Universally Free Logic

Consider the argument:

1) Vautrin is a man.

2) If Vautrin is a man, Vautrin is mortal.

Therefore, Vautrin is a mortal.

Such an argument is valid, because, were the premises 1) and 2) true, the conclusion would have to be true.

Yet what would it be for these sentences to be true when the name fails to refer? For our purposes, let’s assume that the name ‘Vautrin’ refers to no actual person. Narrowing our focus to one sentence, what would it take for “Vautrin is a man” to be true?

Following Aristotle’s definition of truth, the saying “Vautrin is a man” would have to say (something) of what is (i.e., of reality), that it is. (Aristotle defines truth as “To say of what is that it is, or of what is not that it is not.”)

However, as it stands, the sentence rather says of what is not (Vautrin) that it is (is a man). And this is precisely meets Aristotle’s definition of what is false. (Aristotle defines what is false as “To say of what is that it is not, or of what is not that it is.”)

One route towards considering what it would take for the sentence to be true lies on the side of the sentence itself. We could suppose that, in order to be true, the sentence has an implicit meaning that we need to make explicit.

For example, we could include, what very well may be meant in some informal contexts, the implicit consideration that Vautrin is a fictional character. Thus, “Vautrin is a man” is a disguised form of “Balzac’s story goes that Vautrin is a man.” This could apply similarly to the rest of the argument. This reading of what the premises say would indeed make the premises true, and in this case, the conclusion, which is similarly modified, would have to be true.

However, we are relying here on just one meaning of “Vautrin is a man”. An alternative meaning, which we may call the “literal meaning”, is to read it as taking the name ‘Vautrin’ to refer to an actual, non-fictional person.

Bertrand Russell, in his famous On Denoting (1905),  took such singular terms as ‘Vautrin’ to really be descriptors, akin to (in my own words) the one who_, where the blank is filled by a description that fits the term. Thus, ‘Vautrin’, depending on which fictional character we mean, is just whatever it is that meets a sufficient conjunction of descriptions, such as being the one who is an escaped convict, lived with Rastignac for a time, and is also known as “Cheats-death”.

Moreover, propositions that have such singular terms are supposed to have embedded existential claims. Thus, “Vautrin is a man” is, for Russell, logically the same as “There is someone who is an escaped convict, lived with Rastignac for a time, and is also known as ‘Cheats-death’, and this same person is a man”. This sentence is then true if something meets the description, and is false otherwise (For more on Russell’s theory of descriptions for singular terms, see On Denoting). And so, what would make “Vautrin is a man” true, is the existence of someone who both meets the description of Vautrin and is a man.

When I first read Russell’s account of singular terms, I thought it was a very appealing way to account for the intuition that such sentences are false, precisely because the thing (perhaps implicitly) considered to exist does not exist.

Tellingly, Aristotle also considers such sentences false. Regarding the contraries “Socrates is sick” and “Socrates is well”, he says, “if Socrates exists, one will be true and the other false, but if he does not exist, both will be false” (Categories 13b17–19).

In general, it is clear that there is a reading of such sentences with non–referring terms that renders them false, and that such a reading is even typical. However, I now sense some problems with Russell’s account in particular. For one, perhaps a fictional character does not have much that describes it. If all we have to describe Vautrin is what I included above, then two of the three descriptions of Vautrin have singular terms that would themselves embed further descriptions. Moreover, “Cheats-death” would plausibly have, at most, the same descriptors as Vautrin (and perhaps it is not generally known that Cheats-death is Vautrin; this is an interesting question that is too much of a detour to pursue here; in particular, it is interesting that, while, in the story “Cheats-Death” is not known to be Vautrin, he is known to be so by the reader). If so, then how much of a description of Vautrin can be provided by the fact that he is known also as “Cheats-Death”? These difficulties are, I think in the end, ultimately tractable by Russell’s theory. However, they nonetheless point out the flaw that Russell’s theory readily becomes cumbersome. Singular terms can easily become laden with descriptions, leading almost inevitably to descriptions that have other singular terms and thus descriptions for the things that must fulfill such further terms. Thus a singular term on the surface becomes a complex web of descriptions at the level of logic. This seems like an implausible read on singular terms to me now.

Moreover, such a theory has a larger but simpler problem. Consider again that the singular term ‘Vautrin’ in ‘Vautrin is a man’ includes such a web of descriptions, including that something meets them. Well, wouldn’t the simple description that Vautrin is a man be one of them? If so, then ‘Vautrin is a man’ would be logically redundant.

But this suggests an even worse result. ‘Vautrin’ itself, being equivalent to a set of descriptions and the claim that something meets them, is now itself a sentence that is either true or false. But that’s absurd. ‘Vautrin’ is neither true nor false. It’s just a name, one that may fail to refer, but a name nonetheless.

The difficulties with Russell’s theory suggest that a different track is needed for how we are to give a literal reading of “Vautrin is a man”, one which indicates that the sentence is false, but also that it would be true if Vautrin actually existed as a man.

Enter free logic. I still have more to become acquainted with with regard to free logic. But the little I know suggests it is quite capable of keeping the notion that sentences with non-referring terms are false, without requiring a Russellean analysis names and other singular terms. Classical quantified logic (with identity) presumes that all names and singular terms refer to objects in the domain of discourse (roughly, the domain of discourse is just all of the things within the vicinity of what is being talked about by the sentences). This sort of logic is exactly the kind that Russell uses in his theory of descriptions. Free logic is just like classical quantified logic, except it drops this assumption that all the terms must refer things in the domain of discourse. Names and singular terms now may reference things outside of the domain of discourse, or they may even fail to refer to anything at all, as in the case of ‘Vautrin is a man’. (As with classical logic, however, names are supposed to reference at most one thing.)

Most free logics furthermore drop the requirement of classical logic that the domain of discourse be non-empty (i.e. that it have at least one object). Such logics that allow for domains of discourse of nothing at all are called universally free logics.

Karel Lambert developed the underlying logic of free logic in the 1960’s and coined its name. Universally free logics come in three semantic types: negative, positive, and neutral. The negative version treats sentences with non-referring names as false, and is what I prefer and will use to analyse the validity of our introductory argument. The positive version, on the other hand, treats some non-referring sentences as true, such as that of the identity relation that Vautrin is identical with Vautrin. But this is simply to start off on the wrong foot, as far as I can tell: a thing’s being self-identical implies that it exists (and, conversely, a thing’s existing implies its being self-identical). Being and identity are inextricably linked. If this is right, then a sentence of the form of one term’s being identical with itself is true only if (and, indeed, if) it exists. Zeus is not identical with Zeus if ‘Zeus’ refers to nothing (perhaps ‘Zeus’ refers to some collective idea, but again this is not the meaning I am considering now).

The other semantics is neutral: sentences that contain non-referring terms are meaningless, neither true nor false. That in some contexts non-referring terms render their sentence meaningless is uncontroversial as far as I’m concerned. I’m very open about the potential for one and the same sentence, as a bit of syntax, to mean different things according to different contexts, and even sometimes according to speaker meanings, insofar as such meanings can be made explicit (that is, a speaker can just say that their terms either refer, or that they’re speaking nonsense, thus rendering the precise meaning of what they say explicit). Peter Strawson famously feuded with Russell in considering non-referring terms to render their sentences meaningless, rather than false. So there may be a viable program in a logic that renders such sentences meaningless in some cases. However, I consider logic to be primarily bivalent (i.e. either true or false). If a sentence is rendered meaningless, then it is just not susceptible to translation into a logical language. All translation therin is improper. So while I’m open to the neutral semantics, I think another viable route would be to just discern when non-referring terms render a sentence altogether non-logical. Maybe these projects end up being more equivalent than not in the end. I won’t ponder the subject further here.

My primary motivation for free logic is to account for the validity of arguments whose sentences have terms that do not refer. They are clearly valid, but classical logic can only explain how they are valid by appealing to Russellean descriptions, which makes such terms themselves have sentence-like qualities as we have seen. Free logic is beneficial in that we need not accept a Russellian analysis of names. Names can be a blank canvas for any old object, and the description of such entities as the names refer to can now be handled properly by the descriptive or part of the sentence, rather than by the name itself.

So a negative free logic keeps the intuition that, were the terms to refer (and the predicate to apply to the term) then the sentence would be true. And were all the premises true, the conclusion would have to be true. Thus, the argument is valid. Account finished.

Reading up on negative, universally free logic, I came across some alleged problems given by the Stanford Encyclopedia of Philosophy. I was actually going to devote much space here to addressing them. However, I think that I either do not sufficiently understand the problems raised, and so cannot address them, or I do understand the problems raised, but they are such obvious non-problems that I hardly see a reason to address them at all.

Despite my initial assessment, let me address the two most common criticisms against universally free logic, from my side of negative semantics. One common criticism is that predicates that are defined within free logic in terms of other, primitive predicates end up being true in precisely the cases where the negative semanticist would want them to be false.

For example, if the domain of discourse is people, then we may have a primitive predicate A_, which means that the term that fills in the blank is an adult. But say that we want to define another predicate, that of being a minor, using this predicate. Being a minor is just the negation of being an adult. Thus, being a minor will be defined in our language as ~A_.

Notice now that when the term put in the blank position fails to refer, it will be true that such a term is a minor rather than an adult! That is, let v name what ‘Vautrin’ names, which is nothing. Av is false, under negative semantics, while ~Av is true. But of course Vautrin is not a minor, because Vautrin doesn’t exist. How will negative universally free logic answer this anomaly!?

What ought to be obvious is happening here is that we are just doing a straight port of our habits from classical logic about how to define predicates and hoping everything works out for the best. However, in free logics, we no longer just assume that a name refers. Thus, if we are going to define a new predicate within a free logical language, we need to take this into account. Defining the predicate of being a minor as simply the negation of being an adult is not good enough, because we say nothing, as we should say when using free logic, about whether such a term in the blank position exists. And so, adding an existence clause is simply a part of defining predicates in terms of primitive predicates in free logic. Get used to it! (If introductory courses to free logic existed in place of those of classical logic, then such an exercise would be done during translations early on in the course.)

Skip this paragraph if you do not know the formal symbolism of free logic. For a more formal treatment, using the symbols of free logic, where A_ is a primitive predicate that means ‘_ is an adult’, defining a minor as ~A_ is incorrect in free logic, while ∃x ( x = y ^ -Ax) is correct. With this, a sentence that expresses Vautrin’s being a minor is false, just as the one that expresses his being an adult. All predicates defined in terms of other predicates need to take the term’s existing or failing to exist into account in order to give a proper definition.

Other criticisms have to do with ‘valid substitutions’ being rendered invalid in free logic. However, as far as I can tell, the examples I’ve seen are just baseless accusations of invalidity. If Ap is true in negative free logic, then p must refer to something, and something in the domain of discourse at that. Thus, from Ap, it follows that there is something in the domain that p refers to (we can express this claim in free logic with ∃x (x = p)).

Also, from the fact that Ap, it follows that Ax, for some x in the domain of discourse. However, it does not follow that if Ax for some x in the domain of discourse, then there is something in the domain that p refers to. But so what? Consider that If the whole thing is blue then this implies that some part of the whole thing is blue. Also, that the whole thing is blue implies the whole thing is colored. However, that the whole thing is colored doesn’t imply that some part of the whole thing is blue. But there is nothing wrong with these sentences or their implications. But I may be missing a larger point of the criticism embedded in this observation. In first order logic (i.e. classical quantified logic), there is a ready method for making equivalent substitutions, since substituting embedded formulas with their equivalencies will preserve the truth, and hence the validity of what follows, of such formulas. There might very well need to be more details worked out for an effective method for granting the equivalencies of classical logic that we would also want in free logic. This is no doubt a complicated subject, and I will have to examine its progress another time. Suffice it to note for now that while we cannot just port the habits of FOL straight into negative free logics, as we noted above, there has to be a way of rendering explicit they types of substitutions we are after or would want. This is so if free logic just is FOL minus some assumptions regarding the reference of terms, but which can nonetheless be made explicit as true in free logic (that is from free logic one should be able to completely build the assumptions of FOL, and from there be able to say everything that FOL says, which includes making all the relevant substitutions).

I cannot resist getting into a final criticism, in light of my recent post on Descartes’ Cogito. This criticism goes that there are some important metaphysical claims about existence that cannot be rendered explicit by free logic. One of these is Descartes’ Cogito (the other examples given are similar). Consider a rendition of Descartes’ Cogito as follows:

I think.

If anything thinks, it exits.

Therefore, I exist.

Solid argument. And the charge here is that free logic cannot properly translate this argument, because, to be valid in free logic, the conclusion would have to be rendered explicit in the premises, and thus render such an argument question-begging.

Just one problem: there is nothing special about existence as it relates to thinking, and Descartes never says there is; he makes no such argument. Descartes’ focuses on thinking due to its epistemic value, that of being a certain knowledge. If Descartes had certain knowledge, say, that “here is a hand, and here is another” then he would have had certain knowledge of the existence of hands as well.

The broader metaphysical point, and one that a negative free logic renders explicit, is that a thing’s being in some relation or other implies that thing’s existence. This is not an argument that Descartes nor anyone else needs to make. A hand’s being here implies the hand’s existence, and thinking implies that what thinks exists. Descartes’ again rather argues that its existence is a certain type of knowledge. It is telling that, in Descartes’ own words, he sometimes renders the cogito as an argument with one premise: “I think; I am.” (“Je pense; donc je suis” in his Discourse on the Method). Descartes knew that the one implies the other, and not because there is some special relation between thinking and existing (again, Descartes’ rather saw that ‘I think’ is certain, where he would deny that ‘here is a hand’ is certain), but because anything’s being related in any way implies that the thing so related exists.

Thus, from “I think”, symbolized as Ti, we can infer the existential that there is something x of the domain of discourse that is identical with i, ( ∃x (x = i)). The semantics of negative free logic clearly allows for this inference, because such a sentence as Ti is true, only if i refers to something in the domain of discourse. And this is reflective again of the fact that a thing’s being related in some fashion implies that thing’s existence.

Not only does free logic allow for a clear route for sentences with terms that do not refer to be false, as Aristotle claims and Russell agrees, but the commonly asserted problems against a negative free logic look readily answerable.

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