({\mathrm {O'_r}})\) and \(({\mathrm {S_r}})\). For convenience, the principles are reproduced here:

\(({\mathrm {T_r}})\) :

\(\forall p\forall t[t](p\leftrightarrow T_r(p,t))\)

\(({\mathrm {O'_r}})\) :

\(\forall p\forall q(\forall t[t](p\leftrightarrow q)\rightarrow \forall t[t](Op\leftrightarrow Oq))\)

\(({\mathrm {S_r}})\) :

\(\forall p\forall t(T_r(p,t)\leftrightarrow [t]T_r(p,t))\)

\(({\mathrm {O_r}})\) :

\(\forall p\forall q(\forall t(T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t(T_r(Op,t)\leftrightarrow T_r(Oq,t)))\)

Proposition 1

\(({\mathrm {T_r}}),({\mathrm {O'_r}}),({\mathrm {S_r}})\vdash _r({\mathrm {O_r}})\).

Proof

To make the derivation of this claim more readable, we split it up using the following intermediate conclusion:

\(({\mathrm {O''_r}})\) :

\(\forall p\forall q(\forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t[t](T_r(Op,t)\leftrightarrow T_r(Oq,t)))\)

We first show that \(({\mathrm {T_r}}),({\mathrm {O'_r}})\vdash _r({\mathrm {O''_r}})\):

(1) \(({\mathrm {T_r}})\vdash _r\forall t[t](p\leftrightarrow T_r(p,t))\) \(\mathrm {UI}\)
(2) \(({\mathrm {T_r}})\vdash _r\forall t[t](q\leftrightarrow T_r(q,t))\) \(\mathrm {UI}\)
(3) \(({\mathrm {T_r}})\vdash _r\forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t[t](p\leftrightarrow q)\) (1,2), \(\mathrm {TPD}\), \(\mathrm {TPG}\)
(4) \(({\mathrm {O'_r}})\vdash _r\forall t[t](p\leftrightarrow q)\rightarrow \forall t[t](Op\leftrightarrow Oq)\) UI
(5) \(({\mathrm {T_r}})\vdash _r\forall t[t](p\leftrightarrow q)\rightarrow \forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\) (1,2), \(\mathrm {TPD}\), \(\mathrm {TPG}\)
(6) \(({\mathrm {T_r}})\vdash _r\forall t[t](Op\leftrightarrow Oq)\rightarrow \forall t[t](T_r(Op,t)\leftrightarrow T_r(Oq,t))\) (5), \(\mathrm {US}\)
(7) \(({\mathrm {T_r}}),({\mathrm {O'_r}})\vdash _r\forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t[t](T_r(Op,t)\leftrightarrow T_r(Oq,t))\) (3,4,6)
(8) \(({\mathrm {T_r}}),({\mathrm {O'_r}})\vdash _r({\mathrm {O''_r}})\) (7), \(\mathrm {UGC}\)

We now show that \(({\mathrm {O''_r}}),({\mathrm {S_r}})\vdash _r({\mathrm {O_r}})\):

(1) \(\vdash _r[t](T_r(p,t)\leftrightarrow T_r(q,t))\leftrightarrow ([t]T_r(p,t)\leftrightarrow [t]T_r(q,t))\) \(\mathrm {PC}\)
(2) \(({\mathrm {S_r}})\vdash _rT_r(p,t)\leftrightarrow [t]T_r(p,t)\) \(\mathrm {UI}\), \(\mathrm {TI}\)
(3) \(({\mathrm {S_r}})\vdash _rT_r(q,t)\leftrightarrow [t]T_r(q,t)\) \(\mathrm {UI}\), \(\mathrm {TI}\)
(4) \(({\mathrm {S_r}})\vdash _r(T_r(p,t)\leftrightarrow T_r(q,t))\leftrightarrow [t](T_r(p,t)\leftrightarrow T_r(q,t))\) (1–3)
(5) \(({\mathrm {S_r}})\vdash _r\forall t(T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\) (4), \(\mathrm {TGC}\), \(\mathrm {TD}\)
(6) \(({\mathrm {O''_r}})\vdash _r\forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t[t](T_r(Op,t)\leftrightarrow T_r(Oq,t))\) \(\mathrm {UI}\)
(7) \(({\mathrm {S_r}})\vdash _r\forall t[t](T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t(T_r(p,t)\leftrightarrow T_r(q,t))\) (4), \(\mathrm {TGC}\), \(\mathrm {TD}\)
(8) \(({\mathrm {S_r}})\vdash _r\forall t[t](T_r(Op,t)\leftrightarrow T_r(Oq,t))\rightarrow \forall t(T_r(Op,t)\leftrightarrow T_r(Oq,t))\) (7), \(\mathrm {US}\)
(9) \(({\mathrm {O''_r}}),({\mathrm {S_r}})\vdash _r\forall t(T_r(p,t)\leftrightarrow T_r(q,t))\rightarrow \forall t(T_r(Op,t)\leftrightarrow T_r(Oq,t))\) (5,6,8)
(10) \(({\mathrm {O''_r}}),({\mathrm {S_r}})\vdash _r({\mathrm {O_r}})\) (9), \(\mathrm {UGC}\)

Combining the two results, the claim follows with truth-functional reasoning. \(\square \)

Turning to the arguments in Sect. 5, recall the following principles:

\(({\mathrm {E_m}})\) :

\(\forall p(T_mp\rightarrow AT_mp)\)

\(({\mathrm {T_m}})\) :

\(\forall pA(p\leftrightarrow T_mp)\)

\(({\mathrm {Ver_m}})\) :

\(\forall p(p\rightarrow Ap)\)

\(({\mathrm {O_m}})\) :

\(\forall p\forall q(A(p\leftrightarrow q)\rightarrow A(Op\leftrightarrow Oq))\)

\(({\mathrm {TF_m}})\) :

\(\forall p\forall q((p\leftrightarrow q)\rightarrow (Op\leftrightarrow Oq))\)

We first substantiate the claim that \(({\mathrm {Ver_m}})\) follows from \(({\mathrm {E_m}})\) and \(({\mathrm {T_m}})\):

Proposition 2

\(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m({\mathrm {Ver_m}})\).

Proof

(1) \(({\mathrm {T_m}})\vdash _mA(p\leftrightarrow T_mp)\) \(\mathrm {UI}\)
(2) \(\vdash _mA(p\leftrightarrow T_mp)\rightarrow (p\rightarrow T_mp)\) \(\mathrm {AT}\)
(3) \(({\mathrm {T_m}})\vdash _m p\rightarrow T_mp\) (1,2)
(4) \(({\mathrm {E_m}})\vdash _m T_mp\rightarrow AT_mp\) \(\mathrm {UI}\)
(5) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m p\rightarrow AT_mp\) (3,4)
(6) \(\vdash _mA(p\leftrightarrow T_mp)\rightarrow (AT_mp\rightarrow Ap)\) \(\mathrm {AD}\), \(\mathrm {AG}\)
(7) \(({\mathrm {T_m}})\vdash _mAT_mp\rightarrow Ap\) (1,6)
(8) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _mp\rightarrow Ap\) (5,7)
(9) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m({\mathrm {Ver_m}})\) (8), \(\mathrm {UGC}\)

\(\square \)

Using this observation as a lemma, we substantiate the claim that \(({\mathrm {TF_m}})\) follows from \(({\mathrm {E_m}})\), \(({\mathrm {T_m}})\) and \(({\mathrm {O_m}})\):

Proposition 3

\(({\mathrm {E_m}}),({\mathrm {T_m}}),({\mathrm {O_m}})\vdash _m({\mathrm {TF_m}})\).

Proof

(1) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m({\mathrm {Ver_m}})\) see above
(2) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _mp\rightarrow Ap\) (1), \(\mathrm {UI}\)
(3) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _mq\rightarrow Aq\) (1), \(\mathrm {UI}\)
(4) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m\lnot p\rightarrow A\lnot p\) (1), \(\mathrm {UI}\)
(5) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m\lnot q\rightarrow A\lnot q\) (1), \(\mathrm {UI}\)
(6) \(\vdash _m(Ap\wedge Aq)\rightarrow A(p\leftrightarrow q)\) \(\mathrm {AD}\), \(\mathrm {AG}\)
(7) \(\vdash _m(A\lnot p\wedge A\lnot q)\rightarrow A(p\leftrightarrow q)\) \(\mathrm {AD}\), \(\mathrm {AG}\)
(8) \(({\mathrm {E_m}}),({\mathrm {T_m}})\vdash _m(p\leftrightarrow q)\rightarrow A(p\leftrightarrow q)\) (2–7)
(9) \(({\mathrm {O_m}})\vdash _mA(p\leftrightarrow q)\rightarrow A(Op\leftrightarrow Oq)\) \(\mathrm {UI}\)
(10) \(\vdash _mA(Op\leftrightarrow Oq)\rightarrow (Op\leftrightarrow Oq)\) \(\mathrm {AT}\)
(11) \(({\mathrm {E_m}}),({\mathrm {T_m}}),({\mathrm {O_m}})\vdash _m(p\leftrightarrow q)\rightarrow (Op\leftrightarrow Oq)\) (8–10)
(12) \(({\mathrm {E_m}}),({\mathrm {T_m}}),({\mathrm {O_m}})\vdash _m({\mathrm {TF_m}})\) (11), \(\mathrm {UGC}\)

\(\square \)

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Fritz, P., Hawthorne, J. & Yli-Vakkuri, J. Operator arguments revisited. Philos Stud 176, 2933–2959 (2019). https://doi.org/10.1007/s11098-018-1158-8

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Keywords

  • Semantics
  • Modal logic
  • David Kaplan
  • Tense logic
  • Relativism
  • Propositions