Validity in logics with ambiguous terms

First, I have to give a little background about free logic; the point of this post comes towards the end. In model theory for classical logic, every name (= individual constant) is interpreted by exactly one object (in the domain of quantification); colloquially: the name ‘Chicago’ picks out exactly one thing -- the spatiotemporal object Chicago. Free logic relaxes this assumption, by allowing individual constants to be associated with no objects (in the domain of quantification). The rationale here is that some names do not successfully pick out anything in the world (think about ‘Santa Claus’, ‘Pegasus’, or ‘Planet Vulcan’), and since we don’t always know which of our terms are genuinely referential and which ones fail to refer, perhaps we should not build it into our logic that every name in fact refers. (There is another motivation for free logic as well: that we should allow models whose domain of quantification has cardinality zero, since it’s not a matter of logic that at least one thing exists in the universe.)

When we allow names that can refer to nothing into our language, the usual deduction rules have to be modified. In particular, the following (classically valid) argument form is invalid (where a is an individual):

All x are F
Thus, a is F

If a is a non-denoting name, then the premise can be true and the conclusion false or truth-valueless (depending on your preferred semantics for sentences containing non-denoting names). So in short: allowing non-denoting names forces us to give up the usual ‘All’-elimination rule (a.k.a. ‘Universal Instantiation’) just above.

I have recently been thinking about languages in which we relax the classical univocality assumption for names in the ‘other direction’: that is, languages containing terms that refer to more than one thing. (I gave them the uninspired tag ‘Multiply-Referring languages.’) The point of the formal exercise is to model ambiguous or confused terms. I have already developed a family of model theories for such languages (published in JPL last year), and am currently thinking about proof systems.

Back to free logics for a moment. There are three species of semantics for free logics: negative, neutral, and positive. They are distinguished by how they treat atomic sentences containing non-referring names. Negative: all false; neutral: all truth-valueless; positive: at least one atomic sentence with a non-referring name is true. For example, consider ‘Pegasus=Pegasus’: negative semantics declares this false, neutral semantics declares it truth-valueless, and positive semantics declares it true. I take this tripartite characterization straight over into multiply-referring languages.

Now here, finally, is my point. In positive multiply-referring languages without identity, the above rule of Universal Instantiation is valid. And, as far as I can see, all the other classical rules are valid (=truth-preserving) as well. Which means, surprisingly (to me), that the classical introduction and elimination rules for FOL without identity all are also valid rules for positive multiply-referring langauges. More simply, allowing ambiguous names into an otherwise classical language without identity makes no difference to validity. (At least in the sense of truth-preservation; it does mess up ‘backwards-falsehood-preservation,’ for reasons I won't detail here.)

But things change once an identity predicate is introduced into the language. Universal instantiation becomes invalid: ‘Everything is not identical to a’ is true if a is multiply-referring, but ‘a is not identical to a’ is not true. This raises a question for me about the best way to construct the proof system here: put roughly, is the problem with universal instantiation, or with identity? Both of them have to be present to generate the invalid argument form, so which is the one that should be altered? I’ve never tried to make my own proof theory before, so I don’t know how one should proceed under such circumstances. Any thoughts?


Jesse said...

Hey Greg, this is Jesse (from some of your old classes).

I know close to nothing about multiply referring terms, but the issue of free logic came up in a modal logic class, and maybe this is old news, but we modified our universal instantiation and existential generalization rules. Our classical rules were (θ: any formula, τ: any closed singular term, θ[τ/v] results from θ when τ replaces every occurrence of v that is free in θ):

∀vθ θ[τ/v]
------ ------
θ[τ/v] ∃vF

to which we added the following premise to both rules:

∃v[v=τ]. In the end, it turned out the easiest way to formalize the rules was to make a distinction between constants and proper names. We ditched the above version, and instead we used the old rule with the following constraint: we must use constants that are old (already appeared in the derivation, either through universal derivation or existential instantiation).

This became kinda interesting with descriptions since they can be multiply referring across worlds, and they can fail to refer at some worlds. In order to deal with this, we had the following two-way introduction and elimination rules for iota-expressions(ignore that the iota is right-side up):


Where c must be a constant (not a proper name).

Now that I think about it, it seems like there should be some similarities between descriptions in modal logic and multiply referring terms in free logic.

Greg Frost-Arnold said...

Hi Jesse --

Good to hear from you! Thanks for stopping by. I hope grad school is treating you well.

Thanks for your comment. The reason you don't know anything about logics with multiply-referring terms is that I basically made them up. Another general thing to say is that descriptions are (1) probably one of the more interesting things about setting up a multiply-referring language (there aren't that many ambiguous individual constants in natural language -- though ambiguous predicates are more common), and (2) a topic where I have not really developed anything substantive yet. My strategy is to go semantics, then proofs, then descriptions, since having a completed syntax and semantics will presumably be useful for figuring out what to say about descriptions.

But I have a couple further questions for you about what you wrote:

1. How exactly does the restriction to using 'old' constants obviate the need for adding the existence assumptions to universal instantiation and existential generalization? How do you keep the non-referring constants from appearing at a previous line of the proof?

2. I'm not immediately seeing why descriptions in modal logic would be especially relevant/ similar to descriptions in multiply-referring languages. What's the thought there?

Again, thanks for stopping by. It's good to hear from you again.

Jesse said...


I'll have to get back to you on (1), but for (2): I didn't meant to say that descriptions in modal logic are similar to descriptions in multiply referring languages.

What I meant to say was that descriptions in modal logic are multiply referring across worlds, and they seem to be an example of a singular term in a more standard logic that's already multiply referring. Being non-rigid, a def description picks out some individual at the actual world, another individual at some other world, and no one at yet another world. Just seemed like a similarity.

ben said...

Prof. Frost-Arnold,

I listened to your talk today at UCSD and I have some questions regarding non-referring names, which relate to this blog post:

1.) Lets consider 'Santa Clause' as a proper name and "Santa Clause wears a red suit" as a proposition, call it P.

2.) As you stated, under free logic there would be three treatments: 'Santa Clause = Santa Clause' is false; 'Santa Clause = Santa Clause' is without truth-value; and 'Santa Clause = Santa Clause' is true in at least one possible world.

3.) Now, let us take the positive kind of free logic and allow that 'Santa Clause = Santa Clause' is true, but it is true only in one possible world and so is contingently true rather than necessarily true.

4.) Additionally, viewing 'Santa Clause = Santa Clause' using the either the neutral or the positive free logic would allow for it to have a sense---that is, it expresses something, it has semantic content. The same can be said for P.

5.) So by (4), non-referring names and their instance in atomic statements of the form A=A can have a sense--that is, they express some semantic content--even if they are either taken in the positive way (being contingently true) or neutral way (being neither true nor false).

6.) If I understood your presentation today, which I do not think that I did since I am just a lowly undergraduate, then I THINK your claim is this: if we can show that semantic anti-realism results from PI, by means of showing that past theories (and theoretical names such as phlogiston) failed to refer and so failed to have a truth-value, then so much the worse for PI since semantic anti-realism is intolerable.

7.) Turning back to (1) through (5) above, I think confusion arises when we forget that we are talking about scientific language as opposed to ordinary language. Ordinary language has the freedom to construct names and propositions which fail to refer to objects and truth values, respectively. On the Fregean view, poetic and artistic names and propositions (e.g. Odysseus) about fictional characters have a sense but no referent; propositions containing them have a sense--they express some sort of semantic content--but have no truth value. Accordingly, we can coherently speak of Odysseus doing such and such, while still communicating semantic content, without really saying anything true or false, and it is because (on the Fregean view) that any SCIENTIFIC language must refer to entities in the world, not in stories. Using only ordinary language, however allows us to say things like "Odysseus blinded the Cyclops" and this proposition would be "true" only insofar as it communicates content which coheres with what actually happened in the story. But of the proposition "Odysseus blinded the Cyclops" our intuition would not regard this proposition as saying anything that happened in the world.

8.) So, at most, non-referring names and propositions containing non-referring names CAN have semantic content in that they have a sense and express something, respectively, but they do not speak about the actual world, and so, fail to have a truth value on the Fregean view.

9.) Coupled with Kripke's Causal theory of reference, we can see how non-referring names and sentences containing them can be persistently used by speakers who just use names and sentences without the need for these names to refer to actual objects or for the sentences to actually be true. People just continue to use some name based on how they learned about that name and what it refers to--whether or not it actually refers.

10.) Returning to Santa Claus and propositions in ordinary language versus scientific language: propositions in scientific language MUST genuinely refer to objects in our ACTUAL world to be true, whereas propositions in ordinary language (e.g. Santa Claus wears a red suit) can refer to objects in a possible world, thus being either contingently true or neither true nor false.
11.) If a proposition in ordinary language refers to something in a possible world, but does not refer to something in the actual world, it can be either true (positive free logic) or neither true nor false (neutral free logic), but it is false (negative free logic) with respect to our ACTUAL world. So, when someone says something of the sort "Santa Clause wears a red suit", this statement is agreeably true if the discourse is the looser ordinary kind of discourse, in which case we can say with a smile, yes, it is true that Santa Claus wears a red suit. But if we were to give an appraisal of "Santa Claus wears a red suit" with respect to the actual world, we can see that this statement is flatly false on the grounds that Santa Claus does not exist and so, there is no thing in the world such that it is uniquely named Santa Claus and it wears a red suit.

12.) Driving a clear distinction between ordinary language and natural scientific language is useful in identifying these different semantics. The allegedly natural scientific language produced by Priestley resulted in names and statements that failed to refer to anything in the actual world, so they are neither true nor false.

13.) But, if a name fails to refer to something in the actual world and its author's intent was to refer to something in the actual world (unlike poetic and artistic intents, which seek to refer to fictional representations), does that necessarily entail that it is FALSE?

14.) My intuition leans towards the claim that past theories--past theories which have since been supplanted by our contemporary ones--were neither true nor false with reference to the actual world, but still had semantic content because they could have been true in some possible world; and, these past theories could have been true in some possible world because they are logically consistent, internally coherent, and self-contained (which is a point Prof. Clinton Tolley brought up). But I don't think they can be FALSE with regard to the actual world either, since these past theories did make stunningly successful predictions during their tenure and are still used today in practical applications (Newtonian mechanics).

15.) Triggered by what Prof. C. Wuthrich said about models, I think we can say that the older theories were neither true nor false, but due to their success in making predictions (aside from the anomalous ones) they served as useful models but not true pictures of the world (due to leaving patches of the world unexplained or unnamed or with properties that the world did or did not have). Since these past theories were logically consistent (that is, they violated no principles of logic) they could have been true in at least one possible world, which allows them to have semantic content and so allows them to be useful even though they do not fully or accurately picture the actual world.

16.) I don't know. I am sitting on the fence still about this, and to stop my rambling, I would have to ask you what the conditions are for something to be "false" because it is unclear to me how the past theories were 'falsified' or flatly "wrong" if they failed to refer. How can a proposition be 'false' or 'wrong' if it fails to refer, and so, does not have a truth-value? And, how can a name be false or wrong if it is just a name? Only propositions can be true or false, valid or invalid. So non-referring names would just be semantically devoid of content with respect to the actual world, but still possess semantic content within the crisscross network of the set of propositions of the theory which it is embedded in. This is how we are still able to learn about, think about, and say things about phlogiston, or ether, or life-force, without these having reference to the real world: they carry semantic content insofar as they are embedded in the ordinary (not scientific) language that underwrites their respective theories. The same can be said about Santa Claus: it "makes sense" to talk about Santa Claus only because we're able to talk about Santa Claus with respect to those attributes that we learned about what makes Santa Claus who he is (wears a red suit) and what he does (delivers gifts).

ben said...

oh so I forgot to draw the relation between the propositions in scientific language versus the propositions in ordinary language:

The propositions in scientific language describe NECESSARY entities, relations and properties in all possible (natural) worlds, whereas the propositions in ordinary language just describe possible entities, relations and properties some possible world. Here's an example of a necessary entity: the sun. Here's an example of a possible entity: a unicorn.

This is how some authors (e.g. David Chalmers) are able to use possible world semantics to argue against the neurocomputational perspective of consciousness, by positing that consciousness does not necessarily have to be a feature of only the human brain; Chalmers imagines that there can be zombies who possess a human brain but without consciousness, and so since there is no contradiction in this relation (between having a brain and having no consciousness), there is at least one possible world in which knowledge about the brain does not entail knowledge about consciousness. But Chalmers irresponsibly confuses ordinary and natural scientific language, by thinking that he can explain a natural phenomena (consciousness) using ordinary language, which allows him to run buck-wild and posit things that do not violate the principle of non-contradiction.

Anyway, back to philosophy of science and language: since the propositions of natural science need to be about necessary entities, their necessary relations, and their necessary properties across all worlds, these propositions must refer and must be either true or false if they do. If they do not refer at all, then they cannot be true or false. When Priestley communicates his thoughts about phlogiston, sure, he's communicating something with semantic content because he's saying things that can possibly be true in at least one possible world, but these thoughts fail to refer to anything in the actual world, and so, are neither true nor false.

I think it is ONLY by applying modal semantics and showing that natural scientific languages must abide by the more stringent constraint of necessity as opposed to possibility that we can see how past theories--now disposed from their throne--were neither true nor false even though they had semantic content by not violating any laws of logic, and so, were communicable in ordinary language, just like how we are able to publicly speak about Santa Claus wearing a red suit or Chalmer's zombies without consciousness.

Greg Frost-Arnold said...

Hi Ben --

Thanks for a very rich set of comments. I'll give you a few first reactions now, and write again if anything more substantive occurs to me later.

I'll use your numbering system.
3. In standard free logics, 'Santa=Santa' CAN'T be true in only one possible world -- if it's true, it has to be a logical truth (and thus a necessary truth). More on this topic shortly. This is different from the case 'Santa wears a red suit' -- the most popular version of positive free logic would not certify that as true. (There is a semantics out there that could make that true, but you'd have to accept something that sounds close to contradictory: there exist things that do not exist.)
7 and following: I would've thought that, for someone who accepts the Fregean view that a sentence has both a sense and a referent, SCIENTIFIC sentences can also express senses yet have no reference -- that is not restricted to ordinary and/or artistic language.

14: Newtonian success is completely compatible with its falsity (or untruth): remember: for a conjunction to be false, we only need ONE false conjunct -- all the rest of the theory can be true.

You have a view that is very intuitive, but at variance with Kripkean tenets: You say 'Santa lives at the North Pole' is contingently untrue. A Kripkean view of names holds this is necessarily untrue. This seems counterintuitive, but it is a direct consequence of Kripke's claim that a name is a rigid designator, i.e., a name refers to the same object in all possible worlds. If that's right, and 'Santa' refers to nothing in our world, then 'Santa' refers to nothing in all possible worlds. Kripke claims that descriptions are very different from names, so he certainly admits that "There is a jolly fat man who lives at the North Pole" is contingently false.

Re: 16, just to make my view as clear as I can: If an atomic sentence has a non-referring term, then that sentence is neither true nor false. I take no stand (for present purposes) on whether there is any such thing as Fregean sense.

Finally, on your second comment: I think your view about science dealing only with necessity is very unorthodox, to say the least. I think most people think the laws of nature could've been different form what they actually are, and I think almost everyone thinks there could've been no Sun, or no electrons, etc. -- so that 'It is possible that the Sun never existed' is true... which is the same thing as saying it is no necessary that the Sun exists.

Again, thanks for all your comments. I really appreciate you taking the time to think through this material with me.

ben said...

Hi Prof, thanks for your reply; it was helpful in clarifying some of the mistakes I made.

RE: 3) Agreed.

RE: 7) I agree, Scientific sentences can express a sense yet have no reference to a truth-value, but the logically perfect language Frege envisions would consist of only sentences that express one sense and one reference. However, one can make purportedly scientific claims that express a sense but no reference, and these purportedly scientific claims are often branded as pseudo-science.

RE: 14) Yes, the truth or falsity of a theory is independent from its success. That is, success is not a predictor of a theory's truth or falsity. PI commits a base-rate fallacy since we do not know the probability of true theories in a population of current theories (Callender, Magnus). I made a mistake by saying that Newtonian science was not false due to its success and usefulness. My point here, especially by using Newtonian mechanics as an example, is that there is a difference between how a theory is used and what it claims about the actual world. An interesting parallel between the philosophy of language and the philosophy of science is that in each, there is this tension between semantics and pragmatics. For something to be true or false it has to make some sort of semantic claim on the world, not just be a pragmatically useful tool to speak about or to manipulate the world. The intuition is that for something to be true or false it has to actually have said something about the actual world.

RE: Finally..) I fully appreciate and agree with what you say here, but I don't think I made the claim that scientific propositions are necessarily true. However, I can see how I made the mistake of saying that science is about describing necessary entities, properties and relations across ALL possible worlds. You are right, this kind of necessity is too stringent and wrong, since things could have been otherwise in at least one possible world. So in revision of this mistake, I would think that science concerns itself with necessary entities, properties and relations in THIS actual world in which we reside, though as you pointed out, these entities, relations and properties are not necessary across all possible worlds. I think this is probably the most intuitive view of science and natural laws. It is not necessary for the sun to have existed in the universe of discourse spanning all possible worlds, but it is necessarily the case that the sun exists in our actual world. So what I meant is that scientific claims must make reference to this actual world, and those in the past which did not were neither true nor false.