Placeholders, Variables, and Logical Form

London Ed refers us to Understanding Arguments: an Introduction to Informal Logic, Robert Fogelin and Walter Sinnott-Armstrong, and provides this quotation:

Perhaps a bit more surprisingly, our definitions allow 'roses are red and roses are red' to be a substitution instance of 'p & q'. This example makes sense if you compare it to variables in mathematics. Using only positive integers, how many solutions are there to the equation 'x + y = 4'? There are three: 3+1, 1+3, and 2+2. The fact that '2+2' is a solution to 'x + y = 4' shows that '2' can be substituted for both 'x' and 'y' in the same solution. That's just like allowing 'roses are red' to be substituted for both 'p' and 'q', so that 'roses are red and roses are red' is a substitution instance of 'p & q' in propositional logic.

In general, then, we get a substitution instance of a propositional form by uniformly replacing the same variable with the same proposition throughout, but different variables do not have to be replaced with different propositions. The rule is this:

Different variables may be replaced with the same proposition [Ed: Let's call this the London rule], but different propositions may not be replaced with the same variable.

Suppose I am given the task of determining whether  the conditional English sentence 'If roses are red, then roses are red' is a tautology, a contradiction, or a contingency.  How do I proceed?

Step One is translation, or encoding.  Let upper case letters serve as placeholders for propositions.  Let '-->' denote the truth-functional connective known in the trade as the material or Philonian conditional.  I write 'P --> P.'

Step Two is evaluation.  Suppose for reductio that the truth value of 'P -->P' is false.  Then, by the definition of the Philonian conditional, we know that the antecedent must be true, and the consequent false.  But antecedent and consequent are the same proposition.  Therefore, the same proposition  is both true and false. This is a contradiction.  Therefore, the assumption that conditional is false is itself false.  Therefore the conditional is a tautology.

Now that obviously is the right answer since you don't need logic to know that 'If roses are red, then roses are red' is a tautology. (Assuming you know the definition of 'tautology.')  But if if Fogelin & Co. are right, and the 'P -->Q' encoding is permitted, then we get the wrong answer, namely, that the English conditional is a contingency.

I am assuming that if 'P-->Q' is a logical form of 'If roses are red, then roses are red,' then 'P -->Q' is a legitimate translation of 'If roses are red, then roses are red.'  As Heraclitus said, the way up and the way down are the same.  The assumption seems correct.

If I am right, then there must be something wrong with the mathematical analogy.  Now there is no doubt that Fogelin and his side kick are right when it comes to mathematics.  And I allow that what they say is true about variables in general.  Suppose I want to translate into  first-order predicate logic with identity the sentence, 'There is exactly one wise man.'  I would write, '[(Ex)Wx & (y)(Wy --> x = y)].'  Suppose Siddartha is the unique wise man.  Then Siddartha is both the value of 'x' and the value of 'y.' 

So different variables can have the same value.  And they can have the same substituend.  In the example, Siddartha is the value and 'Siddartha' is the substituend.  But is a placeholder the same as a variable?  I don't think so.   Here is a little argument:

No variable is  a constant
Every placeholder is an arbitrary constant
Every arbitrary constant is a constant
-------
No placeholder is a variable.

A placeholder is neither an abbreviation, nor a variable.  It is an arbitrary constant.  Thus the logical form of 'Al is fat' is Fa, not Fx. Fa is a proposition, not a propositional function.   'F' is a predicate constant.  'a' is an individual constant.  We cannot symbolize 'Al is fat' as Fx.  For Fx is not a proposition but a propositional function.  If 'a' were not an arbitrary constant, then Fa would not depict the logical form of 'Al is fat,' a form it shares with other atomic sentences.

Here is another argument:

Every variable is either free or bound by a quantifier
No placeholder is either free or bound by a quantifier
-------
No placeholder is a variable.

Here is a third argument:

Every variable has a domain over which it ranges
No placeholder has a domain over which it ranges
-------
No placeholder is a variable.

A fourth argument:

There is no quantification over propositions in the propositional calculus
-------
There are no propositional variables in the propositional calculus
If there are no propositional variables in the propositional calculus, then the placeholders in the propositional calculus cannot be variables
-------
The placeholders in the proposition calculus cannot be variables.

Punchline: because placeholders are not variables, the fact that the different variables can have the same value and the same substituend does not show that different placeholders can have the same  substituend.  'If roses are red, then roses are red' does not have the logical form 'P -->Q' and the latter form does not have as a substitutution-instance 'If roses are red, then roses are red.'

As I have said many times already, one cannot abstract away from the fact that the same proposition is both antecedent and consequent.

What one could say, perhaps, is that 'P --> P' has the higher order form 'P --> Q.'  But this latter form is not a form of the English sentence but a form of the form of the English sentence.

Ed can appeal to authority all he wants, but that is an unphilosophical move, indeed an informal fallacy.  He needs to show where I am going wrong.

Related articles

Comments on London Ed's "Towards a Positive Theory"

Logical Form and the Supposed Asymmetry of Validity and Invalidity: A Defense of Symmetry

Logical Form and the Symmetry Thesis

How to 'Derive' Ought from Is

Abbreviations, Place-Holders, and Logical Form

Comments

[Ed]

I wasn't appealing to authority at all, and it's unfair to say I was. Much earlier I asked which of the following positions ws yours.

(1) That it is not OK to make that substitution, and that we have misunderstood the textbooks, or that the textbooks we are quoting (including Geach) have got it wrong. (It's not unknown for textbooks to get things wrong.

(2) That we are understanding the textbooks correctly, and correctly quoting them, but that orthodox logic is wrong? That's a very bold claim.

(3) That we and the textbooks are literally correct, but there is some 'other conception' of logic which transcends this? I have no view on that. I am simply interested in agreeing what the textbook 'received opinion' is.

(4) Something else?

Initially you seemed to suggest (1), i.e. that London had made some elementary mistake that could be corrected by looking at any authority on the subject. So the argument wasn't about whether the authorities are right, but about what the authorities say.

Now you are settling on (2). The authorities are wrong. I don't really have a view on that. In mathematics, it's a general rule that if you are questioning authority, then you provide pretty strong reasons.

[Ed]

Do you agree that 'Socrates is sitting and it is not the case that Socrates is sitting' and 'Socrates is sitting or it is not the case that Socrates is sitting' and the plain 'Socrates is sitting' can all be represented by the placeholder 'p'?

Do you agree that the first is necessarily false, the second necessarily true, and the first contingent, i.e. not necessary?

Perhaps I shouldn't get drawn into this though.

[BV]

Ed,

Since you ignore the specific things I say and my wealth of distinctions, I will ignore what you say.

But I have a solution for you.

Take the conjunction of any argument's premises and symbolize that with 'P.' Symbolize the conclusion as 'P'. Reduce the argument to its corresponding conditional, 'P --> P.' Symbolize that conditional with 'P.

You have now simplified logic greatly. Every argument has the form, P!

[Ed]

>>You have now simplified logic greatly. Every argument has the form, P!

No, wrong. Every conditional has the form p (as well as the form ‘if q then r’, we have already agreed this). But every argument has the form ‘p, therefore q’.

You still need to say whether (1) you think Londonistas have got orthodox logic wrong or whether (2) you think orthodox logic is wrong. Do you agree these are separate positions?

Addressing exactly what you say above on variables versus placeholders, note that Fogelin et all compare variables with placeholders. They don't say they are the same thing. And in any case you are simply wrong about placeholders. All standard definitions of 'well-formed-formula' allow for any token formula to be substituted into any placeholder (and thus allow for same into different).

[Ed]

And addressing your 'punchline'.

>>because placeholders are not variables, the fact that the different variables can have the same value and the same substituend does not show that different placeholders can have the same substituend.

But placeholders are analogous to variables precisely in that you can put any wff into them. Note the word 'any'. Not 'any except ones you have substituted into other placeholders'.

We are arguing about standard logic here, not anything particularly exotic.

[Ed]

Willard van Orman, Methods of Logic p.32 (Truth Functions)

"It is permissible to put the same or different schemata for different letters [i.e. the London permission], but we must always put the same schema for recurrences of the same letter".

Perhaps you will then say I am arguing from authority, but I am not. I am arguing about what the authorities say.

So we have Geach, Quine, Fogelin, Cheyne and others on the London side. Can Phoenix cite any on their side?

I repeat, we are not arguing whether the authorities are right or wrong, we are not arguing about what the authorities actually say. End of.

[Peter Lupu]

I still have one question to the Londoners. It is in line with Bill's comment. What is the advantage to logic for preferring the Londoners' rule rather than the one Bill and I prefer? It is very clear what is the advantage in Math. Examples were given. But what is the advantage of doing so in Logic? Give one example where the more restrictive rule Bill and I propose would result in a loss of some kind? "generality" for its own sake cannot be the only basis for preferring the Londoners' rule unless you can show some conceptual or tangible gains in logic by opting for the more general rule.

[Ed]

Wilfrid Hodges (and who quarrels with Hodges?) Logic, 1977 edition p.134, gives the example deriving ~Q -> [Q->Q] from ~P -> [P->Q]. He doesn't mention the 'London permission', but clearly uses it. I gave a similar example very early on in this tortuous series of threads. Why ignore it*?

Again, if you say that this is argument form authority, please give reasons why authority is wrong. Your argument that 'placeholders are not variables' is itself an appeal to authority. Who says that placeholders are not variables? Editions and page numbers please.

*See this comment. Note in that comment I mention it was not the first time I had said it, so I go on like a broken CD.

[Ed]

>>What is the advantage to logic for preferring the Londoners' rule rather than the one Bill and I prefer?

According to Cheyne (by email), avoiding unnecessary complication for proofs in meta-logic.

I have already given the example of deriving "P → (P → P)" from "A → (B → A)". (I think this is the 4th of 5th time).

And generally, when you are allowed to substitute anything, rather than just things that you haven't substituted earlier, makes life easier. Same as in mathematics, no? (I already said this before, too).

[BV]

Ed,

You are confusing different things. The Hodges argument is valid and can be shown to be such by constructing a derivation using the accepted rules of inference, Association, Addition, etc. But this has nothing to do with the 'London permission.'

Nobody is questioning any aspect of the propositional calculus (PC). What we have been talking about is translation and substitution. Translation of an OL argument into the PC, and whether or not a certain OL argument is a substitution-instance of a PC form.

I've had enough of this topic. I've clarified the matter to my own satisfaction. Further discussion will be unproductive.

By the way I gave four arguments why placeholders are not variables -- which you ignored. And you call that an appeal to authority?

I will post your latest missive and we'll see if we can make any progress on that. But nobody should hold his breath.

[Ed]

>>By the way I gave four arguments why placeholders are not variables -- which you ignored.

Not at all. But as you say, you don't want to discuss the subject any more.

[David Brightly]

Bill,
You ask where you are going wrong. No one, I think, will gainsay your arguments that placeholders are not variables in the technical sense of the predicate calculus. But you go on to say 'There are no propositional variables in the propositional calculus.' I'm not sure this is right at all. Fogelin talks about 'variables', as we can see here, and my old copy of Alan Hamilton, Logic for Mathematicians, CUP 1978, uses 'statement variables' in his account of the 'statement calculus', as here. The justification for 'variable' is surely that statements have values, namely truth and falsehood. The truth value of a compound statement is calculated from the truth values of its component simple statements by composition of the truth functions corresponding to the logical connectives. This is analogous to the evaluation of an arithmetic expression by composition of arithmetic functions applied to the values of arithmetic variables.

You also argue that London must wrongly decide that 'if roses are red then roses are red' (RR) is a contingency, because we say it can be seen as having the form 'P-->Q' and in general statements of this form are contingencies. Indeed they are. But we don't so decide. We say this is a special case in which P and Q stand for the same simple sentence, 'roses are red', not different ones. P and Q are therefore either both true or both false and either way the truth function for --> returns true. Hence this special case is tautologous. We disagree that the move from RR to 'P-->Q' must be seen as an abstraction. We retain the information that P and Q stand for specific substatements within RR, which may themselves have internal structure. 'Form' is a device for making such structure explicit.