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Friday, April 25, 2014

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D3. A particular argument A is valid =df A instantiates a valid form. (This allows for the few cases in which an argument has two forms, one valid and one invalid.)

D4. A particular argument A is invalid =df there is no valid form that it instantiates. Equivalently: there is an invalid form that it instantiates.
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David B might already have pointed out that

(i) 'there is no valid form that it instantiates' is not the same as 'there is an invalid form that it instantiates'. For it can be true both that there is some valid form it instantiates, and some non-valid form that it instantiates.

(ii) You have already pointed this out when you say "This allows for the few cases in which an argument has two forms, one valid and one invalid".

I have a separate post on this: http://maverickphilosopher.typepad.com/maverick_philosopher/2009/11/validity-invalidity-and-logical-form.html

>>I have a separate post on this

Right, and I remember that you did. But you are wrong to say in this post that "Equivalently: there is an invalid form that it instantiates". "There is no valid form that it instantiates" is not the same as "there is an invalid form that it instantiates", and this is elementary.

You are also criticising David for saying something which is perfectly correct. David says

(a) The argument "Socrates is a man; all men are mortal; ergo Socrates is mortal" is valid. Correct.

(b) It also instantiates the invalid form "a is F; all Hs are G; ergo a is G". Correct.

You object that it does not instantiate the invalid form. But it does. I have a textbook that states 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". Again, this is elementary.

Thanks, Ed. I too remembered the old post but couldn't locate it in reasonable time.

Bill, in the 2009 post you say, correctly I think, that 'every valid syllogism has the invalid form p, q, therefore r'. In pattern talk we would say that every syllogism matches the pattern p, q, r or just a pattern of three propositions. I don't think pattern talk commits us to physical patterns, ie, patterns in physical stuff. We might equally say that the first ten natural numbers bear a pattern of alternating evens and odds. Is there another way of explaining what we mean by 'instantiating an argument form'?

We need to agree on the issue of validity of form, I think, before we can discuss arguments involving equivocation.

OK, now I get it. You are objecting to the equivalence. Only one half of the equivalence is true, namely,

a. If an argument has no valid form, then it has an invalid form;

but the following is false

b. If an argument has an invalid form, then it has no valid form.

But the repair is easy: I simply delete the second sentence in (D4, which I will now do.

>>I have a textbook that states 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". Again, this is elementary.<<

If you have the textbook, then presumably you can tell me its name, author, etc.

If S. is wise, then S. is wise
S is wise
Ergo
S. is wise.

I am sorry, but what you are saying strikes me as crazy. Obviously, the argument involves exactly one proposition. Therefore, you cannot use two different placeholders to symbolize it.

The form is

If A, then A
A
Ergo
A

and that is obviously valid.

Will you also say that the argument has the form:

If A, then B
C
Therefore
D?

(I fear you will answer in the affirmative.)

See Logic: The Laws of Truth By Nicholas Jeremy Josef Smith. “It is not true in general that every instance of an invalid argument form is an invalid argument”. They give the example

B
A implies B
:. A

which is obviously invalid. “But this does not mean that every set of three propositions of the forms B, A implies B, and A respectively, are such that the first two can be true while the third is false”. And they give the example of any argument which has the form “P, P implies P, therefore P”. For example, the Socrates example I gave earlier.

Or try A Concise Introduction to Logic, by Patrick Hurley (who I have corresponded with in the past).

“Now, since the form is invalid, can we say that any argument that has this form is invalid? Unfortunately, the situation with invalid forms is not quite as simple as it is with valid forms. Every substitution instance of a valid form is a valid argument, but it is not the case the 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.”

>> what you are saying strikes me as crazy. Obviously, the argument involves exactly one proposition. Therefore, you cannot use two different placeholders to symbolize it.
<<

Obviously if the placeholders are the same, one cannot use different concrete terms to instantiate them. But it is not the same the other way round, otherwise all mathematics would be impossible. Often in mathematics we have two different variables x and y, and we may want to prove that x = y, for example 6 = 6, or whatever. The same token can instantiate different variables or placeholders.

What is more puzzling, Bill, is that you have alluded to this very fact in the 2009 post.

"Similarly, every valid syllogism has the invalid form p, q, therefore r. "

Correct. Every valid syllogism instantiates an invalid form.

DB "We need to agree on the issue of validity of form, I think, before we can discuss arguments involving equivocation."

Agree.

Consider the following two rules:

(R1) Each token sentence shall be represented by the SAME sentence-variable (regardless of the number of times it is represented in the complex sentence or its position).

(R2) Each token sentence shall be represented by a distinct sentence-variable.

The Socrates sentence is mapped into Bill's valid argument above if R1 is applied. It will be mapped into the invalid second argument, if R2 is applied. The trouble is that R2 cannot be the basis of a clear definition of validity and, therefore, cannot capture the distinction between valid vs. invalid arguments. I cannot see a logic based on R2.

Bill, your objection is discussed in this excellent presentation here.

“ The appeal to Logical Form also attempts to block valid arguments from being instances of an invalid form by placing a restriction on what counts as a legitimate substitution. The only allowable substitutions are into argument forms which exemplify the (true, unique?) logical form of the resulting argument. For example, the argument: “If cats like cream then cats like cream, Cats like cream, Therefore, cats like cream” can arise from substitution into both A ⊃ B, B ∴ A and A ⊃ A, A ∴ A but only the latter, it is claimed, exemplifies the argument’s logical form.”

It's an instance of a general problem known as the 'asymmetry problem'. We can test for a valid argument if it has a valid form. But we can't test for an invalid argument in the same way. It may have an invalid form, but may still be valid.

Ed says that

B
A implies B
:. A

is obviously invalid. I agree. It is the formal fallacy called Affirming the Consequent.

But why can't we play the same game with argument forms that Ed plays with arguments?

Why can't we say that the above form instantiates the valid form

A
A implies A
Ergo
A?

If
Socrates is wise
ergo
Socrates is wise

has the form
A
ergo
B

then the latter has the form
A
ergo
A?

Ed,

I used to teach out of Hurley. Small world. He's a Copi clone.

In any case the quotation from him doesn't tell against me. I agree with what he says based on the following example:

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

That instantiates Affirming the Consequent. But it also instantiates a more specific form that is valid.

This is quite unlike the chicanery of replacing 'Cats like cream' with different placeholders.


Ed sez: >>Obviously if the placeholders are the same, one cannot use different concrete terms to instantiate them. But it is not the same the other way round, otherwise all mathematics would be impossible. Often in mathematics we have two different variables x and y, and we may want to prove that x = y, for example 6 = 6, or whatever. The same token can instantiate different variables or placeholders.<<

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.

Naturally I am completely confused. The post began by quoting Brightly, who said "We can't say that an argument is invalid because it instantiates an invalid form. " This is correct. Since then I looked at the literature and this seems agreed on all sides. We can say that an argument is invalid if there is no valid form that it instantiates. But the mere fact it instantiates an invalid form does not mean there is no valid form it instantiates. Next, Brightly claimed that "Socrates is a man; all men are mortal; ergo Socrates is mortal instantiates the invalid form a is F; all Hs are G; ergo a is G". Is this wrong or not? The textbook examples I gave, and Cheyne's presentation, all seem agreed that David's example is correct. Moreover Bill, in a 2009 post said ""Similarly, every valid syllogism has the invalid form p, q, therefore r. ""

Peter: "I cannot see a logic based on R2. " Peter, are the textbook examples correct or not? Cheyne says that the argument: “If cats like cream then cats like cream, Cats like cream, Therefore, cats like cream” can arise from substitution into both A ⊃ B, B ∴ A and A ⊃ A, A ∴ A. Is Cheyne right or wrong? This is pretty binary.

We all agree on what I said in 2009, namely, "every valid syllogism has the invalid form p, q, therefore r." The Socrates argument illustrates this phenomenon.

The question is whether the Socrates syllogism has or instantiates the invalid form a is F; all Hs are Gs; ergo a is G". This is what I have been denying.

As I see it the two cases are different. Both argument forms are of course invalid. The difference, I claim, is that the Socrates syllogism has the first form but does not have the second. Why not? Well, the first example does not violate (something like) Lupu's R1. But the second does.

If a sentence or a term occurs more than once in an argument, then that sentence or term must have the same placeholder in the form schema.

Why? Well, you are given an argument and you want to know whether the reasoning it embodies is correct. That is, you want to know whether the argument is valid. To determine this, you must first determine the form of the argument since validity is a question of form. To get the form, you replace the non-logical expressions with placeholders. Obviously, you must use the same placeholder for the same non-logical expression! Does this need explaining? I don't think so. An example:

Maximus is a cat
Maximus is a gladiator
Therefore
Some cat is a gladiator

Here is where we start, with an actual, particular argument, not with a form diagram. Why? Because logic is concerned with the evaluation of actual arguments. Now we replace the non-logical expressions with placeholders and we get:

a is an F
a is a G
Therefore
Some F is a G.

This is obviously the most specific form of the original argument. Now we ask whether the form is valid. It is if it impossible to substitute terms for the placeholders -- the same term for the same placeholder obviously -- in such a way that the premises are both true and the conclusion false. And that is impossible.

Therefore the form is valid.

Therefore the original argument is valid.

The fact that the Maximus argument also has the invalid form p, q, therefore r is irrelevant since this propositional form does not capture the specific reasoning in the original argument.

Ah right so we are in fact arguing about one case rather than two. I now agree you are not being inconsistent in your post here and in your post in 2009.

So the question is whether "If a sentence or a term occurs more than once in an argument, then that sentence or term must have the same placeholder in the form schema."

The textbook and the paper (by Cheyne) I quoted suggest that this does not have to be the case. Cheyne says

“ The appeal to Logical Form also attempts to block valid arguments from being instances of an invalid form by placing a restriction on what counts as a legitimate substitution. The only allowable substitutions are into argument forms which exemplify the (true, unique?) logical form of the resulting argument. For example, the argument: “If cats like cream then cats like cream, Cats like cream, Therefore, cats like cream” can arise from substitution into both A ⊃ B, B ∴ A and A ⊃ A, A ∴ A but only the latter, it is claimed, exemplifies the argument’s logical form.”

This seems to be exactly what you and Peter are arguing here, Bill. The question is whether this is simply wrong on Logic 101 (as you seem to be implying) or whether it is wrong but not for elementary reasons, or whether he is right. His page is here. “Colin studied mathematics at Otago and then taught high school mathematics for some years. He returned to Otago and completed a series of degrees in philosophy before joining the department in 1992. His research is mostly in epistemology and the philosophy of mathematics. He is the author of Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism (Kluwer, 2001).”

Cheyne has written an awful lot about the ‘Asymmetry Problem’.

Bill,

I think the account you give at 11:19 is circular. You say that to show an argument is valid we have first to derive its form and then show that no substitution instance of that form is such that its premises could be true and its conclusion false. But the original argument is a substitution instance of the form and showing that its premises can't be true and conclusion false is just to show it's valid. But that is what we set out to do in the first place. The detour through forms has got us no further forward.

There are several cases to consider.

(a) Cases restricted to propositional logic. Here Logical Form is a structure where (i) the set of logical constants are fixed either by enumeration or defined from prior logical constants that are given; (ii) The set of propositional variables are given; (iii) Logical form is defined over the set of logical constants such that all logical constants that occur in sentences are clearly displayed (i.e., none are absorbed within propositional variables); (iv) All token sentences of the same sentence type within a given argument receive the same propositional variable.

It follows that sentences of the form "If it is raining, then it is wet" cannot be represented as a logically simple formula; it must be represented instead as a complex having the structure "A -> B". by (iii). Similarly by (iv) arguments such as "If Socrates is wise, then Socrates is wise; Socrates is wise; therefore, Socrates is wise" cannot be represented any other way but "A -> A; A; therefore, A". Any other proposal violates restriction (iv).

Examples that violate (iii) and (iv) that were proposed above were completely unmotivated and result in not having any cogent system of logic. I do not see what is the point of considering them seriously.

(b) Quantification cases: too complicated to outline here. What is clear is that David's example will be ruled out by a similar clause regarding predicate variables (The predicate 'man' and 'men' will be both represented by a single predicate variable. David's example capitalizes on grammatical differences that do not reflect logical distinctions. The sentences can be rephrased to eliminate all such problems.

Now consider the following argument:

1. John is tall
Therefore,
2. Someone is tall.

This argument can be represented in propositional logic as the invalid:

1*. P
Therefore,
2*. Q

and in quantification theory as the valid:

1**. Fa
Therefore,
2**. (Ex) Fx.

Which one is it? It depends. If the intention is to display quantification structure, then the argument must be represented in the second form and therefore it is valid. If the intention is not to represent quantification structure, then the argument is not valid, since it is represented in the first form. Thus, there is a sense in which validity must be relativized to a given system. However, since the system is typically understood, there is no need to state it explicitly.


David,

I don't see the circularity. Let's avoid the modal words you use. They only muddy the waters.

An argument form is valid iff no argument having that form has true premises and a false conclusion.

We are given an argument whose validity/invalidity is in question. We extract the form -- which is a one over many -- and test it for validity using a Venn diagram or by trying to find a counterexample (an argument of that form with true premises and a false conclusion). If the form is valid, then the original argument is valid.

It is true that the original argument is an instance of the form, but we are not arguing that the form is valid because it is the form of the original argument which is valid. That would be circular. We shift from the argument, which is a particular, to its form, which is a universal, and then test that universal form for validity.

1. John is tall
Therefore,
2. Someone is tall.

It makes some sense to say that this argument has two forms, an invalid one from the propositional calculus, and a valid one from the predicate calculus.

But I think it makes more sense to say that the argument does not have the form:

1*. P
Therefore,
2*. Q

My reason is that this invalid form does not capture the reasoning embodied in the argument. Obviously, when one concludes that someone is tall from the fact that John is tall, one is not moving from one proposition to a wholly different one.

My talk of reasoning gives no aid and comfort to psychologism. I am not saying that logic studies reasoning as a psychological process. What it studies is abstract patterns of reasoning that are what they are whether or not there are any reasoning beings, any discursive intellects.

Cats like cream
Therefore
It is not the case that cats do not like cream.

Now wouldn't it be absurd to say that this has the form

p
Therefore
~~q

??

But why is it any less absurd when Cheyne says that

If cats like cream, then cats like cream
Cats like cream
Therefore
Cats like cream

has the form

If p, then q
q
Therefore
p

??

What is to stop him from saying that it has the form

If p, then q
r,
therefore
s

??

Bill, I emailed you some extracts from apparently authorative texts which contradict what you say.

I am quoting second hand, but apparently in Thomas Aquinas: God and Explanations, pp. 161-2) Geach claims that we represent any valid argument as instantiating at least one invalid form. For there is nothing to stop us linking the premisses of any argument together with "ands" or other connectives, and representing the long sentence thus formed by the letter "p". Representing the conclusion of the argument by "q", we are thus able to represent any argument as a whole [my emph] as instantiating the form "p, therefore q", which is about as invalid an argument form as you could get. I.e.

(1) Geach’s claim is about any valid argument
(2) The argument ‘Socrates is wise therefore Socrates is wise’ is a valid argument
(3) Therefore Geach’s claim, if true, is true of the ‘Socrates’ argument above.
(4) Geach’s claim is that we can represent this argument by the form ‘p, therefore q’, which is about as invalid a form as you can get.

Can we accept the authority of Geach or not?

>>My reason is that this invalid form does not capture the reasoning embodied in the argument.
It may not capture it, but the question is whether the argument is an instance of the invalid form.

@Peter – so are you saying that

P → (P → P)
is not an instance of
A → (B → A) ?

@Peter – so are you saying that
P → (P → P)
is not an instance of
A → (B → A) ?

And generally, can we say that A → (B → A) is true of any proposition substituted for A and any proposition substituted for B? Or shouldn’t we rather say any proposition substituted for A except the one substituted for B, and conversely?

In which case
P, P → P, therefore P
is not an instance of modus ponens?

Bill,

Regarding circularity, let me explain what I think is going on. The appeal to argument form is an appeal to an axiom schema. As such it needs no further justification or proof. Consider this form/schema:

P
-----
P or Q
We can't doubt the validity of inferences of this form because the form captures the meaning of the logical constant 'or'. No matter how complicated P and Q might be, provided they are properly formed sentences/propositions, the inference holds. Likewise,
a is F
a is G
-----
some F is G
captures an aspect of how proper names (when used unambiguously---but that is a caveat applied to all replacement tokens for placeholders everywhere) interact with the logical constant 'some' and with predicates.

This tells us, I think, that we don't have to cudgel our minds into understanding complicated arguments in order to convince ourselves of their validity. We can simply follow a mechanical process of attempting to match up bits of argument with the patterns found in the finite catalogue of axiom schemas.

Gentlemen:

There are a number of criss-crossing issues here. One issue concerns the very nature of logic. I should make a position statement about that.

For now, I'll refer you to Gerald J. Massey, "The Fallacy behind Fallacies," *Midwest Studies in Philosophy* VI (1981), 489-500. I just remembered this old article which I studied carefully (as my annotations attest) back in '81. And now I realize that one of the examples I give of an argument that has an invalid and a valid form is from Massey!

I'll also toot my own horn once again by reminding you of a 2010 discussion of the asymmetry thesis.

http://maverickphilosopher.typepad.com/maverick_philosopher/2010/12/validity-invalidity-and-contravalidity.html

>>There are a number of criss-crossing issues here.

Notwithstanding, the main issue is your (and Peter's) central claim, against Brightly that the argument Socrates is a man; all men are mortal; ergo Socrates is mortal does not instantiate the invalid form a is F; all Hs are G; ergo a is G.

David and I agree it does instantiate that form. We need to agree on the particulars, before we move to the universals, no?

If Brightly is right, then
Tom is tall
ergo
Tom is tall

instantiates the invalid form

p
ergo
q.

I say that this is a reductio ad absurdum of his view.

As I have said many times, a sentence is not a proposition, and arguments are composed of propositions. The fact that there are two sentence tokens above does not justify using two different placeholders. Only a difference in propositions justifies a difference in placeholders.

You guys are also not noticing the difference between the above example and this one:

Tom is tall
ergo
Someone is tall.

This has two form, one valid the other invalid. The first example has only one form, a valid one.

You are confusing these two claims:

A. Some valid arguments have an invalid form. (True)

B. All valid arguments have an invalid form. (False)

> >>My reason is that this invalid form does not capture the reasoning embodied in the argument.
It may not capture it, but the question is whether the argument is an instance of the invalid form.<

You can't divorce an argument form from a pattern of reasoning. An argument form just is a pattern of reasoning. Now no one who reasons from 'Tom is tall' to 'Someone is tall' reasons according to the pattern *p ergo q.* So a case can be made that the argument does not instantiate this invalid form. For it is not the pattern of reason anyone would follow who inferred the conclusion from the premise.

Bill,

Just to clarify, Ed and I are operating with the following notion of form instantiation. We say that argument A instantiates form F iff A results from substituting tokens for the placeholders in F, subject to the requirement that like placeholders are replaced by like tokens. Thus, starting with form 'P, ergo Q' and replacing P with 'Tom is tall' and replacing Q with 'Tom is tall' we arrive at the argument 'Tom is tall, ergo Tom is tall'. Hence we say that the latter instantiates the former. This is the only notion of 'instantiating a form' that we need to define validity and invalidity of form and argument. We are careful not to use the locution 'argument A has the form F'. Where we do, we mean just 'argument A instantiates form F', as I have defined it above.

Although like placeholders must be replaced with like tokens, there is no requirement that distinct placeholders be replaced with distinct tokens. More asymmetry!

>>If Brightly is right, then "Tom is tall, ergo, Tom is tall" instantiates the invalid form "p ergo q"

Precisely. Geach apparently also says the same, and numerous textbooks agree. The placeholder 'p' means 'any sentence whatever', not 'any sentence except the one substituted into any other placeholder such as q'.

>>I say that this is a reductio ad absurdum of his view.

Clearly not. I see what you want to say, namely that there is a stronger sense of 'instantiate' which would rule out the examples you dislike. But it's not clear how much logic you would have to rule out in order to do this. For example, if you want to prove "P → (P → P)", you show that it is an instance of "A → (B → A)". Under your improved rule, you would not be allowed to substitute the same sentence for a different placeholder. Or are you suggesting there are exceptions to your improved rule, where sometimes you are allowed to substitute the same sentence, sometimes not? But then you need a rule do deal with the exceptions.

>>B. All valid arguments have an invalid form. (False)
(True)

I have located a copy of the Geach paper and will consult when I have time.

>>no one who reasons from 'Tom is tall' to 'Someone is tall' reasons according to the pattern *p ergo q.* So a case can be made that the argument does not instantiate this invalid form. For it is not the pattern of reason anyone would follow who inferred the conclusion from the premise.

If it were the pattern of reasoning anyone would follow who inferred the conclusion from the premise, then it wouldn't be an invalid form, would it?

What are you trying to establish here?

Bill,
Regarding Validity, Invalidity, and Contravalidity, Sanford says, via Google Books,

Jumping ahead a couple of centuries to the early sixteenth century. we find an instructive tripartite classification in Johannes de Celaya. Instead of the more usual two-way division of consequentiae into valid and invalid, Celaya considered a symmetrical three-way division: valid, contravalid, and neither. This subdivides invalid argument forms into contravalid argument forms and those that are neither valid nor contravalid.
Celaya's division is thus,
Valid: None of its instances have true premises and a false conclusion.
Neither invalid nor contravalid: Some of its instances have true premises and a false conclusion, and some do not.
Contravalid: All of its instances have true premises and a false conclusion.
This is consistent with what Sanford goes on to say. The summary you give wrongly labels the middle division as the invalid forms.

I can't get the label right either! Neither valid nor contravalid.

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