5/28/2021

I have just started working on an 'opinionated introduction' to free logic for the Cambridge Elements series in Philosophy and Logic. In classical logic, for a character or strong of characters to be a name, it must refer to exactly one individual. Free Logic relaxes this assumption. Names can be 'empty' in Free Logic.

I'm currently working on the section that motivates Free Logic. I wrote up what I think are the 'standard' reasons in favor of Free Logic, and then added the following one, which I'm not 100% sure about yet. I don't think I've seen it anywhere else before, but please correct me if someone else has made this argument already!

Suppose s is a string of characters with all the grammatical or syntactic markers of a name of an individual. We explicitly leave it open whether s refers to exactly one individual, i.e. we leave it open whether s is like 'Angela Merkel', or is instead like 'Zeus'. For the classical logician, 'Zeus' cannot be a name. So the classical logician holds that some strings of characters with the form s=s are not true (e.g., ‘Zeus = Zeus’), while other strings with the same form are true (e.g., ‘Angela Merkel = Angela Merkel’). However, this fact about classical logic conflicts with the widely-accepted principle that a logical truth is true in virtue of its logical form. That is:

(FORMAL) If a string of characters is a logical truth, then every string with the same grammatical or syntactic form is also true.
Since ‘Angela Merkel = Angela Merkel’ has the same grammatical or syntactic form as ‘Zeus = Zeus’, and classical logic classifies the first but not the second as a logical truth, classical logic violates this widely-held principle (FORMAL). [EDIT/ UPDATE (June 1 2021): This argument might be improved by replacing every instance of 'logical truth' with 'theorem', and 'logical form' with 'syntactic form'; see the 7th comment in the comment thread to this post]

'Positive' free logicians can avoid this problem: they make every instance of s=s, including `Zeus = Zeus', a logical truth (again, where s has all the grammatical features of a name). And 'negative' and 'neutral' free logics make no instances of s=s logical truths. (However, as Nolt points out in the Free Logic SEP entry, "[I]n negative or neutral free logic [it] is not the case [that] ... any substitution instance of a valid formula ... is itself a valid formula"; see the reference there for explanation.)

Daniel Lindquist said...

The move from "grammatical or syntactic form" to "logical form" seems to smuggle in a lot. Without it, the classical logician can deny that "Zeus = Zeus" has the form "a=a" since they can claim that "a" in the schemata only stands for names, and "Zeus" is not one. And I suspect no defender of classical logic will feel too bad if they have to agree with Frege and Russell and Wittgenstein etc. that surface-grammar and logical-grammar are not the same, that "the apparent logical form of the proposition need not be its real form" as TLP 4.0031 puts it.

P.D. Magnus said...

I was thinking along the lines Daniel suggests, but I don't think it quite answers the point. It's one thing if surface-grammar and logical-grammar come apart. It's a further (bad) thing if the only way to figure out the logical-grammar is to add knowledge of what the world is like. One would like to be able to read a sentence like "Zeus=Zeus" and encode it without having to go off and enquire into whether "Zeus" successfully refers or not.

Daniel Lindquist said...

That does not strike me as a desideratum, because I don't see why logical-grammar is supposed to be independent of how the world is, such that it is only by "adding" knowledge of the world that it could become clear if a name is a name. The reason "proper names" are called "proper" is just because they don't leave open whether they refer or not.

Greg Frost-Arnold said...

@Daniel-- "Logical form =/= syntactic, grammatical form" is definitely one very reasonable move a classical logician could -- and probably should -- make. I was planning on including something like your objection in the eventual book, but I think you've put it in an even sharper, crisper way than I was thinking about it. So thank you very much!

That said, I had a couple of thoughts about this objection; I guess now is as good a time as any to try to spell them out more clearly.

(1) I am sympathetic to P.D.'s point, but I wonder if it is actually a slightly different argument. (Part of my concern here is: are there 2 arguments here, or really just one?) I'll try to spell out that other(?) argument:
Most philosophers would agree that logical truths are a priori (if anything is -- of course some famously deny that). That is, if a string of characters is a logical truth, then that string can be known to be true a priori (if anything can). But classical logicians cannot accept this principle. Why? Classical logic classifies `Angela Merkel = Angela Merkel' as a logical truth. But whether Angela Merkel or Zeus exist is an empirical, a posteriori matter. So the classical logician must deny the principle that all logical truths are a priori, on pain of holding that logic alone determines that Merkel exists and Zeus does not.

In your reply-comment to P.D., it sounds to me like you would accept that some logical truths (at least those involving '=') are a posteriori? (I'm looking at your "I don't see why logical-grammar is supposed to be independent of how the world is." If someone says that, then it seems like they have to say logic is not a priori? Or am I missing something?)

(2) As a general principle, I think everyone should agree that logical form can't always be read off of 'surface' grammatical form (e.g. donkey sentences etc.).

But what is the hidden logical form in strings of the form s=s (again, where s passes every grammatical test for name-hood)? My initial thought is that the only plausible candidate for 'hidden' information here is that there exists exactly one individual referent of 's'. But that information is precisely what the free logician wants to make explicit (that is: in free logic, 'b=b' logically follows from 'b exists'). (I don't quite have clear in my head why that would be an argument for the superiority of free logic, but I suspect it might?)

Daniel Lindquist said...

On (1): This is what lead Russell to deny that ordinary names are "logically proper"; he accepted that it is false (IIRC he actually says "nonsense", but I can't find a citation for that -- I think it may be in the Philosophy of Leibniz?) to deny "a exists" for any 'a' which is really a name. Russell's way has gone out of fashion of course, but I think his arguments for it are generally ignored rather than dealt with.

I think it's a fine option to reject the apriority of logic (as Quine does, and Quine is responsible for a lot of the popularization of first-order logic as "classical"); it's still simple enough for the classical logician to state that propositions which instantiate logically valid schemata are different than those which do not, which keeps a line drawn between "Merkel is Merkel" and "Merkel is Prime Minister", and lets the logician say that the one truth is "obvious" in a way that the other is not.

But I think that Quine's way is also going out of fashion, so I think this is a fine point to press on classical logic. The classical logician who wants to hold on to the apriority of logic can, I suppose, hold on to the generalization (x)(x=x) as simply a priori, and then only admit that it is empirical what names we have to slot in for 'x' (which even the free logician will have to admit, in some sense, or else there's no way to distinguish between nonsense flagging an equals-sign and an instance of the law of identity -- between "Zeus=Zeus" and "asdfg=asdfg" or "is flat=is flat"). But once the names are settled, no more from experience is needed; I suspect a lot of classical logicians would feel that the apriority of logic was secure enough on that sort of ground.

On (2): I think the "hidden" logical form of a string such as "s=s" for the classical logician is just "a=a"; that this seems to not be hidden at all is just because of the conventions we have for writing identity-statements in logic and related statements in ordinary English, which makes them look the same. I think a classical logician is actually on reasonably firm ground here; "red is red" is superficially like "Merkel is Merkel" or "Zeus is Zeus", but both the classical and the free logician will want to distinguish "red is red" from an identity-statement. At the syntactic level, "red is red" and "Zeus is Zeus" are both tautologies (in the old-fashioned sense of stuttering or repetitive statements; https://philpapers.org/rec/DRETHN is a wonderful paper on the history of this word, and the funny story of how it became a synonym for "logical truth" when the two used to be contrasted with each other, as in early Russell). The classical logician doesn't reject that a string of the form "s=s" is often of the form "a=a", but does reject many such cases that the free logician accepts: but neither of them can accept all instances of this common syntactic form as having a common logical form. And the classical logician also holds that "a exists" follows from "a=a". So I actually don't see much ground for free logic from these sorts of considerations, either.

Greg Frost-Arnold said...

Hi again Daniel--

Thanks for your very thoughtful and detailed replies! I really appreciate it.

I need to think about all this more, but just so that I have a record here of my initial reactions:

- On Russell's way: if virtually nothing (except 'this' and 'that' referring to immediate sense-data, which iirc is what Russell said were the truly proper names) is a proper name, then logic (involving names) becomes not very useful/ applicable for actual investigations/ reasoning. So it seems like this might actually be convertible into an argument for a Russell-sympathizer to accept free logic: by adopting free logic over classical logic, we get applicability of logic to our everyday names.

- On Quine: he is the reason I wanted to separate the 'Classical logic violates "Every sentence with the same syntactic form is a logical truth" ' argument from the "Classical logic makes logic not a priori" argument -- because Quine would accept the latter but not the former. (So that's part of why this discussion has been very helpful to me -- I want to keep those apart, but our discussion is making me suspect that the syntactic-form argument ultimately collapses into (or at least depends for its plausibility on) the "Logic's not a priori" argument.

Final note: maybe I missed something, but I didn't really find the 'red is red' point very compelling -- there are strictly syntactic tests that distinguish e.g. 'red' (as a predicate) from 'Zeus' and 'Merkel' as names. That was why I explicitly included in the OP and my previous comment the phrase "s is a string of characters with all the grammatical or syntactic markers of a name of an individual." E.g. going all the way back to Aristotle's Categories, 'red' can be predicated of other objects, whereas 'Angela Merkel' can't.

Final, final note: I will definitely check out that Dreben and Floyd paper you mentioned! It sounds very much up my alley.

And again, thanks very much for helping me think/write through these things!

Greg Frost-Arnold said...

Just recording here a further thought about the main argument in the OP: would it maybe strengthen that argument if I replace every instance of "logical truth" with "theorem" (of first-order logic with identity)? So the inconsistent triad becomes:

- "Merkel = Merkel" is a THEOREM in classical logic.

- If a string of characters is a THEOREM, then every string with the same grammatical or syntactic form is also a THEOREM.

- (And leaving unchanged: ‘Angela Merkel = Angela Merkel’ has the same grammatical or syntactic form as ‘Zeus = Zeus’)

The reason I was thinking this might be better than the formulation in terms of 'logical truth' is that theorems are, even more than logical truths, thought to be determined by 'grammatical or syntactic form'. That is, the class of theorems is supposed to be characterized in a completely syntactic, semantics-free way.

(That said, the completeness of FOL with identity maybe means that this 'logical truth' -> 'theorem' replacement is really a distinction without a difference. I need to think that over.)

Daniel Lindquist said...

I think the shift to theorem-talk makes the disagreement more pronounced, but doesn't help the case for free logic. Nothing can be proved about "Zeus" syntactically if it is not provided that "Zeus" functions syntactically like a name, and the free logician and the classical logician do not agree about this matter (for instance, whether from "Zeus is the father of Athena" it follows that "Someone is the father of Athena" will be settled in any adequate formalization of these ordinary English expressions, but the free logician and the classical logician will settle this in very different ways -- they will disagree about the adequacy of a formalism that rules the "wrong way" about the inference). But making this explicit does, I think, make the issue more pronounced, as people are generally sloppy with that part of predicate logic.

I think someone who likes the Russellian way about names will not be happy with free logic, because free logic will still seem to give the wrong verdict about some issues: for instance, if "Zeus" is really a disguised definite description (or something of that sort) then "Zeus=Zeus" can end up false if there is nothing which satisfies the relevant description (because this ends up analyzed as "There exists some x which Fs and nothing which Fs other than x, and this x is identical with the thing which alone Fs", and this is false if there is no such x). For Russell, ordinary "proper names" are not proper names, and so do not behave logically like them; they are incomplete symbols, and logically proper names are complete symbols.

Greg Frost-Arnold said...

Thanks again! I think you're right about Russell. The only remaining sticking point is your claim: "Nothing can be proved about "Zeus" syntactically if it is not provided that "Zeus" functions syntactically like a name, and the free logician and the classical logician do not agree about this matter."

Maybe I'm just wrong about this, but my intuitive reaction is: a classical logician (?does? ?should? ?takes themselves to?) offer a purely syntactic characterization of what is a name and what isn't. That is, they should allow "that 'Zeus' functions syntactically like a name," even if it isn't a genuine name, i.e. it doesn't meet the semantic conditions for a name. If classical logicians can't do that, then they don't have a formal/ syntactic notion of proof -- but a syntactic notion of proof one of the central goals of (a) logic. In other words: saying 'Zeus' does not function syntactically as a name, ONLY BECAUSE of semantic information about it, seems to give up on a purely syntactic formalization of logic.

That said, I'm (finally) starting to lean towards your point of view. Maybe the following is just exactly what you meant in your last couple of comments, and I just needed the following formulation for it to sink in.

What it is to be a name, syntactically, just is to obey the classical rules for ∀-Elimination and ∃-Introduction. These rules are taken as defining/ characterizing the class of names in the language: if a (string of) character(s) does not satisfy those (purely syntactic) inference rules, then it's not a name. Thus, 'Mars' is a name but 'Zeus' isn't -- for purely syntactic reasons.

To re-phrase the core idea: the classical quantifier rules perfectly capture, in a purely syntactic fashion, the Principle of Univocality (which is a semantic principle).

(1) First, one might worry about circularity (or some other sort of problem of definitional order), if we define names in terms of quantifier rules. In a typical Intro to Logic textbook, names are presented very early on, and quantifier rules much later -- so it seems like the quantifier rules are defined in terms of an ANTECEDENTLY given notion of the class of names. So the concern is something like 'If we need the quantifier-rules to figure out which things are names and which aren't, then we can't know when we're allowed to apply the quantifier-rules' (or some other sort of circularity-ish worry).

(2) I can't 100% shake the feeling (mentioned at the end of my first comment) that the classical logician is somehow not making their inference completely explicit. My feeling is analogous to the following:

Premise: Paris is bigger than London.

Conclusion: London is smaller than Paris.

This entailment is necessarily truth-preserving, but it is not fully logical. It leaves implicit the (necessary) truth that if one thing is bigger than a second, then the second is smaller than the first. So we can transform the above argument into a LOGICALLY valid argument, by making the implicit explicit:

Premise 1: Paris is bigger than London.

Premise 2: Whenever one thing is bigger than a second, then the second is smaller than the first.

Conclusion: London is smaller than Paris.

And to me, the first argument feels like classical logic's rules for ∀-Elimination and ∃-Introduction. Free logic, by requiring the addition of the additional premise ('Zeus exists' or 'Merkel exists' etc.) to the free rules for ∀-Elimination and ∃-Introduction, is doing the analog of the second argument just above. The free logician is making explicit a key thing that classical logic leaves implicit.

Again, thanks for all of this! Now I am really on the fence about including anything like this argument for free logic at all in the little book; and if I do include it at all, whether I will ultimately endorse it or not.

Greg Frost-Arnold said...

Oh, and I also wanted to mention that the Dreben and Floyd paper you recommended has been really useful and interesting. There's a 1913 Wittgenstein letter to Russell quoted in it that is (I think) helpful for the OP argument:

"It is the peculiar (and most important) mark of NON-logical propositions that one is not able to recognize their truth from the propositional sign alone. If I say, for example, "Meier is stupid", you cannot tell by looking at this proposition whether it is true or false. But the propositions of logic - and only they - have the property that their truth or falsity, as the case may be, finds its expression in the very sign for the proposition." (p.33)

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