Logical Form, Equivocation, and Propositions

Ed Buckner wants to re-fight old battles. I'm game. The following post of his, reproduced verbatim, just appeared at Dale Tuggy's site:

The concept of logical form is essential to any discussion of identity, and hence to any discussion of the Trinity. Here is a puzzle I have been discussing with the famous Bill Vallicella for many years.

(Argument 1) ‘Cicero is a Roman, therefore Cicero is a Roman’

(Argument 2) ‘Cicero is a Roman, therefore Tully is a Roman’

My puzzle [is] that the first argument is clearly not valid if the first ‘Cicero’ means the Roman, the second the American town, yet the argument seems to instantiate a valid form. Bill objects that if there is equivocation, then the argument really has the form ‘a is F, therefore b is F’, which fails to instantiate a valid form.

I then ask what is the form of. Clearly not of the sentences, since the sentences do not include the meaning or the proposition. Is it the form of the proposition expressed by the sentences? But then we have the problem of the second argument, where both ‘Cicero’ and ‘Tully’ mean the same man. Then the man is contained in both propositions, and if the form is of the proposition, the argument has the true form ‘a is F, so a is F’, which is valid. But I think no one would agree that the second argument is valid.

So logical form does not belong to the sentences, nor to the propositions expressed by them. So what is it the form of?

My answer is that the logical form of the argument is the form of the Fregean propositions expressed by the sentences that make up the argument. Let me explain.

I agree with Ed that logical form is not the form of an array of sentence-tokens. It is rather the form of an array of propositions expressed by the sentences. (To be painfully precise: it is the form of an array of propositions expressed by the assertive utterance, and thus the tokening, of a series of sentence-types by a speaker or thinker on a given occasion. A sentence-token buried in a book does not express anything by itself!)

To solve Ed's puzzle we need to distinguish three views of propositions: the Aristotelian, the Fregean, and the Russellian. This would be a good topic for an extended post. Here I will be brief.  Brevity is the soul of blog.

An Aristotelian proposition is an assertively uttered meaningful sentence in the indicative mood that expresses a complete thought.  What makes such a proposition 'Aristotelian' as opposed to 'Platonic' is that the meaning of the sentence is not something that can subsist on its own apart from the assertive tokening of the sentence.  The meaning of the sentence depends on its being expressed, whether in overt speech or in thought, by someone. If there were no minds there would be no Aristotelian propositions. And if there were no languages there would be no Aristotelian propositions. In this sense, Aristotelian propositions are linguistic entities.

In brief: An Aristotelian proposition is just a declarative sentence in use together with its dependent sense or meaning. Suppose I write a declarative sentence on a piece of paper. The Aristotelian proposition is not the string of physical marks on the paper, nor it is the producing of the marks; it is the marks as produced by a minded organism on a particular occasion together with the meaning those marks embody.

A Fregean proposition is a nonlinguistic entity that subsists independently of minds and language. It is the sense (Sinn) of a declarative sentence from which indexical elements have been extruded. For example, 'I am blogging'  does not express a Fregean proposition because of the indexical 'I' and because of the present tense of the verb phrase.  But 'BV blogs at 10:50 AM PST on 4 September 2017' expresses a Fregean proposition.

Fregean senses are extralinguistic and extramental 'abstract' or 'Platonic' items.  They are not in time or space even when the objects they are about are in time and space. This is what makes Fregean propositions 'Platonic' rather than 'Aristotelian.' Fregean propositions are the primary truth-bearers; the sentences that express them are derivatively true or false.

A Russellian proposition is a blurry, hybrid entity that combines some of the features of a Fregean truth-bearer and some of the features of a truth-maker. A Russellian proposition does not reside at the level of sense (Sinn) but at the level of reference (Bedeutung).  It is out there in the (natural) world. It is what some of us call a fact or 'concrete fact' (as in my existence book) and others a state of affairs.  

Now consider a singular sentence such as 'Ed is happy.'  For present purposes, the crucial difference between a Fregean proposition and a Russellian proposition is that, on the Fregean view, the subject constituent of Ed is happy is not Ed himself with skin and hair, but an abstract surrogate that represents him in the Fregean proposition, whereas in the Russellian proposition Ed himself is a constituent of the proposition!

We needn't consider why so many distinguished philosophers have opted for this (monstrous) view.  But this is the view that seems to have Ed in its grip and that powers his puzzle above.

If we take the relatively saner (but nonetheless problematic) view that propositions are Fregean in nature, then the puzzle is easily solved.

Ed asks: What is the logical form the form of?  He maintains, rightly, that it cannot be the form of an array of sentences. So it must be the form of an array of propositions. Right again. But then he falls into puzzlement: 

. . . ‘Cicero’ and ‘Tully’ mean the same man. Then the man is contained in both propositions, and if the form is of the proposition, the argument has the true form ‘a is F, so a is F’, which is valid.

The puzzlement disappears if we reject the Russsellian theory of propositions. A man cannot be contained in a proposition. and so it cannot be the same man in both propositions.

‘Cicero is a Roman, therefore Tully is a Roman’ is plainly invalid. Its form is: Rc, ergo Rt, which is an invalid form. If we adopt  either an Aristotelian or a Fregean view of propositions we will not be tempted to think otherwise.

‘Cicero is a Roman, therefore Cicero is a Roman’ is plainly valid. ‘Cicero is a Roman, therefore Tully is a Roman’ is plainly invalid. The logical forms are different! If, on a Russellian theory of propositions, the forms are the same, then so much the worse for a Russellian theory of propositions!

Comments

[David Brightly]

Is there a puzzle here? I don't see it. The argument,

Cicero is a Roman
-----------------------
Cicero is a Roman

is valid. Full stop. If one wants to say,

Cicero-the-man is a Roman
-----------------------
Cicero-the-town is a Roman

then that is a different, and invalid, argument.

Logic expresses the meanings of the logical connectives independently of the meanings of categorematic terms. But there is a prerequisite that like terms have like meanings. This enables us to detect equivocation. Consider,

Cicero is a Roman
Cicero is a town
-----------------------
Some Roman is a town

This is valid. With the usual meanings of 'Roman' and 'town' the conclusion is false. Valid arguments are truth-preserving. Hence at least one of the premises is false. Suppose I think the first premise is true (I've heard of the famous Roman orator) and the second is false. If you insist that the second is also true then I will challenge you on the meaning of the second 'Cicero'.

I don't think Ed has given us any reason to think that logical form has to do with anything but sentences, perhaps with some shallow notion of grammar thrown in. How do we explain instantiation of logical form except by substitution of grammatically acceptable token strings for placeholders within schematic sentences?

[BV]

Hello, David. I basically agree with you.

>> But there is a prerequisite that like terms have like meanings.<<

Yes. I would put it like this: It is (defeasibly) presumed that multiple occurrences of the same term have the same meaning. On this presumption, the form of the first argument is Rc, ergo Rc, which is valid. If it should turn out that 'Cicero' is being used equivocally, then the form is Rc, ergo Rd, which is invalid.

[The London Ostrich]

David says

>> How do we explain instantiation of logical form except by substitution of grammatically acceptable token strings for placeholders within schematic sentences?

Bill has explained exactly how. What we must substitute (according to him) is Fregean senses. Logical form is not the form of the sentences, but of Fregean propositions.

Bill says:
>> If it should turn out that 'Cicero' is being used equivocally, then the form is Rc, ergo Rd, which is invalid.

This seems inconsistent with David’s ‘full stop’ (= US ‘period’). Both tokens of ‘Cicero’ are a substitution of ‘grammatically acceptable token strings for placeholders within schematic sentences’, in David’s words

>> But there is a prerequisite that like terms have like meanings.
This is also inconsistent with the ‘full stop’. You may object that we can design a special language where the same term always has the same meaning. I agree. But ordinary language was not designed this way. The same proper name can have different meanings, even within the same text. Acts 1 contains three occurrences of the name ‘Judas’. The first is ‘Judas the son of James’ (1:13), referring to an apostle present at a meeting in Jerusalem after the crucifixion. The second is in 1:16, which at first sight is to the same Judas. However it is clear that this is a reference to a man who (according to the narrator) fell down in a field he bought and ‘burst open in the middle and all his bowels gushed out’. So the first Judas is alive, being present at the meeting, the second is dead.

David says that the likeness of terms ‘enables us to detect equivocation’. Now I agree that validity must be made manifest, must be recognisable, must be capable of being detected. But I suggest it is not likeness of linguistic terms, nor sameness of senses (how do we detect that the senses are the same) that enables this. Consider why readers of Acts 1 have no problem with the apparent contradiction ‘Judas is alive … Judas is dead’. They first apply the rule that consecutive tokens of the same name have the same reference. Then they realise that this is contradictory. Then they apply the rule that a narrator who is aiming to write clearly (as all scriptural writers will do) will not assert a contradiction. This is one of Grice’s maxims, I don’t have the reference to hand. So they infer that the tokens have a different meaning. So it’s not that they use likeness of term, or of sense. Rather, they infer the intended sense by discarding another possible sense.

[BV]

Well, this is a messy discussion, as usual. Not sure quite what is going on. But Ed's question is a good one: What is logical form a form of?

A form of arguments.

What is an argument? A sequence of sentences. But the following are two different sentence sequences but the same argument:

No cat is a dog
-----------
No dog is a cat

Keine Katze ist ein Hund
-----------
Kein Hund ist eine Katze.

This suggests that an argument is a sequence of propositions, not sentences.

[The London Ostrich]

PS I should have said I thought your post was a lucid summary of the three types of propositions, and you are to be commended.

'Same argument' is interesting.

[The London Ostrich]

>>Ed's question is a good one

Thank you for that. Yes, it is a good one.

[The London Ostrich]

Here is an attempt at resolving the ‘Hund’ problem.

The argument ‘Keine Katze ist ein Hund, so Kein Hund ist eine Katze’ is the argument that no cat is a dog, so no dog is a cat. The argument ‘No cat is a dog, so no dog is a cat’ is also the argument that no cat is a dog, so no dog is a cat. So they are the same argument.

Generally if a1 is an argument in one language, a2 and argument in another language, then if a1 argues that p and a2 argues that p, they are the same argument. Also, they are valid if and only if ‘p’ instantiates a valid form.

[The London Ostrich]

Here is a problem for the Fregean view. Let’s grant (which is dubious) that the meaning of the words in the argument is manifest to the reader. I.e. the reader has same way of detecting what the meaning is, even when there is equivocation. When he reads ‘Cicero is Roman, so Cicero is Roman’, he just grasps if the meaning of the second proper name is the same as, or different from that of the first.

So assume the correct sense is always available. But the validity of the arguments with proper names is not to do with sense, but reference. ‘Cicero is Roman, so Cicero is Roman’ is valid only if the two tokens of ‘Cicero’ refer to the same individual. But how is this common reference manifest to the reader? How is he able to detect it? If from the reference, then we can dispense with the Fregean sense altogether. But then we get the problem of ‘Cicero is Roman, so Tully is Roman’. If validity is manifest from the reference, this is valid, which it isn’t.

Nor can it be detectable from the sense alone, if this is all that is available to the reader. ‘Cicero is Roman, so Cicero is Roman’ has tokens of the same sense, but how is the common referent available to us?

You argue that the sense itself points to the referent, like a laser beam. I reply, if so, then the sense of ‘Tully’ will point, like a laser beam, to the same referent that the sense of ‘Cicero’ points to, so the ‘Cicero is Roman, so Tully is Roman’ is valid, which it isn’t.

The underlying question is: how is validity made manifest to the receiver of the argument? If it is not manifest, then there is no point in talking about logical form or whatnot. There is simply no way that we will be able to recognise the validity of any argument. But if it is manifest, how does language achieve this? The logical form, i.e. equiformity of language is not enough, given the possibility of equivocation. So how is this possible at all? That is my underlying problem.

[David Brightly]

Ed is worried about how we are to understand argument form in the presence of equivocation. What is the right form for the argument,

Cicero is a Roman
-----------------------
Cicero is a Roman

when the two tokens 'Cicero' have different meanings? I say this is a needless worry. Logical form is a shallow syntactical business quite independent of the meanings of categorematic terms. It 'floats above' such meanings, as it were. I think we are agreed that we have a notion of valid argument forms, what it means for an argument to instantiate an argument form, and that an argument is valid iff it instantiates a valid form. My next point is one we have not yet brought out. Valid arguments are truth-preserving. Yes, but only in the absence of equivocation in the categorematic terms. We need meanings to reach truth and equivocation on meaning kills truth-preservation. The above argument is valid, instantiating the valid form,

P
---
P

but if we equivocate on the meaning of 'Cicero' (or indeed 'Roman') it may fail to preserve truth. In summary: equivocation impacts truth-preservation but not validity. That's what I meant by 'full stop' in my first comment. Considerations of form and validity do not 'extend down' into considerations of (categorematic) meanings. The two issues are nicely decoupled.

I think this answers Ed's underlying problem in his 02:53 PM comment. At 06:57 AM Ed also takes issue with my 'But there is a prerequisite that like terms have like meanings.' I should have been clearer. This is not a prerequisite for ascertaining validity. It is a requirement for a valid argument to be truth-preserving. Note that Ed's account of his resolving 'Judas' in Acts 1 into two distinct meanings is just like the account I give of 'Cicero is a Roman/Cicero is a town' in my first comment.

[The London Ostrich]

David says:
>>Valid arguments are truth-preserving.

Can you clarify this? Do you mean that a valid argument cannot have true premises and false conclusion?

[BV]

What else could he mean?

If an argument is valid, then the truth of the premises (assuming that they are all true) is transmitted to the conclusion.

My problem is with this: >>In summary: equivocation impacts truth-preservation but not validity. <<

On one reading this is true. It is argument forms, not arguments, that are primary vehicles of validity/invalidity, and no question of equivocation arises with a mere argument form.

On the other hand, if one of the three terms in a syllogism is equivocal, then we have an invalid syllogism, one with four terms. The equivocation induces a formal defect which destroys truth-preservation.

[David Brightly]

Bill,
I prefer to say either,
(a) we have a valid three-term syllogism whose truth-preserving property is broken by equivocation,
or,
(b) we have a four-term syllogism, with no equivocation, but an invalid argument.
We must try to not mix these cases up.

[The London Ostrich]

So does ‘valid’ mean ‘having a valid form’ or ‘truth preserving’? I quick check on online textbooks suggests there is a confusion here too.
http://logic.philosophy.ox.ac.uk/tutorial1/Tut1-07.htm ‘An argument is valid just if it would be impossible for its premises all to be true and its conclusion false simultaneously.’ So truth preservation is all that is needed.
http://www.iep.utm.edu/val-snd ‘A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.’ Here, it is a prerequisite that it has the right form.
https://www.britannica.com/topic/validity ‘Validity, In logic, the property of an argument consisting in the fact that the truth of the premises logically guarantees the truth of the conclusion. Whenever the premises are true, the conclusion must be true, because of the form of the argument’

The problem with this ‘because of the form’ stuff is that no expository syllogism, i.e. ‘a is F, so a is F’ is ever valid. If there is the possibility that the singular term is ambiguous, then the argument is never truth preserving because of its form.

>>either, (a) we have a valid three-term syllogism whose truth-preserving property is broken by equivocation, or, (b) we have a four-term syllogism, with no equivocation, but an invalid argument. We must try to not mix these cases up.<<

Good point, although the puzzle remains.

[David Brightly]

Ed, you have to understand 'truth-preserving' as 'truth-preserving in the absence of equivocation'. Equivocation is utterly deadly to logic. And it doesn't have to be in the singular term. How can there be any discussion at all possible if the meaning of 'Roman' changes from one token to the next? It can render even 'Cicero is a Roman, ergo Cicero is a Roman' truth-corrupting.

[David Brightly]

So does ‘valid’ mean ‘having a valid form’ or ‘truth preserving’?

The property of primary interest is 'truth-preserving' (TP). It turns out (ie, there is a theorem to the effect) that there is a set of forms we call the 'valid' forms such that every TP argument instantiates a valid form and every argument that instantiates a valid form (call these the 'valid' arguments) is TP. So the valid arguments are exactly the truth-preserving ones.

[The London Ostrich]

‘Every argument that instantiates a valid form is TP in the absence of equivocation’. I agree with that. Where ‘argument’ means a set of sentences, not Russellian propositions, and ‘form’ means the form of the words (and not of the constituents of Russellian propositions, or sense or whatnot).

But here we are talking about a special kind of language where words have fixed meanings, and where all proper names are defined in advance. The problem is (1) that even common names have different dictionary meanings and (2) worse, that proper names have no dictionary meaning at all. They are defined in the text as we move along. So ordinary language is utterly different from ‘logical’ language, and yet we got the whole idea of logic from ordinary language! It’s baffling.