Negative existential claims, definite descriptions, empty names, and presentism.

Philosophers have long been puzzled by statements about or involving non-existence. For example, the pre-Socratic Parmenides famously held that it is impossible to think about or talk about something that does not exist. For if you are literally thinking or talking of something that does not exist, then you are thinking of nothing; that is, you are not thinking about any thing at all. Classical Indian philosophers like Uddyotakara presented a very similar argument, in order to refute their Buddhist contemporaries who argued that the self does not exist (Chakrabarti, Reference Chakrabarti1997, p. 211ff.). Roughly this same argument is studied nowadays under the title ‘The problem of negative existential statements’ (Clapp, Reimer, & Spire, Reference Clapp, Reimer, Spire, Gundel and Abbott2019).

Put briefly, the problem is this. Consider the sentence ‘Atlantis does not exist.’ Most people (who are not conspiracy theorists) would say that sentence is true. But ‘Atlantis’ does not refer to anything that exists. However, one can now ask: What exactly is that true sentence talking about? Which entity is being discussed? If there is no entity being discussed, then it seems that our original sentence fails to convey the specific meaning we hoped to convey by uttering it. If we say that there is an entity being discussed, then it seems Atlantis does exist, and the sentence we initially thought was true has to be false instead. As we shall see in Section 3.4.4, Meinongianism is an attempt to block that inference: The Meinongian says that we can meaningfully and legitimately discuss an item or thing, without committing ourselves to that item existing in any sense. As Priest (Reference Priest2016, p. xxvii), who currently works in the Meinongian tradition, puts it: “Some things do not exist” (Priest, Reference Priest2016, p. xxvii). People trained in the standard ways of regimenting English sentences into the symbolism of quantificational logic may hear that claim as a logical contradiction, despite it sounding like common sense to people who have not read logic textbooks. Priest, however, distinguishes between ‘exists’ and ‘some.’ He introduces a new quantifier for ‘some’ ($S$), and keeps the familiar existential quantifier with its usual meaning. He also introduces a predicate $E\left(x\right)$, which formalizes ‘ $x$ exists.’ So the English sentence ‘Some things do not exist’ is symbolized as $Sx\neg E\left(x\right)$.

Many readers will know that one response to the problem of negative existentials, inaugurated by Bertrand Russell, is that there are not (or at least need not be, in scientifically respectable language) any genuine names, that is, terms that pick out individuals. Instead, words that have the surface appearance of names in our everyday language (‘Angela Merkel,’ ‘Zeus’) should instead be understood as having some other deep logical-grammatical nature. Russell thought what we typically consider names should be understood as definite descriptions (‘the Chancellor of Germany from 2005 to 2021,’ ‘the divine being who lives atop Mount Olympus,’ ‘the author of the Iliad’), and that definite descriptions were not singular terms. That is, they should not be understood as referring to individuals. Instead, they are a special kind of quantifier (see three paragraphs below).

Quine suggested that we simply take apparent names as predicates: ‘is Angela Merkel’ would then be in the same logical category as ‘is red’ and ‘is an apple.’ Quine suggests introducing a predicate ‘Angela-Merkelizes,’ which is true of only Angela Merkel. This position, known as predicativism about names, is not the currently dominant view in philosophy of language, but in the last several years it has attracted some new defenders, after many years of unpopularity.Footnote ¹³

This is an enormous issue, which is at the heart of central developments in twentieth-century Anglophone philosophy, so there is not space to deal with it fully here.Footnote ¹⁴ If all apparent names are replaced by predicates or by definite descriptions that are not construed as singular terms, then there cannot be any violations of the Principle of Univocality for names, and thus the philosophical/ logical problems about negative singular existence statements will disappear (since they become no more logically puzzling than ‘There are no humans over ten feet tall’). However, note that the uniqueness-free languages described in this Element allow non-univocal predicates, and thus also for non-univocal definite descriptions, such as ‘the current US senator from California’ (since each US state has two senators).

Furthermore, another motivation for pursuing free logics is the desire for alternatives to Russell’s theory of definite descriptions. As is well known, this theory states that a sentence of the form ‘The $F$ is $G$’ is true exactly when (i) there is at least one $F$, (ii) there is at most one $F$, and (iii) that one $F$ is $G$. Formally, Russell’s theory states (where ‘ $℩xF\left(x\right)$’ symbolizes ‘the $F$’):

$G\left(℩xF\right(x\left)\right)\leftrightarrow \exists x\left[F\right(x)\wedge \forall y(F\left(y\right)\to y=x)\wedge G(x\left)\right]\left(RussellDD\right)$

One criticism of Russell’s theory of definite descriptions is that from a linguistic point of view, ‘The $F$’ appears very much to be a singular term, that is, a term that picks out an individual, but Russell’s theory does not treat it as such, contrary to the linguistic evidence. So one might want a theory of definite descriptions that can classify them as singular terms (Morscher & Simons Reference Morscher, Simons, Hieke and Morscher2001, p. 20), (Lambert & van Fraassen, Reference Lambert and van Fraassen1972, p. 152). Since the core idea of free logic is to allow singular terms that fail to refer to exactly one individual, free logic appears naturally positioned to provide an alternative to Russell’s theory.

The problem of negative existentials is closely related to the more general ‘Problem of Empty Names.’ In general, the truth-value of a sentence is determined by the referents (extensions) of the phrases in it, plus the syntactic arrangement (i.e., logical form) of those phrases. So, in a grammatical sentence with a (purported, apparent) name, if that name does not refer, then there will be a ‘blank’ or missing semantic component in that sentence – a missing component that would prevent the sentence from having any truth-value. To take a simple example, the true sentence ‘Quito is a capital’ has the grammatical form: name + copula + one-place predicate. For this sentence to have a truth-value at all, it seems very plausible that the name must refer to something, since a string of words that is just a copula and a predicate (such as ‘is a capital’), considered by itself, cannot express a (full, whole) proposition. The Problem of Empty Names is the set of difficulties involved in interpreting sentences that seem to have a semantic ‘blank’ where a (single) individual should be.

The Problem of Empty Names threatens to make communication between people who have different beliefs about what exists impossible: If failures of univocal reference make sentences unable to (be used to) communicate full, meaningful propositions, then ontological disagreements threaten to become linguistically impossible. As Greg Restall (Reference Restall2019, p. 11) puts it, “We would very much like to allow the use of logical techniques in a discussion where we have participants who disagree not only about what is the case, but also disagree about what there is – in a shared vocabulary with an agreed upon syntactic regimentation.” One advantage of a free logic over classical logic is that free logics allow for such direct disagreements in a common language.

For example, consider the debate in philosophy of time between presentists and eternalists: Do only present things exist, or do past and future things exist, as well as present things? The presentist says that, for example, Napoleon Bonaparte does not exist (though of course he once did); the eternalist allows that Napoleon does exist (for he is not a fictional or mythical character). Without using a free logic as the background logic in which to frame the debate, the presentist’s claim ‘Napoleon does not exist’ either does not express a complete proposition (if the presentist says ‘Napoleon’ lacks a referent) or is logically false (if ‘Napoleon’ is treated as a legitimate name).Footnote ¹⁵ And even the critics of presentism do not think that this is the right kind of reason to reject presentism: The real debate lies elsewhere.Footnote ¹⁶

Teleosemantics.

Uniqueness-free logics can be used to articulate more precisely philosophical debates that crucially involve conflation, confusion, and certain types of indeterminacy. For example, so-called teleosemantic theories of content are often thought to face indeterminacy problems (Neander, Reference Neander2017, ch. 7). Teleosemantics attempts to provide a completely naturalistic theory of representation, that is, a scientifically respectable account of under what circumstances one thing represents (i.e., is about) another; the most commonly discussed case is how a particular mental state in a particular organism represents a particular state of affairs in the world rather than another. There are a wide variety of teleosemantic theories of content (Schulte & Neander, Reference Schulte, Neander and Zalta2022, §3), but the core idea is that the content of a representation is determined (at least in large part) by its function. And functions are explained naturalistically, for example, by natural selection (the function of an organ is whatever it was selected for) or learning processes. Why do some people think this would lead to indeterminacy of content? A commonly used example here involves the fly-catching system of the frog. “The frog’s visual system is designed so that a fast moving dark object flying through the visual field will trigger a certain response; it shoots out its tongue and attempts to capture the object” (Agar, Reference Agar1993, p. 2). Many people find it very likely that there is some cognitive state in the frog that comes between the visual stimulus and the tongue snapping. But what, exactly, does that cognitive state represent (if anything)? Indeterminacy appears here, because there seem to be multiple plausible candidate answers teleosemantics could give to this question. It could represent a fly (of a particular species? A particular genus?), or frog-food, or a small, fast-moving, dark object. Each of these would play a role in increasing the frog’s fitness in typical frog environments (‘typical’ is necessary, since in an environment with many small, fast-moving, dark objects that are poisonous to frogs, such a representation triggering a tongue-flick would not be adaptive). Different versions of teleosemantics are distinguished, in part, by adding further conditions that pick out just one of the candidates. There is a massive amount of debate about which of these various proposals is best. But perhaps, as Karl Bergman (Reference Bergman2023) suggests, we should stop trying to figure out which one of these candidate contents is correct, and instead simply think of the content of the frog’s representation as genuinely indeterminate between the candidate contents. We could use a uniqueness-free language to say, in a logically regimented way, that the frog’s cognitive state signifies all three (and others besides). Some philosophers may fear that representational indeterminacy threatens some sort of conceptual incoherence or other cognitive disaster. But if we have a logic that allows for limited indeterminacy as a matter of course, then Bergman’s proposal to embrace the apparent representational indeterminacy of teleosemantics can be considered as a genuine, conceptually coherent alternative.

Transparency of mental content and slow-switch cases.

Another area in philosophy of language and mind in which uniqueness-free logic could be relevant is semantic externalism, and in particular, debates about so-called slow-switch cases. To understand slow-switch cases (and semantic externalism), we need the concept of Twin Earth. Twin Earth is a place where things have the same easily observable properties as things on our Earth do, but are chemically very different. So for example, on Twin Earth, there is a shiny yellow ductile metal that is often fashioned into jewelry and used for exchanging wedding vows. However, this stuff, at the atomic level, is not what we call ‘gold,’ that is, an element with seventy-nine protons. Instead, this stuff on Twin Earth has a different chemical formula, which we will abbreviate ABC. Confusingly for us on Earth, the word for ABC on Twin Earth is also ‘gold.’ But (at least according to the semantic externalist) the word ‘gold’ has a different meaning for the inhabitants of Twin Earth than it does for us here on Earth, and thoughts involving the concept gold are different for inhabitants of the two planets.

The following is an example of what Tyler Burge (Reference Burge1988) calls a ‘slow-switch’ case. Imagine a person just like you who is born on (regular) Earth, lives there for several years, and is exposed to a normal amount of gold, and a normal amount of people talking about gold. Then, one night, interstellar visitors from Twin Earth abduct this person in their sleep, taking them to Twin Earth. But when the traveler wakes up, it appears to them that they are in their usual bed, surrounded by the usual people, in their usual neighborhood. In other words, from their point of view, there have been no detectable changes. This is called a ‘slow-switch’ case, because it seems that the contents of our traveler’s words and thoughts slowly switch from their Earth-contents to the new Twin-Earth-contents. For example, when the traveler wakes up on their first day on Twin Earth, before they have interacted with any ABC, or talked to anyone on Twin Earth, they may wonder ‘Is my favorite gold necklace in the drawer?’ Many people have the intuition that ‘gold’ here is still referring to the element with seventy-nine protons, not ABC. But after our traveler has lived on Twin Earth for twenty years, then their utterances of ‘gold’ will refer to ABC instead.

Now imagine our traveler, after first living on Earth for twenty years, and then living on Twin Earth for twenty years, engages in the following simple reasoning:

1. (P1) When I was a child, my mother’s favorite necklace was made of gold.

2. (P2) My current favorite ring is made of gold.

   Thus, my current favorite ring, and my mother’s favorite necklace when I was a child, are both made of gold.

How should we understand this reasoning? If we accept the intuitions described in the immediately preceding paragraph, then both premises are true, but the conclusion cannot be true. Thus, the traveler commits a fallacy of equivocation: ‘gold’ means Earth-gold in P1, but means Twin-gold in P2. Fallacies of equivocation of course happen in everyday life. But in the slow-switch case, unlike everyday cases, the traveler is “in principle not in a position to notice” this equivocation (Boghossian, Reference Boghossian1992, p. 22). And if this is correct, then according to Boghossian, logical reasoning is no longer a priori: Whether a particular inference is valid or invalid (because equivocal) depends on whether or not the reasoner has been kidnapped and taken to Twin Earth without their knowledge.

Furthermore, a principle called ‘Epistemic Transparency’ about your own concepts seems to be violated by the slow-switch case. This principle states: “[P]rovided that you are minimally rational, you simply cannot mistake one conceptual content for another” (Schroeter, Reference Schroeter2007, p. 597). Here is an alternative formulation, also drawing on Schroeter: “If it seems to you that two tokens [e.g. ‘gold’ in P1 and ‘gold’ in P2] ‘obviously and uncontrovertibly’ mean the same, then they do mean the same and [because meaning determines reference] co-refer (if they refer at all)” (Recanati, Reference Recanati2012, p. 117–118). And from the traveler’s point of view, the concept gold occurring in the P1-thought is no different from the concept gold occurring in the P2-thought; they “seem to” the traveler to “obviously and incontrovertably mean the same” – if anything does. And if epistemic transparency fails, then you cannot tell if $F\left(a\right)$ and $F\left(a\right)$ are the same belief or different beliefs. And if you can’t determine that, then you cannot tell whether two claims contradict each other: $\neg F\left(a\right)$ might not contradict $F\left(a\right)$, since you do not know if those two ‘ $a$’s have the same meaning, or if those two ‘ $F$’s have the same meaning.

People have proposed various ways to avoid drawing this unwelcome conclusion from the slow-switch cases. Recanati offers one: He proposes that ‘gold’ in the above argument fails to refer in both P1 and P2. On his view, the traveler “is confused: he uses a single mental file [roughly, a concept of an individual] to refer to two distinct objects. This can only generate reference failure: the file the subject deploys in thought does not refer” (Recanati, Reference Recanati2012, p. 142). As a result, both premises are neither true nor false (Recanati Reference Recanati2012, p. 131). Thus the validity of the argument is saved, so logic can still be a priori. This also allows Recanati to save epistemic transparency, by relying on a proviso in the statement of the principle. Recall one version of the principle: “If it seems to you that two tokens ‘obviously and uncontrovertibly’ mean the same, then they do mean the same and co-refer (if they refer at all)” (Recanati, Reference Recanati2012, p. 117–118) . Since Recanati holds that both appearances of ‘gold’ in the slow-switch argument do not refer to anything, they are not counter-examples to epistemic transparency on his view, since the italicized condition is not met.

It is clear that Recanati assumes arguments involving confused words and concepts follow a neutral semantics for free logic (Section 3.2). He does not entertain the alternative options that P1 and P2 should be understood as false (as negative free logic would have it), or that ‘gold’ in all three lines of the argument could refer both to the element with seventy-nine protons, and to the stuff with molecular composition ABC. By itself, this is of course not an objection to Recanati’s position. But it does open up further lines of inquiry about slow-switch cases. Since Recanati is (implicitly) using a neutral semantics, his view will have all the advantages and disadvantages of neutral semantics more generally (Section 5.1). Situating his position within the discussion about the pros and cons of various semantics for free logics immediately delivers a fuller evaluation of his position on slow-switch cases in particular, and confusion more generally. Additionally, if someone is broadly sympathetic to Recanati’s overall position, but finds some of the particulars unpalatable (is the traveler really failing to have any thought whatsoever?), then alternative understandings of confusion are available, which may avoid some of the more undesirable consequences of Recanati’s specific proposal.

Free logic should not be expected, on its own, to solve the presentism versus actualism debate, the problem of negative existentials, how best to formulate a teleosemantic theory of content, or how to understand slow-switch cases. But free logic can nonetheless be of use in these debates. It can help with precise formulations of various positions one might take in this debate, and perhaps more importantly with tracing out the logical consequences of the various positions. In particular, a position that might seem fairly plausible when initially characterized in everyday language might nonetheless logically entail some highly unintuitive consequences, which only become apparent after the position is stated rigorously. Put otherwise, free logic can help make explicit the various positions in the debates over empty or ambiguous names, thereby also making clear each position’s costs and benefits. And it can also provide a framework that does not make opposing sides in disagreements over what exists incomprehensible to each other.

The Pessimistic Induction.

The Pessimistic Induction over the history of science is an argument against the view that current scientific theories are probably approximately true. Scientific realists typically argue that current scientific theories are approximately true, on the grounds that these theories make predictions about observable things that turn out to be incredibly accurate. However, a critic of this argument could point out that, for example, Newtonian mechanics and gravitation were (considered) extremely well confirmed in 1800, because they made extremely accurate predictions about a variety of observable events. Nonetheless, Newton’s theories have since been supplanted by quantum mechanics and general relativity, and the fundamental picture of the physical world provided by those two theories is very different from that of Newton. The Pessimistic Induction is a generalization of this example: From the (purported) fact that most past scientific theories turned out to be incorrect in important ways, the pessimistic inductor concludes that our currently accepted theories will also turn out to be importantly untrue as well.

The Pessimistic Induction has been widely discussed; a short Element on free logic is not the place to delve into these debates. But which type of free logic one adopts – positive, negative, or neutral – will have important consequences for debates about the Pessimistic Induction. As explained briefly in the final paragraph of Section 2.3.2, positive, negative, and neutral free logics are distinguished by how they view atomic sentences containing defective terms (like ‘Vulcan’ and ‘Charley’): Negative free logics say they are all false, positive free logics say at least one is true, and neutral free logics say such atomic sentences are neither true nor false. So, if one accepts neutral free logic or one of the positive free logics that allow some sentences to be neither true nor false, and is also committed to the anti-realist thesis that there are many non-referring and/or ambiguous terms in the language of superseded scientific theories (for example, ‘phlogiston,’Footnote ¹⁷ ‘caloric,’Footnote ¹⁸ or ‘is simultaneous with’Footnote ¹⁹), then one will believe that past theoretical claims containing such terms are not false, but rather truth-valueless.Footnote ²⁰ Now, one might think this distinction is unimportant: Both ways count as being incorrect, neither is true. However, this difference matters, because the vast majority of current scientific anti-realists would characterize themselves as epistemic anti-realists, but semantic realists. That is, most anti-realists today wish to draw a contrast between their anti-realism and (versions of) instrumentalism of the early twentieth century. This instrumentalism maintains that scientific theories are merely tools for prediction and control, tools which are not the kinds of things that can be true or false: A hammer is neither true nor false, but rather more or less useful for a particular task. That is, these earlier instrumentalists were semantic anti-realists. But at least since van Fraassen (Reference van Fraassen1980), the dominant anti-realist view has been that theoretical claims are either true or false (semantic realism), but that the evidence available to us does not justify our believing those theories to be even approximately true (epistemic anti-realism).Footnote ²¹ Many people, several anti-realists included, would reject a position that entails semantic anti-realism. As the saying goes, one person’s modus tollens is another’s interesting modus ponens, so a committed anti-realist could view this as a way to construct an argument for semantic anti-realism.

So free logic bears on the scientific realism debates: Which version of free logic one adopts (positive, negative, or neutral) affects whether you are a semantic realist or not (Frost-Arnold, Reference Frost-Arnold2014). Finally, it should be noted that this is not only an issue for anti-realists; scientific realists also need to have some semantic account of apparently defective terms like ‘caloric’ and ‘phlogiston.’ Free logics, and in particular positive free logics, could be used to defend a realist position, since positive free logics allow speakers to state truths even when the language they are using harbors incorrect presuppositions (from their successors’ point of view).

Semantic incommensurability.

One of Thomas Kuhn’s characteristic ideas is the claim that, after a scientific revolution occurs, the pre-revolutionary theory and post-revolutionary theory are incommensurable. Kuhn discusses different types of incommensurabilities, including incommensurability of methodologies and incommensurability of weights of criteria for choosing a theory. (For example, if simplicity is very important to me but scope of application is not, whereas broad scope is important to you but simplicity is not, then you and I may have an irresolvable disagreement about which of two theories is better supported by our shared available evidence; see Kuhn [Reference Kuhn1977].) That said, the type of incommensurability that Kuhn increasingly stressed later in his career was semantic incommensurability (Sankey Reference Sankey1993); Paul Feyerabend (Reference Feyerabend1981) also famously argued for this type of incommensurability between earlier and later scientific theories. The core idea is that certain terms in the language of pre-revolutionary science cannot be expressed in the post-revolutionary language, or vice versa, and this entails that we cannot fully compare the two theories, that is, they are incommensurable. For example, there is simply no way to capture perfectly, within a relativistic conceptual framework, the Newtonian assertion that the Sun is at absolute rest, since neither the term ‘absolute velocity,’ nor sentences containing that term, can be translated into relativistic language without changing the meaning of the original, pre-relativistic assertion. In short, if a term (or sentence) cannot be translated without loss from the pre-revolutionary language into the post-revolutionary language, or conversely, then the two theories cannot be meaningfully compared (at least on those claims only expressible in the contested language: There might be other parts of the languages where an acceptable translation is possible). There is no shared, mutually comprehensible medium for expressing thoughts.

However, generalized free logic can arguably reduce some (though not all) of the incommensurability that results from untranslatability. That is, the fact that an assertion stated in pre-revolutionary vocabulary cannot be translated without loss into an equivalent assertion in the post-revolutionary language does not entail that the truth-value of a pre-revolutionary assertion cannot be assessed within post-revolutionary language. Hartry Field (Reference Field1973) pioneered this idea, so we will use his central example. He noted that special relativity deploys two concepts of mass (relativistic mass, which is the total energy divided by the speed of light squared, and proper mass, which is the non-kinetic energy of a body divided by the speed of light squared), where pre-relativistic physics used just one. In other words, from a special-relativistic point of view, the single Newtonian word ‘mass’ conflates ‘relativistic mass’ and ‘proper mass.’ Thus, there is no way to translate claims involving the term ‘mass’ between the two languages. However, Field proposes a way, from within the special-relativistic language, to declare certain Newtonian sentences involving ‘mass’ to be true or false. When a Newtonian says ‘The mass of the Earth is less than that of the Sun,’ a relativist can declare that to be true, on the grounds that the relativistic mass and the proper mass of the Earth are both less than those of the Sun. And on Field’s proposal, the sentence ‘The mass of the Earth is greater than the mass of Jupiter’ is false within the relativist framework, because both the relativistic mass and the proper mass of the Earth is less than the relativistic mass and proper mass of Jupiter. The only time, on Field’s proposal, a relativist cannot assign a truth value to a Newtonian claim involving ‘mass’ is when the Newtonian sentence is true of proper mass and false of relativistic mass, or vice versa (for example, the Newtonian sentence ‘A body’s mass does not change when its velocity changes,’ which is true of proper mass but false of relativistic mass). Generalizing from this case, Field’s proposal is the following. For a sentence containing a confused or ambiguous term, that sentence is (i) true if and only if it is true on all disambiguations, (ii) false if and only if it is false on all disambiguations, and (iii) truth-valueless if that sentence is true on some disambiguations but not on others. This is called a ‘supervaluational’ semantics; if one does not want to identify ‘truth on all disambiguations’ with ‘truth’ itself, the former is labeled ‘supertruth.’ Supervaluational semantics like Field’s are one kind of positive semantics for free logic. They will be discussed at greater length in Section 3.4.2.

In sum, generalized free logic is interesting from a logical point of view, and there are plausible reasons to study it for its own sake. Imposing the Principle of Univocality is at odds with actual human languages, unnecessarily restricts the languages whose logical behavior we can study, and makes certain sentences into logical truths that, at least at first glance, do not appear to be logical truths. But there are also ‘instrumental’ or applied reasons to be interested in free logics as well: Significant questions in metaphysics, philosophy of language, and philosophy of science depend on how the details of one’s free logic are developed. And certain ontological questions cannot even be coherently, intelligibly asked within classical logic.

Supervaluations for ambiguous terms.

Let us first consider the case of a sentence containing an ambiguous name or predicate, that is, a name that refers to more than one individual, or a predicate that refers to more than one set of individuals in the domain. To oversimplify things slightly, that sentence will be supervaluated true iff it is true on every uniform disambiguation, it will be supervaluated false iff it is false on every uniform disambiguation, and it will lack a truth-value if it is true on some disambiguations and false on others. (The official definition of ‘supervaluation’ is on p. 43 below.)Footnote ⁴⁰ So for example, on this semantics, ‘Charley is an ant’ will be supervaluated true (since both Ant A and Ant B are ants), and ‘Charley is over one meter long’ will be supervaluated false (since Ant A and Ant B are both much shorter than one meter long). If we imagine a situation where Ant A is awake, and Ant B is asleep, then at that moment, ‘Charley is awake’ is neither true nor false, as is ‘Charley is asleep.’ Note that the definition of supervaluational truth above includes the modifier ‘uniform.’ This means that, in a particular formula, if there are multiple instances of the same ambiguous term, then every instance of that term is disambiguated in the same way. Otherwise, ‘Charley $=$ Charley’ would not be supervaluated as true, for we could disambiguate the first occurence of the name ‘Charley’ in that sentence as Ant A, and the second ‘Charley’ as Ant B, so that this mixed (i.e., non-uniform) disambiguation would be false. Allowing mixed disambiguations would result in a semantics that assigns truth-values in a way that is much closer to the neutral semantics, when the non-univocal terms appear more than once in a single sentence.

This supervaluational semantics ‘saves’ all the propositional classical logical truths. For example, every sentence of the form $F\left(b\right)\vee \neg F\left(b\right)$ is true, regardless of which element(s) in the domain $b$ refers to, whereas in the classical context $b$ must refer to one individual for a sentence of this form to be true. It also saves some first-order logical truths (though of course not all): ‘Charley $=$ Charley’ comes out true on this semantic proposal, because Ant A is identical to itself, and Ant B is identical to itself, and there are no other disambiguations of ‘Charley’ besides those two ants.

At this point, one should object that the supervaluational semantics delivers the wrong verdicts about certain sentences. In particular, no matter whether ‘Charley’ is assigned to Ant A or Ant B, the sentence ‘There is exactly one ant identical to Charley’ ($\exists x(x=c)$) will be true. But as we saw in the previous subsection, there are good reasons to think that sentence is false: no ant is identical to Charley. This potential problem is why so much time was spent in Section 3.4.1, discussing the special anti-satisfaction conditions that should be added for sentences containing ‘=.’ This problem is solved by requiring that a supervaluation must always respect the assignment of truth-values to sentences that $M^R$ makes over the truth-values assigned in a disambiguation – and (AS ${}_{=}$ 1) guarantees that $\exists x(x=c)$ will be false in $M^R$.

Let us spell this out more formally. We begin with the notion of a model that is a ‘complete disambiguation,’ that is, a model that assigns exactly one of the multiple extensions to each multiply signifying term. For example, if a language has exactly three multiply signifying terms, and the first term has two extensions, the second has five extensions, and the third has twelve, then there will be a total of $2\times 5\times 12=120$ complete disambiguation models for this multiply signifying model. We can capture this basic idea model-theoretically in the notion of a complete disambiguation of a multiply signifying model.

To continue with our example of Fred’s ant colony, there are two complete disambiguation models: In $M_1^d$, ‘Charley’ refers only to Ant A, and in $M_2^d$ ‘Charley’ refers only to Ant B. Now, Field holds (in our terminology) that a sentence containing multiply signifying terms is true in $M^U$ if it is true in each $M^d$ constructable from that $M^U$.

However, this cannot be right. For $\exists x(x=c)$ is true in every disambiguation: In the first disambiguation $f{}_1^d\left(c\right)=$ Ant A, so $\exists x\left(x=c\right)$ is true in $M_1^d$, and in the second disambiguation $f{}_2^d=$ Ant B, so $\exists x\left(x=c\right)$ is true in $M_2^d$ as well. However, we saw earlier that this sentence should not be true when $c$ is multiply signifying.Footnote ⁴¹

To avoid this problem, we can follow the strategy used by earlier supervaluation proposals for existentially free logic. The basic idea is that the truth-values assigned by the restricted model $M^R$ trump the truth-values assigned by any disambiguation model $M^d$ built on top of it. We characterize notions of truth and falsity in $M^d$ quotewith respect to $M^R$:

Now we can finally define truth on a supervaluation (i.e., supertruth) in $M^U$:

For example, $\exists x(x=c)$ will be superfalse (in the imagined $M^U$), because even though that sentence is true in every complete disambiguation model, it is false in the restricted model $M^R$ (and the condition $(M^d$ w.r.t. $M^{R}1)$ makes the truth-value assigned by $M^R$ (false) override the fact that the sentence is true in every disambiguation). And $c=c$ and ‘Charley is an insect’ will be supertrue, because they are neither true nor false in $M^R$, but both are true in every complete disambiguation. And any sentence that has no truth-value in $M^R$, and is true in some disambiguations and false in others, will be neither supertrue nor superfalse.

There is one more complication concerning an interpreted identity predicate. Many pet owners give their pets nicknames, in addition to the pets’ ‘official’ names. We can imagine Fred, the owner of the ant colony, sometimes refers to Charley using the name ‘Chuck,’ and other times using ‘Charles,’ as well as the original ‘Charley.’ Intuitively, it seems ‘Charley = Chuck’ and ‘Chuck = Charles’ should be true. However, they will not be true on the above semantics; rather, they will be truth-valueless. Anti-satisfaction condition (AS ${}_{=}$ 2) makes them truth-valueless in $M^R$, and they will be true on some disambiguation models (namely, those models where ‘Charley’ and ‘Chuck’ are both assigned to the same ant) and false on others. At first glance, there is an obvious solution to this problem: Re-write the satisfaction conditions so that $a=b$ is true when $a$ and $b$ each multiply-refer to the same set of individuals, that is,

• $a=b$ is true in $M^R$ (based on $M^U$) iff $\left\{x|R_I\right(a,x\left)\right\}=\left\{x|R_I\right(b,x\left)\right\}$.

This would achieve the desired effect of making ‘Chuck = Charley’ true.

However, imposing this condition has the following consequence: Every identity statement involving two empty names would be true. For example, ‘Pegasus = Zeus = Atlantis’ would be true. Interestingly, this is a direct consequence of one of the stronger theories of positive free logic used for languages containing definite descriptions. Specifically, this logic is known as FD2, and its distinguishing trait is the following:

$(\neg E!(t_1)\wedge \neg E!(t_2\left)\right)\to t_1=t_2\left(FD2\right)$(FD2)

for any singular terms $t_1,t_2$ (including definite descriptions).Footnote ⁴² It is clear that, in a language with multiply referring singular terms, FD2 should not hold. For presumably, if Fred confused two of the other non-large ants in his colony, and called them by the name ‘Andrea,’ we should not say ‘Charley = Andrea’ is true.

If we consider ‘Pegasus = Zeus’ to be a problem, then we could fix it by only applying the above identity condition when $a$ and $b$ both refer to at least one thing:

(Positive satisfaction for = in $M^R$)

$a=b$ is true in $M^R$ (based on $M^U$) iff $\left\{x|R_I\right(a,x\left)\right\}=\left\{x|R_I\right(b,x\left)\right\}$, and neither set is empty.

Finally, there is not yet an established supervaluational theory for definite descriptions in languages containing ambiguous terms; this remains an open area for research. Here is one difficulty. For example, suppose someone sympathetic to a positive supervaluational semantics wanted to construct a semantics in which ‘Charley = The big ant in Fred’s farm’ is true. If they simply followed the satisfaction clause for ‘=’ just above, then we would get:

(Positive satisfaction for $\iota$ and = in $M^R$)

$a=℩yP\left(y\right)$ is true in $M^R$ (based on $M^U$) iff $\left\{x|R_I\right(a,x\left)\right\}=\left\{x|R_I\right(℩yP\left(y\right),x\left)\right\}$.

$R_I\left(℩yP\right(y),x)$

$M^U$

$P$

$℩yP\left(y\right)$

$P$

$M^U$

$R_I\left(℩yP\right(y),x)$

$R_I(P,\Phi )$

A problem arises when we combine this seemingly natural idea with another natural, and quite weak idea: Frege’s only axiom for definite descriptions (Morscher & Simons, Reference Morscher, Simons, Hieke and Morscher2001, p. 21).

$c=℩x(x=c)\left(Freg{e}^{\prime }s℩-axiom\right)$

for any name $c$. And this generates a problem, for we now have: ‘The thing identical to Charley = Charley.’ And whereas $R_I$ (on our above proposal) associates two semantic values (Ant A and Ant B) with ‘Charley,’ it associates nothing with ‘The individual identical to Charley’ (Section 3.4.1 gives the reasoning).Footnote ⁴³ In addition to this problem, things become even more complicated when we allow for definite descriptions with multiply signifying predicates, for the definite descriptions have to then be relativized to particular complete disambiguation models. In Section 4.2.2, the discussion of theories of definite descriptions for positive existentially free logics will highlight some other important differences between the existentially-free cases and the uniqueness-free cases; theories of descriptions for uniqueness-free languages remains an open research area.

Supervaluations for existentially free logic.

One might initially think that it would be simple to re-purpose the supervaluational machinery earlier in this subsection to handle empty names. After all, an existentially free model is mathematically equivalent to the restricted models we used to discuss ambiguity. One might then take a cue from the Science of Logic §133, where Hegel claims that “pure nothing is $\dots$ the same as pure being,” since being and nothing both lack any specific, distinguishing characteristics. Following this line of thinking, we would take empty names to be maximally ambiguous, so to speak. Instead of a name like ‘Charley,’ which refers to only two elements of the domain of discourse, we treat ‘Zeus’ and ‘Vulcan’ as referring to every element of the domain. As a result, ‘Vulcan = Vulcan’ would be logically supertrue, since no matter what element of the domain we assign to the name ‘Vulcan,’ that element will be identical to itself. Similarly, ‘If Vulcan is a planet, then Vulcan is a planet’ and ‘Vulcan is a planet, or it is not a planet’ would be supertrue as well. And ‘Vulcan = Zeus’ would not be supertrue, since some complete disambiguation models would not assign the same element of the domain to both ‘Vulcan’ and ‘Zeus.’ This is all good news for someone who finds the idea of a positive semantics appealing.

However, as it stands, this proposal will not work as a semantics for existentially free logic. Recall that one of the key markers distinguishing free logic from classical logic is that the following are valid in classical logic, but not in free logic:

($\forall$-Elimination)

$\forall xF\left(x\right)$, therefore $F\left(a\right)$

($\exists$-Introduction)

$F\left(a\right)$, therefore $\exists xF\left(x\right)$

$\forall$

$F$

$a$

$F$

$\exists$

$a$

$F$

$\exists xF\left(x\right)$

Free logicians who want to use a supervaluational semantics recognize that ($\forall$-Elimination) and ($\exists$-Introduction) should not count as valid. So they use a model called a ‘completion model’ (or more briefly just a ‘completion’), which is similar but not identical to a complete disambiguation model. A completion model starts from an existentially free model, that is, a model whose interpretation function is partial over the individual constants: Not every name is assigned an element of the domain. A completion model will ‘fill in the blanks’ of the existentially free model. Thus far, it does not differ from a complete disambiguation. The key differences are that, in a completion model, there is a completion domain $D^c$ that contains the original base domain $D$ as a proper subset, and the interpretation function for the completion model can assign elements of $D^c$ to empty names, and the extensions of $n$-ary predicates can involve elements from $D^c$, whereas non-defective names must be assigned to elements of $D$.

Note that ($\forall$-Introduction) and ($\exists$-Introduction) are still logically valid, if we supervaluate over such completion models. Without any further semantic tweaks, both entailments will be truth-preserving in every completion of every existentially free model. There are two ways in the literature to prevent ($\forall$-Introduction) and ($\exists$-Introduction) from being logically valid, in a supervaluational setting. The first and more common way is to change the semantics for the quantifiers: Let $\forall$ and $\exists$ range over $D$ only, not over the whole $D^c$. Then both ($\forall$-Introduction) and ($\exists$-Introduction) will have a completion model where their premise is true but their conclusion is false. To see this for ($\forall$-Introduction), let every element of $D$ be in the extension of $F$ (i.e., $f{}^c\left(F\right)=D$), so that $\forall xF\left(x\right)$ is true, but let $a$ be an empty name assigned to an element in $D^c$ that is not in the extension of $F$ (i.e., $f{}^c\left(a\right)\in (D^c-D$)), so that $F\left(a\right)$ is false. For a completion model where ($\exists$-Introduction) is false, let $f{}^c\left(a\right)$ again be an element of $D^c-D$ and $f{}^c\left(a\right)\in f{}^c\left(F\right)$, so that $F\left(a\right)$ is true. Further suppose that the extension of $F$ contains no elements of $D$ (i.e., $f{}^c\left(F\right)\cap D=\varnothing$), so that $\exists xF\left(x\right)$ is false.

Ermanno Bencivenga proposes a slightly different strategy to prevent ($\forall$-Introduction) and ($\exists$-Introduction) from being supertrue. He uses the same definition of completion models, but he does not alter the usual semantics for the quantifiers, that is, quantifiers still range over every element of a completion model (so ‘all’ really means all of the elements, and not just a proper subset of them). Bencivenga uses a two-stage process for supervaluating a sentence, which is similar to the notion of truth in $M^d$-w.r.t.-${}^R$ that we saw a few pages ago; here, it will be truth in $M^c$-w.r.t.-$M^E$. First, determine all the truth-values in the ‘base’ model $M^E$. Each of these truth-values always trumps any truth-values assigned via supervaluating on the completion models. For example, if there is a sentence that is made true in the base model, but is false in some completion models, then that sentence is simply true, not truth-valueless. So let us apply this to ($\forall$-Introduction) and ($\exists$-Introduction). Suppose we have a existentially free model $M^E$, with at least one empty name $a$. Further suppose that in this model, $\forall xF\left(x\right)$ is true, that is, every member of $D$ is in the extension of $F$, but $F\left(a\right)$ has no truth-value, since $a$ is undefined. When we make completion models that assign a referent to $a$, that referent will have to be in $D^c-D$. And in some completion models, $f{}^c\left(a\right)\in f{}^c\left(F\right)$; in those models, both the premise and conclusion of ($\forall$-Introduction) are true. And in other completion models, $f{}^c\left(a\right)\notin f{}^c\left(F\right)$; in those models, the premise and conclusion of ($\forall$-Introduction) are both false. So there is no completion model where the premise of ($\forall$-Introduction) is true, but its conclusion is not (this is what distinguishes Bencivenga’s method from the one described in the previous paragraph). As a result, we cannot identify ‘supertrue’ with ‘truth in all completion models,’ since that would make ($\forall$-Introduction) truth-preserving, violating the fundamental spirit of free logic. This is why the two-stage evaluation process is necessary. If a sentence has a truth-value in the base model, that truth-value always overrides anything the chorus of completion models says. On this two-stage semantics, ($\forall$-Introduction) is not logically valid. There are models $M^E$ where $\forall xF\left(x\right)$ is true because it is true in the base model $M^E$, but $F\left(a\right)$ is not evaluated as true, because it is not true in all completion models.Footnote ⁴⁴

Lewis-Priest semantics.

Imagine you know that I am spending my lunch break sitting next to the river that runs through our town. Someone then asks you ‘Is Greg at the bank?’. You do not know if this person is asking if I am at the strip of land bordering our local river, or instead if I am at a financial institution where I could deposit checks and make withdrawals. So you reply ‘Well, yes and no.’ This is a perfectly normal response in everyday conversation, though on its face your answer appears contradictory. This response seems completely unexceptional to a typical English speaker, because saying ‘Yes and no’ in reply is conventionally used to signal to the question-asker that the hearer thinks the question is ambiguous. That is, the responder is communicating that if the sentence is interpreted in one way, the answer is ‘yes,’ but if it is interpreted it in another legitimate way, the answer is ‘no.’ Now, if one thinks ‘Yes and no’ is the best way to answer this question, then one might naturally also think that the declarative sentence ‘Greg is at the bank’ should be both true and false.

Thus, ambiguity can be used to motivate a relatively mild version of dialetheism, the view that a single sentence can be true and false. This is often called a ‘truth-value glut’ (contrasted with a ‘truth-value gap,’ when a sentence has no truth-value). And if we keep the standard principles that the negation of a true sentence is false, and the negation of a false sentence is true, then there can be a sentence $A$ where both $A$ and its negation $\neg A$ are true (and also both false). Thus, dialetheism allows for true contradictions. I called this a mild or weak form of dialetheism, because unlike standard, stronger dialetheism, one could accept all of the above and still maintain that there are no propositions (i.e., the semantic contents of sentences) that are both true and false simultaneously. On this weaker form of dialetheism, only sentences can be true contradictions; propositions cannot. For it is reasonable to claim that a single sentence can be ambiguous between two propositions, even if one thinks that propositions themselves cannot be ambiguous: The sentence ‘Greg Frost-Arnold is at the local bank’ is ambiguous between (i) the unambiguous proposition that I am at the land bordering the river running through our town, and (ii) the unambiguous proposition that I am at the local financial institution. David K. Lewis (Reference Lewis1982) uses the fact that ambiguity is a feature of everyday life to motivate the use of truth-value gluts, and thereby to motivate using a relevant logic. Graham Priest (Reference Priest1995; Reference Priest2016) does the same, but for first-order logic, whereas Lewis only deals with the propositional case. See also Ripley (Reference Ripley2018).

Once sentences that have more than one truth-value are allowed, we must also revisit the sentential connectives. The standard approach, advocated by Priest, is to keep the classical characterizations of the connectives unchanged. Thus, for example, consider the conjunction $A\wedge B$. Suppose $A$ is both true and false, while $B$ is true only. Classically, a conjunction is true iff both conjuncts are true. Thus, $A\wedge B$ is true. But classically, a conjunction is false if at least one of the conjuncts is false. And in the sentence under consideration, $A$ is false (as well as true), so $A\wedge B$ is false too. So the conjunction $A\wedge B$ is both true and false, when one conjunct has both truth-values, and the other conjunct is just true.

At this point, you might think that Lewis and Priest’s proposal can be framed using the concept of a complete disambiguation introduced earlier, as follows. A sentence $A$ is true iff it is true on at least one complete disambiguation model, and it is false iff $A$ is false on at least one complete disambiguation model. As it turns out, this does not accurately capture the Lewis-Priest semantics. The reason is that they allow for “mixed disambiguations” (Lewis, Reference Lewis1982, p. 439), that is, when the same expression appears more than once in a single sentence, the different instances of that expression can be assigned different semantic values. Here is why they permit mixed disambiguations. Imagine a situation in which Ant A is eating, but Ant B is not. Consider the following argument:

On the Lewis-Priest semantics, the first premise is true, since Ant A is eating, and the second premise is also true, since Ant B is not eating. (Both premises are also false as well.) However, there is no uniform disambiguation model in which Ant A is both eating and not eating, and there is also no uniform disambiguation model in which Ant B is both eating and not eating. So then the conclusion is not true in any uniform disambiguation model, in which case the proof rule of $\wedge$-Intro fails. Because Priest and Lewis want $\wedge$-Intro to be a valid rule, their response to this situation is to allow mixed disambiguations: In a sentence where the same ambiguous expression appears twice, the two tokens of the expression can be semantically assigned different disambiguations. Thus, the conclusion in the above argument is true and false: It is true, because if we assign Ant A to ‘Charley’ in the first conjunct, and Ant B to ‘Charley’ in the second conjunct, then both conjuncts are true. Thus, there is a mixed disambiguation in which $A\wedge \neg A$ is true. So the $\wedge$-intro rule is saved by allowing for mixed disambiguations. Note that allowing for mixed disambiguations will make ‘Charley = Charley’ both true and false (likewise for ‘Charley $\ne$ Charley’).

McLeish semantics.

Finally , suppose you do not want to accept either the truth-value gluts of the Lewis-Priest semantics, or the truth-value gaps of the supervaluational semantics, but you still want some sentences containing defective terms to be true (and others false), that is, you want a positive semantics. Christina McLeish (Reference McLeish2006) has proposed a semantics which does exactly that. Loosely speaking, McLeish’s proposal combines Lewis and Priest’s truth-definition with the supervaluational one. Lewis and Priest treat truth and falsity symmetrically: To repeat what was said above, a sentence is true iff it is true in at least one complete mixed disambiguation, and false iff it is false in at least one complete mixed disambiguation. McLeish’s proposal, in contrast, handles truth and falsity asymmetrically: It uses the same truth-definition that Lewis and Priest do, but says a sentence is false iff it is false on all complete disambiguations. Thus, on McLeish’s semantics, there are no sentences that are both true and false, distinguishing her proposal from Lewis and Priest’s. But there are also no truth-valueless sentences created by ambiguities, thereby distinguishing it from the supervaluational semantics, which declares a sentence neither true nor false if it is true on some disambiguations but not on others. On first glance, McLeish’s semantics might appear to be the best of both worlds, especially if one is reluctant to accept non-standard truth-value assignments, that is, one wants to avoid truth-value gaps and gluts. But as we shall see later (in Section 4.3.3), this view has its own implausible consequences. In particular, the logical rule of $\wedge$-Introduction is no longer valid.

Meinongianism.

The inner-domain/outer-domain semantics is related to another venerable philosophico-logical topic: Meinongianism. Graham Priest, who has worked to revive Meinongian views based on the work of Richard Routley (Reference Routley1980), identifies the minimal core of this doctrine as ‘Noneism’ (Priest, Reference Priest2016, xxvii):

(Noneism)

Some things do not exist.

Of course, for the doctrine of Noneism to be coherent, we cannot (contra introductory logic texts) read the quantifier ‘some’ in the statement of Noneism as the standard existential quantifier $\exists$ of classical logic: ‘There exists something that doesn’t exist’ is plainly incoherent. So the Noneist introduces a new quantifier for ‘some’ ($S$) which does not carry existential assumptions, and ranges over the entire domain, the outer domain as as well as the inner. As a result, ‘Some things do not exist’ is true in a model just in case there is at least one element in the outer domain. The characteristic Noneist slogan can be precisely expressed as $Sx\neg E!\left(x\right)$ in a language with a unique existence predicate, and as $Sx\neg \exists y(x=y)$ in a language with identity. There is a corresponding universal quantifier ($A$) that ranges over the whole inner and outer domains as well. The semantics for the new quantifiers are precisely the classical ones, over the entire domain $D=D_i\cup D_o$. Note that we can define the old classical quantifiers in terms of their Meinongian counterparts, provided we have an existence predicate:

• $\exists xF\left(x\right)\iff Sx(E!(x)\wedge F(x\left)\right)$

• $\forall xF\left(x\right)\iff Ax(E!(x)\to F(x\left)\right)$

So anything that can be said in a classical language can also be said in this Meinongian language. (However, we cannot define the new, extended quantifiers in terms of the old, standard ones.)

Now, a question any philosophically inclined person would have about Noneism and the inner-domain/outer-domain semantics is: What exactly are the things in the outer domain? What determines what is included in the outer domain, and what is not? An answer to this question goes beyond the bare statement of Noneism, but many people will not be satisfied with Noneism as a brute, unexplained assertion. Meinong himself, as well as many Neo-Meinongians, generally argue for the group of non-existing items to be as large as it can be, without creating disastrous consequences (and different people have different views of what counts as a disaster). For one consideration motivating these thinkers is that the Problem of Empty Names (Section 2.3.1) is a real problem: If I am thinking or talking about Pegasus, then there must be something that I am thinking or talking about. I am not thinking of nothing (as I would be during a dreamless sleep, for example); the thing I am thinking of just happens to lack the property of existence. And human beings are capable of thinking about all sorts of things, existing or not. We are even arguably able to think about things that are not just non-existent but impossible, since we make claims like ‘Penrose’s triangle is impossible,’ which states that a particular thing cannot have the property of existence. If the justification for (Noneism) is that humans can think about lots of non-existent things, then the set of non-existent things, that is, the population of the outer domain, will be everything coherently conceivable. And given any set of properties, humans can posit and entertain some object that has those properties.Footnote ⁴⁵ This is called the (naive or unrestricted) ‘characterization principle’ or ‘comprehension principle’: For any property or set of properties $P$, there is an item or thing $b_P$ (possibly in the outer domain) having those properties, that is, $P\left(b_P\right)$ (Berto, Reference Berto2013, p. 86).

Unfortunately, this simple, naive characterization principle leads to consequences even Meinongians find unacceptable. First, this principle guarantees the existence of every object. Meinongians stress, contra the philosophical orthodoxy, that existence is just one property among others: Some things have it, others do not. But if that is true, then it can appear in instances of the unrestricted characterization principle: The principle guarantees not only an object that has the properties of both being golden and a mountain (which Meinongians are happy with), but also a thing that has the three properties of being golden, a mountain, and existent. And even Meinongians do not want to endorse the obviously false claim that a golden mountain actually exists. Second, the unrestricted characterization principle allows us to prove literally anything (Priest, Reference Priest2016, p. xix). Let $A$ be an arbitrary sentence. Now consider the (admittedly artificial) property $x=x\wedge A$. The unrestricted characterization principle guarantees that there is an object with that property, call it $c$. So $c=c\wedge A$ is true, and $A$ immediately logically follows from that sentence. Since $A$ was arbitrary, this shows that endorsing the unrestricted characterization principle allows you to prove anything.

Philosophers working in the Meinongian tradition have offered several ways to avoid this problem.Footnote ⁴⁶ One route restricts the set of properties that can occur in the characterization principle (Parsons, Reference Parsons1980). In particular, the property of existence is disallowed; this prevents the Meinongian from accepting that a golden mountain, or greatest natural number, actually exists. The key difficulty with this approach is providing a principled distinction between properties that are allowed in the (restricted) characterization principle and those that are not, that is not merely a post hoc rationalization contrived for no reason other than saving Meinongianism from apparent counterexamples (Priest, Reference Priest2016, p. 83). A second way Meinongians have responded to the two problems described in the previous paragraph is via what Berto calls ‘modal Meinongianism.’ This accepts a unrestricted comprehension principle, but instead of holding that $P\left(b_P\right)$ is true in our world, that is, in the actual world (as the original version does), instead holds the much weaker claim that it is true at some world Berto, Reference Berto2013, p.141), (Priest, Reference Priest2016, p. 84). So ‘The existent golden mountain exists’ is true, just not at our world. And if your Meinongianism allows you to think about impossible objects like the round square, your semantics will need to include impossible worlds along with possible ones. Priest (Reference Priest2016) and Berto (Berto (Reference Berto2013)) both pursue this project in detail.

Looking over the options for a positive free semantics canvassed above, one might like the bivalence of the inner-domain/outer-domain semantics, but feel hesitant about allowing non-existent things into the (outer) domain; conversely, one might like that the supervaluational semantics avoids any non-existent things in its (base) domain, but dislike that it allows for truth-value gaps. As a result, one might wonder whether it would be possible to have the best of both worlds, namely, a positive semantics that is both bivalent and does not contain elements meant to represent non-existent things in its technical machinery. Aldo Antonelli (Reference Antonelli2000; Reference Antonelli2007) has proposed just such a positive semantics. I will not present the whole apparatus here, but the central idea can be explained informally. Antonelli’s semantics adds information to the interpretation function for names and predicates, so that, in effect, the interpretation function will encode the information that makes, for example, ‘Zeus is a Greek god’ come out true, and ‘Zeus is a horse’ come out false. For each ordered $n$-tuple of names, and every $n$-place predicate in the language, the interpretation function determines whether that $n$-tuple creates a true sentence when combined with that predicate – regardless of whether all those names refer or not. Antonelli himself calls this maneuver “somewhat artificial” (like other positive free semantics) (Antonelli, Reference Antonelli2007, p. 71), but it does show that it is technically possible to have a bivalent positive free logic that does not need anything to represent non-existent individuals. The price is dissociating true predications from the central, intuitive idea of things having properties.