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

For the 'Londonistas,'  Ed and David, partners in logical investigations.  We are unlikely ever to agree, but clarification of differences is an attainable and worthwhile goal, here, and in every arena of controversy.  Have at it, boys.

.............

1. Suppose someone reasons as follows. 'Some Englishmen are Londoners; therefore, some Londoners are Englishmen.'  To reason is one thing, to reason correctly another.  So one can ask: Is this specimen of reasoning correct or incorrect?  This is the sort of question with which logic deals.  Logic is the study of inference and argument from a normative point of view.   It seeks to articulate the criteria of correct and incorrect reasoning.  It is analogous to ethics which seeks to articulate the criteria of correct and incorrect action.

2. We all take for granted that some reasoning is correct and some incorrect, and we are all more or less naturally good at reasoning correctly.  Almost everyone grasps immediately that if Tom is an Englishman and some Englishmen are Londoners, it does not follow that Tom is a Londoner. What distinguishes the logician is his reflective stance.  He reflects upon reasoning in general and tries to extract and systematize the principles of correct reasoning.  'Extract' is an apt metaphor.  The logician  develops a theory from his pre-theoretical understanding of argumentative correctness.  As every teacher of logic comes to learn, one must already be logical to profit from the study of logic just as one must already be ethical to profit from the study of ethics.  It is a matter of making explicit and raising to the full light of awareness what must already be implicitly present if the e-duc-ation, the drawing out into the explicit is to occur.  This is why courses in logic and ethics are useless for many and positively harmful for some.  But they do make some of us more logical and more ethical.

3.  Correctness in deductive logic is called validity, and incorrectness invalidity.   Since one can argue correctly from false premises and incorrectly from true premises, we distinguish validity from truth.  Consider the following argument:

Some Englishmen are Londoners
-------
Some Londoners are Englishmen.

We say of neither the premise nor the conclusion that it is either valid or invalid: we say that it is either true or false.  And we do not say of the argument that it is true or false, but that it is either valid or invalid. We also speak of inferences as either valid or invalid. 

4.  What makes a valid argument valid?  It can't be that it has true premises and a true conclusion.  For there are invalid arguments that satisfy this condition.  Some say that what makes a valid argument valid is the impossibility of the premises' being true and the conclusion false.  Theirs is a modal explanation of validity.  Equivalently,

D1. Argument A is valid =df necessarily, if A's premises are all true, then A's conclusion is true.

This necessity is plainly the necessity of the consequence (necessitas consequentiae), not the necessity of the consequent (necessitas consequentiis):  in the majority of cases the premises and conclusion are all contingent propositions.

The modal explanation of validity in (D1) is fine as far as it goes, but it leads to the question: what is the ground of the necessity?  If validity is explained by the RHS of (D1), what explains the necessity?  What explains the necessitas consequentiae of the conditional on the RHS of (D1)?

Enter logical form.

The validity of a given valid argument evidently resides in something distinct from the given argument.  What is this distinct something?  It is the logical form of the argument, the argument form.  The form F of an argument A is distinct from A because F is a universal (a repeatable) while A is a particular (an unrepeatable).  Thus the form

All S are M
All M are P
-------
All S are P

is a one-in-many, a repeatable.  It is repeated in every argument of that form.  It is the form of  indefinitely many syllogisms, although it is not itself a syllogism, any more than 'All S are M' is a proposition.  A proposition is either true or false, but 'All S are M' is neither true nor false.  To appreciate this, bear in mind that 'S' and 'M' are not abbreviations but placeholders.  If the letters above were abbreviations, then the array above would be an (abbreviated) argument, not  an argument form.  An argument form is not an argument but a form of indefinitely many arguments. 

Now validity is a property of argument forms primarily, and secondarily of arguments having valid forms. What makes a valid argument valid is the validity of its form:

D2. Argument A is valid =df A is an instance of a valid argument form.

D3. Argument form F is valid =df no  instance of F has true premises and a false conclusion.

Validity is truth-preserving: a valid argument form will never take you from true premises to a false conclusion.  (Exercise for the reader: show that invalidity is not falsehood preserving.)  In sum, an argument is valid in virtue of having a valid form, and a form is valid if no argument of that form has true premises and a false concusion. The logical form of a valid argument is what makes it impossible for the premises to be true and the conclusion false.

5.  If a valid argument is one with a valid form, one will be tempted to to say that an invalid argument is one with an invalid form.  Call this the Symmetry Thesis:

ST. If an argument  is an instance of a valid form, then it is valid, and if it is an instance of an invalid form, then it is invalid.

But there are examples that appear to break the symmetry, e.g.:

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. 

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

So which is it? Is the argument valid or invalid?   It can't be both and it can't be neither.  One option is to abandon the Symmetry Thesis and maintain that having a valid form is sufficient for an argument to be valid, but that having an invalid form is not sufficient for it to be invalid. One would then be adopting the following Asymmetry Thesis:

AT.  Having a valid form suffices for an argument to be valid, but having an invalid form does not suffice for an argument to be invalid.

Another option is to hold to the Symmetry Thesis and maintain that the Massey argument is really two arguments, not one.  But before exploring this option, let us consider the unintuitive consequences of holding that one and the same argument can have two different forms, one valid, the other invalid. 

6. Consider any valid syllogism.  A syllogism, by definition, consists of exactly three different propositions: a major premise, a minor premise, and a conclusion.  So every valid syllogism has the invalid form: p, q, ergo r.  Generalizing, we can say that any argument whose validity hinges upon the internal subpropositional logical structure of its constituent propositions will instantiate an invalid form from the propositional calculus (PC).  For example, any argument of the valid form, Some S are P; ergo, Some P are S, is an instance of the invalid PC form, p, ergo q. 

To think of a valid syllogism as having the invalid form p, q, ergo r is to abstract away from the internal subpropositional logical structure that the syllogism's validity pivots on.  But if this abstraction is permitted, one may permit oneself to abstract away from the requirement that the same terms in an argument be replaced by the same placeholders.  One might then maintain that

All men are mortal
Socrates is a man
-------
Socrates is mortal

has the invalid logical form

All Fs are Gs
a is an H
-------
a is a G

But why stop there?  By the same 'reasoning,' the Socrates syllogism has the invalid form:

All Fs are Gs
a is an H
-------
b is an I.

But if one abstracts away from the requirement that the same term or sentence be replaced by the same placeholder, then we get the result that the obviously valid

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

has the valid form p ergo p and the invalid form p ergo q.  Here we are abstracting away from the fact that a proposition entails itself and ascending to the higher level of abstraction at which  a proposition entails a proposition.  After all, it is surely true that in our example a proposition entails a proposition.

I submit, however, that our example's having an invalid form is an intolerable result.  Something has gone wrong.  Surely the last argument has no invalid form.  Surely one cannot lay bare the form of an argument, in an serious sense of 'argument,'  if one abandons the requirement that the same term or sentence be replaced by the same placeholder. To do that is to engage in vicious abstraction.  It is vicious because an argument in any serious sense of the term is not just a sequence of isolated propositions, but a sequence of propositions together with the idea that one of them is supposed to follow from the others.  An argument in any serious sense of the term is a sequence of propositions that has the property of being putatively such that one of them, the conclusion, follows from the others, the premises.  But no sequence of propositions can have this property if the argument's form allows for different terms/propositions to have different placeholders.

7.  So I suggest that we abandon the Asymmetry Thesis and adopt the Symmetry Thesis according to which no valid argument has any invalid forms.  Let me now try to motivate this proposal.

An argument form is an abstraction from an argument.  But it is also true that an argument is an abstraction from a concrete episode of reasoning by a definite person at a definite time.  Clearly, the same argument can be enacted by the same person at different times, and by the same or different persons at different times.  I can 'run through' the argument that the null set is unique any number of times, and so can you.  An argument in this sense is not a concrete episode of arguing (reasoning) but a sequence of propositions.  A proposition, of course, is not the same as a sentence used to express  it.

Now I grant that an argument taken in abstraction from an episode of reasoning (and as the content of that reasoning) can instantiate two or more argument forms.  But I deny that a concrete episode of reasoning by a definite person at a definite time can instantiate two or more argument forms. So my claim is that while an argument in abstracto can have two or more forms, an argument in concreto, i.e. a concrete episode of reasoning cannot have more than one form.  If this form is valid the argument in concreto is valid.  If invalid, the argument in concreto is invalid.  To illustrate:

Suppose I know that no Democrat supports capital punishment.  Then I learn that Jones is a Democrat.  Putting together these two pieces of information, I infer that Jones does not support capital punishment. By 'the concrete episode of reasoning,' I mean the reasoning process together with its content.  One first thinks of the first proposition, then the second, then one infers the third, and all of this in the unity of one consciousness.  The content is the argument considered in abstraction from any particular diachronic mental enactment by a particular person at a particular time.  The reasoning process as a datable temporally extended mental process is also an abstraction from the concrete episode of reasoning which must include both, the reasoning and its content.

Now the concrete episode of reasoning embodies a pattern.  In the example, I reason in accordance with this pattern:

(x) (Fx --> ~Gx)
Fa
-------
~Ga

Which is also representable as follows:

No Fs are Gs
a is an F
-------
a is not a G.

The pattern or logical form of my concrete episode of reasoning is assuredly not: p, q, ergo r.  This is consistent with saying that the argument in abstracto instantiates the invalid form p, q, ergo r in addition to the valid form above.

The point I am making is this.  If we take an argument in abstraction from the concrete episode of reasoning in which it is embodied, then we may find that it instantiates more than one form.  There is no denying that every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r.  But no one who reasons syllogistically reasons in accordance with that invalid form.  Syllogistic reasoning, whether correct or incorrect, is reasoning that is sensitive to the internal subpropositional logical structure of the syllogism's constituent propositions.  The invalid form is not a form of the argument in concreto.

 One must  distinguish among the following:

• The temporally extended event of Jones' reasoning.  This is a particular mental process.
• The content of this reasoning process, the argument in abstracto as sequence of propositions.
• The concrete episode of reasoning (i.e. the argument in concreto)  which involves both the reasoning and its content.
• The verbal expression in written or spoken sentences of the argument.
• The form or forms of the argument in abstracto.
• The verbal expression of a form or forms in a form diagram(s).
• The form of the argument in concreto.

 My point, again, is that we can uphold the Symmetry Thesis if we make a distinction between arguments in the concrete and arguments in the abstract.  But this is a distinction we need in any case.  The Symmetry Thesis holds for arguments in the concrete.  But these are the arguments that matter because these are the ones people actually give.

Applying this to the Massey example above, we can say that while the abstract argument expressed by the following display has two forms, one invalid, the other valid:

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

there is no one concrete argument, no one concrete episode of reasoning, that the display expresses.  One who reasons in a way that is attentive to the internal subpropositional structure of the constituent propositions reasons correctly.  But one who ignores this internal structure reasons incorrectly.

In this way we can uphold the Symmetry Thesis and avoid the absurdities to which the Asymmetry Thesis leads.

 Related articles

Abbreviations, Place-Holders, and Logical Form

 

Logical Form, Instantiation, and Pattern-Matching

The Stromboli Puzzle Revisited

On Whether Some Arguments from Evil Beg the Question

On the Enforcing and Permitting of Coreferentiality by Argument-Forms

How to 'Derive' Ought from Is

Comments

[Ed]

>>There is no denying that every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r.
<<

Right, so now we are in agreement. You agree that 'p, p, therefore p' instantiates 'p, q, ergo r'.

Earlier, you objected to David's claim about instantiation, and Peter Lupu said something about not instantiating different placeholders with the same token. Now you agree?

>>But no one who reasons syllogistically reasons in accordance with that invalid form.

This is a different and entirely separate claim. Neither I nor I think David made any mention of that.

[BV]

Although I perhaps did not make it sufficiently clear in my post, if one grants the point you quote me as making, then there is no principled way to stop the slide to >> 'p, p, therefore p' instantiates 'p, q, ergo r'.<<

And that I take to be a reductio ad absurdum of the notion that logic is concerned with arguments in the abstract. I am urging that logic is concerned with arguments in the concrete (as I explained, and only in that sense), and that this distinction allows us to uphold the Symmetry Thesis.

[Peter Lupu]

Conversation I:

A. God exists and it is not the case that God exists.
B. Ahhh!! Got ya. You just contradicted yourself.
A. Not necessarily. If you take what I just said as having the logical form

(i) P & ~P

then you are right. One the other hand, you may take what I just said to have the form

(ii) P & Q

or perhaps even just

(iii) R.

in both of the last cases of course you are wrong.

B. Well, then, which one you meant?

A. All three.

B. Simultaneously?

A. Yes!

B. But this makes no sense. For if you could have meant all three simultaneously, then you both contradicted yourself and also did not.

A. Exactly!

B. But, then, no one ever contradicts themselves or for that matter fails to contradict themselves.

A. So what is wrong with that?

B. Everything!!!

Conversation II:

A. God exists or it is not the case that God exists.

B. But this is trivial; you have not asserted anything substantive.

A. That depends. If you construe what I just said as

(i) P v ~P

then you are right. But you do not have to. What I said could also be construed as having the form

(ii) P v Q

or even

(iii) R.

B. So which one you meant?

.......

Other variants are also possible. For instance: A asserts: 'God exists'. B responds: So you contradicted yourself for you just said: 'God exists and it is not the case that God exists'. etc.

All of the above are parodies of the Londoners' position, here illustrating the consequences of their position regarding contradictions and tautologies. Ridiculous!

I think I have said this before. Logical Form must be always relativized to the logical connectives. It cannot be defined in abstraction from the logical connectives; o/w logical form makes no sense. Once it is clear which logical connectives are relevant for the logical form, then logical form is defined relative to them so that all the relevant connectives are explicitly represented and the relevant variables (propositional, , predicate variables, and individual variables, etc.,) are consistently placed. Validity, truth, contradiction, and tautologies are defined relative to logical form so construed. Unless one follows something like this procedure, there is no point to logic at all.

Incidentally, while I certainly agree with most of what Bill says in the above post, I do not think that logic as such should be intimately linked to any mental process. The fundamental logical concepts apply to abstract entities such as propositions and therefore they are themselves abstract concepts. Even if there were no thinking beings at all, logical form and the rest of the logical paraphernalia would have been still true of propositions (not unlike mathematical structures). Of course, if there were no thinking beings, then there would not have been any concrete instances of valid arguments either in thought alone or manifested by way of a physical medium.

[Ed]

>>And that I take to be a reductio ad absurdum

I thought we had reached agreement, but now I am confused again. What do you mean by a reductio ad absurdum? I thought it was: assume P, reach an absurd and impossible conclusion Q, then conclude not-P.

I assume that the P here is your “every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r”. And Q is your “p, p, therefore p' instantiates 'p, q, ergo r'”. For you say that there is no way to ‘stop the slide’ from P to Q. So you seem to be suggesting that since Q is absurd or impossible, so is P.

But then I don’t understand the ‘there is no denying that’ part. You said

>>There is no denying that every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r.<<

So you seem to be arguing “There is no denying that P, there is no way from stopping the slide from P to Q, but Q is absurd, therefore not P”. This is an outright contradiction.

And then you say “I take to be a reductio ad absurdum of the notion that logic is concerned with arguments in the abstract.”

Ah, so P is not “every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r” but rather “logic is concerned with arguments in the abstract.”?

I am now thoroughly confused.

[Ed]

Peter:
---
A. God exists and it is not the case that God exists.
B. Ahhh!! Got ya. You just contradicted yourself.
A. Not necessarily. If you take what I just said as having the logical form

(i) P & ~P

then you are right. One the other hand, you may take what I just said to have the form

(ii) P & Q

or perhaps even just

(iii) R.

in both of the last cases of course you are wrong.
---

Again, I don’t understand. Why would a Londonista suppose that in uttering “God exists and it is not the case that God exists”, we would not contradict ourselves. To contradict yourself is to say something of the form ‘P and not-P’.

And why can’t ‘P & Q’ or just ‘R’ be instantiated by a contradiction?

This whole debate gets weirder.

[David Brightly]

Bill says,

[1] Surely one cannot lay bare the form of an argument, in an serious sense of 'argument,' if one abandons the requirement that the same term or sentence be replaced by the same placeholder. [2] To do that is to engage in vicious abstraction. [3] It is vicious because an argument in any serious sense of the term is not just a sequence of isolated propositions, but a sequence of propositions together with the idea that one of them is supposed to follow from the others. [4] An argument in any serious sense of the term is a sequence of propositions that has the property of being putatively such that one of them, the conclusion, follows from the others, the premises.

Let me respond to these in reverse order. Sentences [3] and [4] seem to be raising the question, When is an argument so bad that it ceases to be an argument and becomes a mere sequence of propositions? Fortunately we don't need to answer this. We say any sequence of propositions counts as an argument. Some sequences are valid arguments, the remainder aren't. A miss is as good as a mile. We can however use the theory of argument form to explain common errors human reasoners make. For example, an 'affirming the consequent' error is an instantiation of the invalid argument form obtained by transposition of the final two placeholders in the valid form 'modus ponens'. Thus

MP: P-->Q; P; ergo Q
AC: P-->Q; Q; ergo P

The reasoner may be 'aiming' at MP, but a common feature of human psychology, a transposition of adjacent symbols, brings him to AC.

We plead not guilty to the 'vicious abstraction' of [2] because we never consider the replacement of tokens by placeholders of the indictment [1]. We have not ceased to beat our wives. The theory talks about a process of 'instantiation' whereby an argument is obtained from an argument form by substitutions of tokens for placeholders. That is, the placeholders are replaced with tokens, not the other way about. This is analogous to the production rules of generative grammars. Contrary to [1] we do not claim to be able to lay bare the form of an argument. The theory does not require a unique form associated with each argument. In principle a valid argument could instantiate two distinct valid argument forms, but I suspect this does not happen in practice (Counter-example?) Is the absence of a specific form for every argument problematic? Well, Bill says.

The validity of a given valid argument evidently resides in something distinct from the given argument. What is this distinct something? It is the logical form of the argument, the argument form.

We would have to deny this. I would say that the validity of an argument resides in its correct use of the logical connectives according to their meanings. Thus, substituting any propositions you like for P and Q, same for same, of course, the arguments

P; ergo P or Q
P and Q; ergo P

are valid and

P; ergo P and Q
P or Q; ergo P

are invalid by virtue of the meanings of 'or' and 'and'. The 'any propositions you like' reflects the context-independence of the connectives. Their meaning is not qualified in any way by the elements they join. This is touching on Peter's requirement that forms be 'relativised to the connectives', perhaps.

[David Brightly]

Peter,

I was amused by your skit, but it works only if one accepts that form conveys both structure and content. Saying that A's opening statement has the form 'R' just says it's a single statement; that it has the form 'P & Q' says it makes two assertions; that it has the form 'P & ~P' says that the second assertion denies the first. Are these three claims inconsistent?

[BV]

Excellent, Peter, with the exception of your last paragraph, which raises a separate topic for separate discussion. I don't understand why our London friends don't get the points that you brilliantly illustrate via your little dialogs.

They want to say that 'Tom is tall, ergo, Tom is tall' has the form 'p ergo q' in addition to the form 'p ergo p.' But if they say that, then 'Tom is tall and Tom is not tall' has the form 'p & ~q' in addition to the form 'p & ~p.' Which is crazy: Is the sentence both a contradiction and a contingency?

So I agree with Ed: this is one weird discussion. What are you Londoners trying to say?

[Ed]

>>So I agree with Ed: this is one weird discussion. What are you Londoners trying to say?

We want to say that 'Tom is tall, ergo, Tom is tall' instantiates the form ‘P implies Q’, and we have been insisting on this for a while. We haven’t been saying that it has the form, which looks like something different.

And AFAICS what we are saying is standard textbook logic 101.

What are you trying to say?

>>They want to say that 'Tom is tall, ergo, Tom is tall' has the form 'p ergo q' in addition to the form 'p ergo p.' But if they say that, then 'Tom is tall and Tom is not tall' has the form 'p & ~q' in addition to the form 'p & ~p.' Which is crazy: Is the sentence both a contradiction and a contingency?
<<

We don’t want to say that. It is an instance of that form. It doesn’t have that form, in the sense of having it as some sort of essential nature.

“Is the sentence both a contradiction and a contingency?” No. 'p & ~q' can be instantiated by a contradiction or a contingency, of course, but not something that is both. Just as ‘triangle’ can be instantiated by either an equilateral or a scalene triangle. And you reply “is the triangle both equilateral and scalene? But that’s impossible!”

Spot the fallacy:

(1) a has the form F

(2) a is G

(3) b has the form F

(4) b is not G

(5) Therefore something is G and not G

[David Brightly]

What are we trying to say?

First, there is no such thing as the unique form of a valid argument from which it derives its validity.

Second, a fortiori, we are not trying to lay bare said form by abstraction from the concrete argument.

The theory cannot be coerced into compatibility with these two ideas to which you both continually return. To see what we are saying requires that you abandon these preconceptions.

Third, and this only helps if you know a little generative grammar, our concept of form is very closely related to the sequences of terminals and non-terminals that appear in the derivation of a sentence via the production rules of the grammar, all of which can be seen as 'forms' of the sentence.

[BV]

>>We want to say that 'Tom is tall, ergo, Tom is tall' instantiates the form ‘P implies Q’, and we have been insisting on this for a while. We haven’t been saying that it has the form, which looks like something different.<<

Not so fast! We have been assuming all along that 'A instantiates F' is just a high-falutin' way of saying 'A has F.' To have a form is to be an instance of a form is to instantiate a form.

But of course you are free to make a distinction so long as you explain it. After all, one of the mottos of the philosopher is *Distinguo ergo sum.*

But you didn't explain the distinction. So please tell us the difference between an argument's having a form and instantiating a form.

I have read and taught out of many logic textbooks, but I have never come across one that said that 'Tom is tall ergo Tom is tall' instantiates 'p ergo q.'

I thought my post was very clear. I am denying the Asymmetry Thesis. Did I not fairly define said thesis?

>>We don’t want to say that. It is an instance of that form. It doesn’t have that form, in the sense of having it as some sort of essential nature.<<

I am sorry, but 'Tom is tall and it is not the case that Tom is tall' is not an instance of 'p & ~q.'

If you think that it is, then you are violating the requirement that each token of a sentence-type or a term-type be replaced by a token of the same placeholder-type.

You are suggesting that 'Tom is tall and it is not the case that Tom is tall' is only contingently of the form 'p & ~p.' I am sorry, but that makes no sense.

In our example there is exactly one proposition expressed by exactly two sentence-tokens of the same sentence-type, and exactly two logical connectives. So the form must be: p & ~p.

[David Brightly]

Bill, regarding the definition of 'instantiation' that Ed and I are working with, please see this comment under an earlier post in this series.

[Jan]

Dear Americans,

Why do you find the following objectionable:

1. 'Tom is tall, ergo, Tom is tall' is a contradiction.
2. 'Tom is tall, ergo, Tom is tall' is a contradiction qua being an instance of the form p&~p.
3. 'Tom is tall, ergo, Tom is tall' is an instance of the form p&q and it is not the case it is a contradiction qua being an instance of this form.

while presumably accepting:

1*. Fido is four-legged.
2*. Fido is four-legged qua being an instance of dog.
3*. Fido is an instance of animal an it is not the case it is four-legged by being an instance of animal.

In particular, (1.-3.) do not imply that the statement in question is not a contradiction.

BV writes: "You are suggesting that 'Tom is tall and it is not the case that Tom is tall' is only contingently of the form 'p & ~p.'"

Not so. The statement is necessarily an instance of p&~p (and also of p&q). It is just that, to use Fregean jargon:

a) The concept p&~p is subordinate to the concept of self-contradiction.
b) The concept p&q is not subordinate to the concept of self-contradiction.

[Ed]

>>I have read and taught out of many logic textbooks, but I have never come across one that said that 'Tom is tall ergo Tom is tall' instantiates 'p ergo q.'
<<

OK so now it's clear what you mean (I was applying the charity principle before). Can you give me an example of a textbook which says you can't do this? We all agree that you can't substitute a different token for the same placeholder. 'Any p' means any sentence that 'p' holds a place for. But as far as I know, 'any p' does not mean 'any p except a sentence you have substituted into a different placeholder'.

I sent you a few examples by email of exceptions to your claim. I wonder why you aren't mentioning these?

[David Brightly]

I am sorry, but 'Tom is tall and it is not the case that Tom is tall' is not an instance of 'p & ~q.'

It is according to our notion of 'instantiation'. Replace P with 'Tom is tall' and replace Q with 'Tom is tall'.

If you think that it is, then you are violating the requirement that each token of a sentence-type or a term-type be replaced by a token of the same placeholder-type.

That is a requirement of your theory, Bill, but not one of ours. As we have said before, we move from forms to arguments by replacing placeholders with tokens. We never move in the opposite direction.

[Ed]

Found it – see below. The odd thing is that I emailed you with this example a week ago. So I was applying the charity principle: on the assumption you had read my email, and had appreciated that you can subtitute the same argument in different placeholders, I assumed you weren’t talking about ‘instantiation’ in any ordinary sense.

To be explicit, Shand says you can substitute the same token for different placeholders. I also emailed you the Geach example at the same time. Perhaps you are confusing it with the converse rule?

---
From Fundamentals of Philosophy, edited by John Shand:

"Here is an example of invalid argument form.

If p, then q. Therefore, p.

This looks a lot like the previous argument form, but it not as good: it has many invalid instances. Here is one of them.

If it's a Tuesday, then it's a weekday. It's a weekday. Therefore, it's Tuesday.

This is an invalid argument, because there are plenty of circumstances in which the premises are both true, but the conclusion is not. (Try Wednesday.) We shouldn't conclude that every instance of this form is invalid. An invalid argument form can have valid instances. Here is one:

If it's a Tuesday, then it's a Tuesday. It's a Tuesday. Therefore, it's Tuesday.

The text adds "You might object that the argument doesn't have the form requested. After all, it has the form

If p, then p; p; therefore p

Which is valid, and that is correct. An argument can be an instance of different forms. This argument is an instance of the first form by selecting "its Tuesday" for both p and q [my emphasis], it is an instance of the second form by selecting "its Tuesday" for p”.

[G. Rodrigues]

So, Consider the argument: "Tom is tall, ergo, Tom is tall" and put P = "Tom is tall".

(1) Then it has the form P |- P. Does it instantiate the (invalid) form P |- Q? Of course it does, since the latter is more general and contains the former as a special subcase. So in this sense, the "Londonistas" are correct.

(2) On the other hand, P |- P is the most *specific* argument form that "Tom is tall, ergo, Tom is tall" instantiates. Each argument instantiates only one most specific form (and it is possible to say which algorithmically, assuming a formalized language. In natural language contexts, things may not be so clear cut.) and in this sense B. Vallicella is correct, since if a string (sticking to standard Hilbert deductive calculus) of sentences instantiates a unique most specific form, it follows that the form is either valid or invalid. If it is valid, it follows that an argument does not instantiate an invalid form in the sense that the unique most specific form it instantiates is valid, not invalid.

Does this sound correct?

[Jan]

Correction, "Tom is tall, ergo, Tom is tall" should read "Tom is tall and Tom is not tall" in my (1.-3.). I was thinking about Peter Lupu's "God exists and God does not exist" and copy-pasted the quote without reading it carefully.

[BV]

G. R.,

Thank you for the astute comment. In fact, this is the best comment so far. If the Londonistas were as clear and pithy as you, this discussion would not be as woolly as it is.

If I remember my Frege, the turnstile is used as a sign of assertion. But I take it you are using it to mean 'therefore.' And if I am not mistaken, in the context of the prop. calculus, it means 'syntactic derivability.'

There is a clear sense in which 'P |- P' is a special case of 'P |- Q.' In the former we are moving from a proposition to the same proposition. In the latter we abstract from its being the same proposition and we move from a proposition to a proposition (leaving open whether it is the same or a different proposition). I said something like this myself above somewhere.

What we have done is make a double abstraction.

First step: We abstract the logical form from the argument 'Tom is tall, ergo, Tom is tall.' This yields: P |- P. I am using upper case 'P' as a placeholder. (I hold that a placeholder is not the same as a variable. A placeholder is a constant, albeit an arbitrary constant.)

Obviously, if anyone actually performs this Mickey Mouse inference, and he is not equivocating on any term, then he executes a valid inference which can only have one form, the valid form 'P |- P.'

Second step: We kick it up a notch by abstracting away from the fact that the premise and the conclusion are the same proposition, and preserve only the notion that we are moving from a proposition to a proposition.

So here is what I say. The only form of the actual concrete argument is the first form, the valid one. The second form is not a form of that argument, but a form of the first form.

On this way of looking at things, the Londonistas make the following mistake: they falsely assume that a form F2 of a Form F1 of argument A is a form of A.

So I am tentatively taxing them with the fallacy of assuming that form-instantiation is transitive. But it seems to me that if x is a form of y, and y is a form of z, it does not follow that x is a form of z.

[london ed]

>>So here is what I say. The only form of the actual concrete argument is the first form, the valid one. The second form is not a form of that argument, but a form of the first form.

This is not what Rodrigues says.

[BV]

Of course not. I am disagreeing with him.

The real problem, Ed, is that you apparently have a different conception of logic than I do. I explained above, clearly, what I take logic to be concerned with. If you disagree, tell me why.

[Peter Lupu]

A Question to the Londoners and their pack:

According to you the argument, call it 'Charlie': 'Tom is tall' ergo 'Tom is tall' has two forms. One is the valid one that has the logical form 'P, therefore P'; the other the invalid one which has the form 'P, therefore Q'.

Question: Is Charlie valid or invalid?

If the answer is that Charlie is valid, then the second form is not a form of Charlie.

If the answer is that Charlie is invalid, then I have no clue what you guys are talking about.

If the answer is both, then I want you to answer the following question: how do you build a logic at the level of abstraction that is required in order to yield both forms and at the same time distinguishes between valid and invalid arguments?

In other words: What is the point of the level of abstraction whereby Charlie instantiates both a valid as well as an invalid form when it comes to a system of logic which attempts to give us a way of distinguishing valid from invalid arguments? After all, such a level of abstraction which the Londoners favor obliterates the very distinction that logic attempts to carefully delineate.

[phil]

Wonderful post and truly intriguing argument. I will examine this in greater detail (since I believe it holds great relevance to my own philosohpical project) and give any feedback that I deem worthwhile. Thank you for posting this.

[london ed]

>>The real problem, Ed, is that you apparently have a different conception of logic than I do.

I don't see how 'different conception' is relevant. This is a question of what is agreed fact.For example, you said "have read and taught out of many logic textbooks" implying it is a Logic 101 issue. And you said at 05:54 >>you [i.e. Londonistas] are violating the requirement that each token of a sentence-type or a term-type be replaced by a token of the same placeholder-type". In which elementary textbook is this requirement specified?

I think you should publish the first comment I made today, please. You are replying above to the second. The first is also relevant. I have a copy.

[Ed]

>>If the answer is that Charlie is valid, then the second form is not a form of Charlie.
<<

Incorrect. It is uncontroversial that an invalid form can have valid instances. See this seminal paper by Oliver 1967 or refer to any textbook.

How do you distinguish? If an argument instantiates a valid form, then it is valid, period. However, if it instantiates an invalid form, then it does not necessarily follow that it is invalid. It is invalid if and only if there is no valid form it instantiates. You agree that ‘there is an invalid form it instantiates’ is different from ‘there is no valid form it instantiates’, right?

Or you can look for a counterexample, a case where the premisses are true and the conclusion false.

[G. Rodrigues]

@Bill Valicella:

"So I am tentatively taxing them with the fallacy of assuming that form-instantiation is transitive. But it seems to me that if x is a form of y, and y is a form of z, it does not follow that x is a form of z."

There is a purely syntactical criterion to decide when a concrete argument is an instantiation of a given form, and with this criterion, it follows that if a instantiates F and F is a special form of G then a instantiates G.

note: I am skipping the (somewhat tedious) task of giving precise definitions, but it should be no trouble giving them.

Now, it seems to me that if you want to block this conclusion, the only way to do it is by prohibiting some moves that to my mathematically inclined mind seem arbitrary and ill-motivated. For example, you could try to argue that "'Tom is tall ergo Tom is tall' [does not] instantiate[s] 'p ergo q.'"; but that would entail arbitrary limits on substitution rules for argument forms that no logician that I know of makes, and for good reason as having a deductive calculus with good formal properties is a Good Thing. And for what? In other words, my question, and excuse me if I am jumping right in the middle not cognizant of prior discussions, is why would you want to? As I showed in my first post, at least prima-facie, you can have your cake and eat it too.

note: oh and yes, I was using |- for "therefore", or "ergo".

[BV]

Here is how I would put Peter's point.

Charlie is obviously valid. That is a Moorean fact. Although its verbal representation involves two tokens of the sentence type 'Tom is tall,' the argument itself involves exactly one proposition. Obviously an inference from a proposition to itself is the ne plus ultra of validity. The form of the inference is 'P ergo P.' 'P' stands in for that one proposition while abstracting from its content. Since we kniow that there is exactly one proposition we know that the form diagram cannot have tokens of different placeholder types.

If you say that the inference also has the form 'P ergo Q,' then you are talking nonsense. For again, there is exactly one proposition in Charlie, and the inference proceeds from that proposition to itself. It does not proceed from a proposition to a proposition that may or may not be identical to the first proposition. For we know that there is exactly one proposition in Charlie, and that the inference proceeds from this proposition to itself.

This ends the discussion of Charlie, as far as I am concerned. The other examples are different.

But I can also explain where some of you are going wrong. If you take the form 'P ergo P,' then you can see that as a special case of 'P ergo Q.' I don't object to that. But just understand what you are doing. You have made a further abstraction. The second form abstracts not only from the content of the proposition but also from the fact that one and the same proposition is both premise and conclusion.

I have already shown that the second form is not the form of Charlie. It is merely a form of the first form. So the mistake some of you are making is to think that if A has Form 1 and Form 1 has Form 2, that A has Form 2.

But even if my diagnosis of the mistake is incorrect, it remains the case that Charlie has exactly one form, the valid form.

[london ed]

As I keep saying liking a broken record, my simple question here is whether the Phoenicians are (1) claiming that standard textbook logic is incorrect or (2) that it is correct, but that the Londoners are making some elementary 101 mistake.

When I look back at the thread, all the indications are that it is (2). For example, when I said I had a textbook saying that "Socrates is wise implies Socrates is wise, Socrates is wise, therefore Socrates is wise" instantiates the invalid form "A implies B, B, therefore A", Bill replies on Sunday, April 27, 2014 at 06:03 PM "If you have the textbook, then presumably you can tell me its name, author, etc”. I.e. he questions whether there is such a textbook, which to me implies he thinks that the Phoenicians have the textbook answer, and the Londoners are ‘wrong’. (I was actually referring to Logic: The Laws of Truth By Nicholas Jeremy Josef Smith, who says that any argument which has the valid form "P, P implies P, therefore P” also has the invalid form "A implies B, and A”).

On Monday, April 28, 2014 at 05:36 AM Peter Lupu mentions the ‘rule’ R2 "(R2) Each token sentence shall be represented by a distinct sentence-variable.” In fact there is no such rule but he clearly implies there is one.

On Monday, April 28, 2014 at 06:49 AM Bill says "It is clear that in mathematics different variables can have the same value. But a placeholder is not a variable. A placeholder is an arbitrary individual constant”. Again, he clearly implies that this is not just his opinion, but established science.

On Monday, April 28, 2014 at 11:19 AM Bill says "If a sentence or a term occurs more than once in an argument, then that sentence or term must [my emph] have the same placeholder in the form schema”. Again, Logic 101 is implied.

So the Phoenicians are clearly saying that their position is supported by elementary logic texts, and that (by implication) the Londoners are not just wrong, but wrong in an embarrassing and elementary way. Do the Phoenicians want to modify their position? E.g. if they now concede that their position is more speculative, and involves a ‘different conception of logic’ from elementary textbooks, that’s fine. But you shouldn’t be implying we are making elementary mistakes. The honour of London is at stake.

>> But even if my diagnosis of the mistake is incorrect, it remains the case that Charlie has exactly one form, the valid form.

On any standard version of logic, what you are saying is completely wrong. Perhaps it is ‘right’ in some other sense (textbooks and standard theory have been wrong before, Ptolemy was wrong). But we need to be clear about the status of your claim, as speculative.

[Peter Lupu]

Ed the Londoner:

I want to know one thing. How do you determine validity if you allow that the Charlie sentence has two forms one 'P, ergo P' and the other 'P, ergo Q'. i.e., giving you the rule whereby any token sentence can be replaced by any placeholder, how would you determine the validity of any concrete argument?

[David Brightly]

London is ever so slightly getting the impression that Phoenix, if she is paying any attention at all to what is coming from across the pond, has preconceptions that are preventing her from seeing what London is saying. Let's try again, but this time using different language.

1. Let's start by immediately saying that this theory is purely syntactical. You may baulk at this thinking the theory must therefore be a non-starter. We ask you to bear with us. Our claim is that the theory gets the right results at the level of words and sentences.

2. A grook is a sequence of sentences.

3. A grook pattern is a sequence of words, punctuation, and labelled, typed, placeholders. Placeholders are of three types: name holders, concept holders, and sentence holders.

4. A grook G derives from a pattern P iff G can be obtained from P by replacing each placeholder in P with an appropriate word or sentence. A name holder must be replaced with a proper name, a concept holder with the name of a concept, and a sentence holder with an entire sentence. Placeholders bearing the same label must be replaced with the same text.

Some examples of patterns:

[P:sentence]; ergo [P] or [Q:sentence].
[a:name] is [F:concept]; [a] is [G:concept]; ergo some [F] is [G].

5. There is a distinguished set of patterns known as the pretty patterns. Both examples above are pretty. A pattern is ugly iff it is not pretty.

6. A grook is good iff it derives from a pretty pattern, else it is bad. A good grook may derive from an ugly pattern.

The London claim is that, under a suitably chosen set of pretty patterns, the good grooks express exactly the valid arguments. That is all.

[G. Rodrigues]

@Peter Lupu:

"I want to know one thing. How do you determine validity if you allow that the Charlie sentence has two forms one 'P, ergo P' and the other 'P, ergo Q'. i.e., giving you the rule whereby any token sentence can be replaced by any placeholder, how would you determine the validity of any concrete argument?"

london ed will surely answer but I would like to say a couple of things.

(1) All london ed needs is that a instantiates the form F can be checked; and it very well can be. One could then do the following (there may well be more ways, I am just listing two):

(1a) The one making the argument specifies which form he is invoking. Since it can be checked whether a instantiates the form F or not, the validity of the argument can be checked.

(1aa) As regards your Pythonesque skit, if anyone happened to argue in that way he would be a douche. One does not argue with douches, one metaphorically punches them in the face.

(1b) Augment the language with a |- symbol say, and lay down the recursive rules to form well-formed-arguments (wfa's) and then specify, among all the possible wfa's, the valid wfa's. Since a wfa has only one most specific form, once again, validity of the argument can be checked.

(2) There is an asymmetry between P |- P and P |- Q. It is a valid syntactical transformation to go from P |- Q to P |- P but the converse is *not*; you can replace possibly unequals for equals but not equals for possibly unequals.

(2a) If your deductive calculus is set up correctly there are no valid forms F and G such that F is a special form of G. The reasons should be obvious.

(3) In regards to (2), the idea behind it is to view substitution as a special case of (function) composition. So now you are in good shape to prove interesting things (if you are a mathematician) like the Curry-Howard-Lambek correspondence.

note to Bill: I am reposting the comment, since by mistake I included my previous comment to you in it. Could you please delete it (you can also delete this note).

[london ed]

@Peter: “I want to know one thing. How do you determine validity if you allow that the Charlie sentence has two forms one 'P, ergo P' and the other 'P, ergo Q'. i.e., giving you the rule whereby any token sentence can be replaced by any placeholder, how would you determine the validity of any concrete argument?”

As mentioned before, the concrete argument is valid iff it instantiates a valid form (which the Tom is tall argument does). So you determine validity by seeing whether there is such a valid form.

Demonstrating invalidity is more difficult, since the fact it instantiates some invalid form does not entail that there is no valid form it instantiates. The only way is to demonstrate the invalidity of a concrete argument is to demonstrate a case where the premisses are true, the conclusion false.