Logical Form and the Symmetry Thesis

"The most conspicuous purpose of logic, in its applications to science and everyday discourse, is the justification and criticism of inference." (Emphasis added, Willard Van Orman Quine, Methods of Logic, 2nd revised ed., Holt, Rinehart & Winston, 1959, p. 33.

Perhaps the dispute in the earlier thread could be resolved if we all could agree on the following.

1. The most specific logical form of a deductive argument A is the form relevant for assessing whether the reasoning embodied in A is valid or invalid.

2. Every deductive argument has exactly one most specific form.

3. Symmetry Thesis:  if the most specific form of A is valid, then A is valid; if the most specific form of A is invalid, then A is invalid.

In case 'most specific logical form' needs explanation, consider the difference between the following valid  form from the predicate calculus and the following invalid form from the propositional calculus:

Fa
Ga
-------
(Ex)(Fx & Gx)

p
q
-------
r.

The former is the most specific logical form of 'Al is fat, Al is gay, ergo, something is both fat and gay.'  The latter, if a form of the argument at all, is less specific: it abstracts from the internal subpropositional logical structure of the constituent propositions.

Now three examples in illustration of (1)-(3).

Example One.  Call the following argument 'Charley':

Tom is tall
-------
Tom is tall.

Although the above display, which is a written expression of the argument and not the argument itself, shows two tokens of the sentence type 'Tom is tall,' the argument consists of exactly one proposition.  Anyone who executes the reasoning displayed infers the  proposition *Tom is tall* from itself.  (I am using asterisks to mention propositions.  So '*Tom is tall*' is an abbreviation of 'the proposition expressed by a tokening of the sentence type "Tom is tall".') 

It is perfectly clear that the reasoning embodied by Charley is valid and that its form is 'P ergo P.'  The reasoning is not from P to some proposition that may or may not be identical to P.  Therefore the concrete episode of reasoning does not have the form 'P ergo Q.' 

But let us irenically  concede that if one wished, for whatever reason, to abstract not only from the content of the argument but also from the plain fact that the argument involves exactly one proposition, one could view the form 'P ergo P' as a special case of 'P ergo Q.'   And I will also concede, to keep peace between Phoenix and London, that the argument instantiates the second invalid form, even though I don't believe that this is the case.

Either way, the Symmetry Thesis stands and the Asymmetry Thesis falls.  For as G. Rodrigues in the earlier thread pointed out, 'P ergo P' is the most specific form of Charley. 

Example Two.  Call the following argument 'Kitty Kat.'

If cats like cream, then cats like cream
Cats like cream
-------
Cats like cream.

Please note that there is no equivocation in this example: 'Cats like cream' has the same sense in all four of its occurrences.

Kitty Kat's most specific form is 'P --> P, P, ergo P.'  This form is valid.  So Kitty Kat is valid, notwithstanding the fact, if it is a fact, that Kitty Kat also instantiates the formal fallacy, Affirming the Consequent: P --> Q, Q, ergo P.  By (1) above, the fact, if it is a fact, that Kitty Kat instantiates Affirming the Consequent is irrelevant to the assessment of the validity/invalidty of the reasoning embodied in Kitty Kat.

Example Three.  Call the following example 'Massey':

If God created something , then God created everything.
God created everything.
-------
God created something.

 

This argument fits the pattern of the formal fallacy, Affirming the Consequent:

 

If p then q
q
-------
p.

 

But the argument also has a valid form:

 

Every x is such that Cgx
-------
Some x is such that Cgx. 

Please note that if an argument is valid, adding  a premise can't make it invalid; this principle is what allows us to disregard the first line.

 

(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy VI (1981), pp. 489-500)

The most specific form of Massey is the predicate logic form above displayed.  Since it is valid, Massey is valid.

Symmetry Thesis vindicatus est.

Is everybody happy now?      

Related articles

The Stromboli Puzzle Revisited

Abbreviations, Place-Holders, and Logical Form

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

Logical Form, Instantiation, and Pattern-Matching

On Whether Some Arguments from Evil Beg the Question

Eric Voegelin's 1948 Definition of Scientism

Comments

[Ed]

>> And I will also concede, to keep peace between Phoenix and London, that the argument instantiates the second invalid form, even though I don't believe that this is the case.<<

You are conceding something you don’t believe the case? How unphilosophical. The philosophical way is to continue the path to the end, however painful.

In any case, I still don’t understand what you are believing or not believing. Earlier I asked you which of these was you position:

(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.

Now I think you are veering towards (3), i.e. you accept authority for what it is, but that you have some other conception of logic in mind. You accept that the argument ‘tom is tall, therefore tom is tall’ instantiates the form 'P therefore Q', as orthodox logicians understand the terms ‘argument’ and ‘instantiate’.

But your conception of logic is different. Rather than talk about ‘arguments’, you want to talk about ‘concrete episodes of reasoning’. The concrete episode of reasoning ‘tom is tall, therefore tom is tall’does not have the form 'P therefore Q'.

Is that correct? You need to be specific about your position. Are we arguing about what is correct per the standard textbooks, the ‘authorities’ as the medieval logicians called them? Or some other conception of logic?

If it is another conception of logic, I would have to think about that. One problem is this: can there be a formal logic of ‘concrete episodes of reasoning’ at all?

Colin Cheyne looked at the previous thread, and told me (by email), that applying Peter’s restriction R2 (i.e. not allowing same into different) “would result in proofs in meta-logic being unnecessarily complicated”.

[BV]

The corresponding conditional for Charley is 'If Tom is tall, then Tom is tall.'

How do we translate this conditional into the PC? First we arbitrarily choose some schematic letter. Let it be 'P.' Then we write ' P --> P.'

One cannot write 'P --> Q.' Why not? Because the antecedent and the consequent of the conditional are the same proposition. 'Q' would allow the consequent to be a different proposition.

The rule here is that there must be a UNIFORM replacement of term/sentence by placeholder. If this rule is not stated in any of your textbooks, then that this presumably because it is too obvious to need stating.

I mean, is it not just perfectly obvious that one has not captured the logical form of 'If Tom is tall, then Tom is tall' if one depicts the logical form as 'P --> Q'?

[BV]

Furthermore it is easy to show using a truth table that 'P --> P' is a tautology while 'P --> Q' is a contingency. Now it is obvious that the English conditional is a tautology. Ergo, etc.

[Ed]

Right, so we are veering back to (1). It's not about a 'conception of logic', but rather an elementary mistake by Londoners.

I have already pointed out that we can derive

P → (P → P)

from

A → (B → A)

>>it is easy to show using a truth table that 'P --> P' is a tautology while 'P --> Q' is a contingency. Now it is obvious that the English conditional is a tautology

Likewise, we can symbolise both 'it is raining or it is not raining' and 'it is raining' as 'p'.

[london ed]

>>If this rule is not stated in any of your textbooks, then that this presumably because it is too obvious to need stating.

The very opposite. See Understanding Arguments: an Introduction to Informal Logic, Robert Fogelin and Walter Sinnott-Armstrong. Is Fogelin good enough?

"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.

The fact that most textbooks do not state the London rule explicitly is presumably because what is obviously the case to you, is obviously not the case for most textbook writers. I.e. you think it obvious that different variables may not be replaced with the same proposition, but most textbook writers think it obvious that different variables may be replaced with the same proposition, and so do not state it.

I freely confess that though I had originally assumed the London rule (by reason of the 2+2 substitution rule the Fogelin et al mention), I hesitated when you and Peter Lupu came back so strongly. I had to search hard for textbook examples.

This suggests that the London rule isn't obvious, and that a good textbook should not hurry through substitution, as many do. And I note that Fogelin says 'somewhat surprisingly', so clearly he thinks it isn't obvious.

[David Brightly]

Bill,

I think it's become clear that we are wrangling over two distinct methods for evaluating the validity of an argument. The London method starts with a finite catalogue of argument forms stipulated to be valid. It has a relation of 'instantiation' or 'derivation' between arguments and forms. London declares an argument valid if it can be derived from one of the valid forms. Conversely, if it cannot be so derived then it is invalid. The Phoenix method works in the opposite direction. It says that every argument has a 'most specific form' (MSF) and declares that an argument is valid iff its MSF is one of the valid forms. Both methods agree on the valid forms. So if Londonistas and Phoenicians do indeed have 'distinct concepts of logic', they do at least agree on what argument forms are valid. That's good, and suggests that the distinct concepts of logic cannot be that far apart. But London remains perplexed because Phoenix hasn't explained how to calculate the MSF from a given argument. We are told that the MSF of 'Al is fat, Al is gay, ergo, something is both fat and gay' is 'Fa; Ga; ergo, (Ex)(Fx & Gx)', paying attention to subsentential structure. By analogy then, the MSF of 'Tom is tall; ergo, Tom is tall' ought perhaps to be 'Fa; ergo Fa', but we are told it's 'P; ergo, P'. We agree that the Phoenix method works informally. Our question is, 'Can the extraction of the MSF be formalised?'

[BV]

David,

Thank you for that very helpful comment. I think you have put your finger on one source of our disagreement. You and Ed start with argument forms and then proceed to arguments. We start with arguments and proceed to forms.

Thus in the simple case of 'Charley,' we try to translate the argument in such a way as to expose its logical form. We peel the flesh off the bones so as to lay bare the bones, the logical skeleton. Proceeding in this way, we arrive at 'P ergo P' and deny that the form is 'P ergo Q' -- for the reasons I gave above. You on the other hand start with the two forms mentioned and subsume Charley under both of them. What justifies your subsumption of Charley under the invalid form is that the use of 'Q' does not rule out the possibility that 'P' and 'Q' stand for the same proposition.

As you say, we agree on which forms are valid, both in the prop calc and the pred calc. We also agree on which forms are invalid. Thus we all agree, for the sake of this discussion at least, to take as unquestionable standard first-order predicate logic with identity. (Relevance logic and dialetheism are off the table.)

You ask: Can the extraction of the MSF be formalized? What exactly do you mean by formalize? Are you asking whether there is a decision procedure in the sense in which truth tables provide a decision procedure for evaluating arguments encoded in the prop calc?

I don't believe one can formalize the procedure for the extraction of the MSF from an ordinary language argument. One has to interpret the meaning of the OL sentences. Man does not live or think by syntax alone. Consider this argument:

Nobody came to the party
Everybody who came to the party received a gift
-------
Nobody received a gift.

Suppose some poor schmuck rejoices under the name 'Nobody.' Then the argument has this form:

Ca
(x)(Cx --> Gx)
-------
Ga

which is plainly valid. But the argument could, with rather more plausibility, be interpreted to have this form:

~(Ex)Cx
(x)(Cx --> Gx)
-------
~(Ex)Gx

which is invalid.

There is no way to extract the MSF except by first determining the meaning of the OL argument. That is the first step, and it cannot be formalized. What this first step does is establish the identity of the argument as sequence of propositions. Sentences can be ambiguous, not propositions.

But if you are a nominalist like Ed, then you will balk at talk of propositions. This is another bone of contention beneath or alongside the other bones we have been worrying.

Your comment is admirably irenic. We now need to pacify Ed.

[Ed]

>>We now need to pacify Ed.

Consider me pacified.

[Peter Lupu]

Londoners:

I apologize to the Londoners for the "Londoners and their pack" phrase above. You do not deserve it!!!

I think we are getting closer to some basis of agreement. Thanks David B. and BV. Ed and I, the stubborn ones, hopefully helped clarifying at least some issues. In any case, the debate was useful. I learned some things, one of which that some basic things in logic may not be as clear cut as one hoped for. Well, I suppose this shows that there is so much still to learn and therefore grow.

Incidentally, I had a *little* time to check and found that textbooks indeed fail to be very specific about the precise method when it comes to replacing various phrases/sentences into a formal notation. I think unstated or vaguely stated assumptions are made. In one book, however, the Londoner's rule was mentioned as in principle one to use, but then it was quickly discarded because there are very few cases in a Natural Language that would require such a rule. The example itself however was not very good because once the argument was displayed, the conclusion was a tautology and, hence, the premises were superfluous.

May be now we can return to the beginning of our journey to Ed's issues with fictional names. I think there is much there to explore.

[BV]

Peter,

Which book are you referring to? If you have time, please reproduce here the relevant portion.

[Ed]

>>Which book are you [PL] referring to?

I don't know what book he is referring to, but I have already excerpted Fogelin & Armstrong. They clearly state the London rule ("Different variables may be replaced with the same proposition [i.e. the London rule], but different propositions may not be replaced with the same variable". All textbooks mention the second part of the rule, very few mention the first.

[Peter Lupu]

Bill,

The Book:

Patrick, J. Hurley, (2008), A Concise Introduction to Logic, (10th. edition).

On page 56 Hurley says:

"Every substitution instance of a valid form is a valid argument, but it is not the case that every substitution instance of an invalid form is an invalid argument. The reason is that some substitution instances of invalid forms are also substitution instances of valid forms.* However, we can say that any substitution instance of an invalid form is an invalid argument *provided* that it is not a substitution instance of any valid form."

The example he gives (in a footnote; bottom of page 56) of a case of a valid argument that is also a substitution instance of an invalid form is:

"All bachelors are persons.
All unmarried men are persons.
Therefore, all bachelors are unmarried men."

The argument is valid because the conclusion must be true given the meaning of 'bachelor'. The argument is invalid, presumably, because it features the invalid form:

All As are Bs.
All Cs are Bs.
Therefore, all As are Cs.

But, Hurley argues in a footnote designated by (*) that given that

Bachelor =df. unmarried man

You get that the argument is also a substitution instance of the valid form:

All As are Bs
All As are Bs
Therefore, All As are As.

Finally, Hurley says:

"However, cases of ordinary language arguments that can be interpreted as substitution instances of both valid and invalid forms are so rare that this book chooses to ignore them." (p.57)

Two points: first, the example is not really cogent, in my opinion, for the linguistic definition should be already factored into the argument about bachelors from the start. Second, I do not see a point to be so permissible about logical form so as to allow substitution instances of both valid and invalid arguments only to discard all such cases due to rare instances of such. I see no gain and much confusion emanating from such a procedure.

But, again, the overall lesson is I think that much more in depth study needs to be undertaken regarding the concept of *logical form* and its role in logic.

(I hope I had fifty more years.)