Invalid form of an argument (Propositional Logic)

[lex]

JeffM
Hopefully my post to Zermelo adds a little more clarity.
As for your own situation, I'll note that the OP was not doubting the validity of modus ponens, which seems to be what you have addressed. Nor indeed was the OP doubting the invalidity of the argument:
[MATH]\mathscr{A} \rightarrow \mathscr{B}\\ \mathscr{B}\\ \therefore \mathscr{A}[/MATH]because of the problematical assignment of True to [MATH] \mathscr{B}[/MATH] and False to [MATH]\mathscr{A}[/MATH] making the 2 premisses True and the conclusion False.

Nor was the OP doubting that "Today is Saturday" [MATH]\leftrightarrow[/MATH] "Tomorrow is Sunday", but for that very fact they were proposing that the particular argument of the above form:
"Today is Saturday" [MATH]\rightarrow[/MATH] "Tomorrow is Sunday"
"Tomorrow is Sunday"
[MATH]\therefore[/MATH] "Today is Saturday"
somehow (in contradiction to what they themselves had just stated) appears to be 'valid' since the problematical truth assignments do not apply to these particular statements.
(True to "Tomorrow is Sunday", False to "Today is Saturday").
The OP saw this as a problem which they wanted resolved. That was the issue in the post.

The resolution is simply to know the definition of the validity of an argument and apply it.
The argument cannot be considered as valid.

(The 'contradiction' disappears when we consider that in their preamble to presenting their argument, they were using the following valid form of argument:
"Today is Saturday" [MATH]\leftrightarrow[/MATH] "Tomorrow is Sunday"
"Tomorrow is Sunday"
[MATH]\therefore[/MATH] "Today is Saturday").

 

[lex]

@Afive
If you mean is the following a tautology, the answer is yes:
[MATH]((\mathscr{A}\rightarrow \mathscr{B}) \land \mathscr{B} \land (\mathscr{A} \leftrightarrow True) \land (\mathscr{B} \leftrightarrow True)) \rightarrow \mathscr{A}[/MATH]
However:
[MATH]((\mathscr{A}\rightarrow \mathscr{B}) \land \mathscr{B}) \rightarrow \mathscr{A}[/MATH]is not a tautology.

 

[JeffM]

@lex I have decided not to ask Subhotosh Khan to delete my post. Some poor soul may come along and be totally confused by this thread.

You are correct that I did not follow what the OP was getting at. He was making the “fundamental” argument that

If an argument is logically valid, its conclusion may be false in the real world, and
If an argument is logically invalid, its conclusion may be true in the real word,
Therefore, (and I now quote the OP’s post #6) “knowing whether an argument is valid or not is useless.”

How do we address that argument about arguments?

If an argument is logically valid and its premises are true in the real world, you know that the conclusion is certainly true in the real world. Knowledge.

If an argument is logically invalid, you do not know whether the conclusion is true or false in the real world, regardless of the truth or falsity of the premises. Ignorance.

To put it a different way, if an argument is valid, all you need to do to determine whether its conclusion is true is to determine the truth of iits premises, a process that is often quicker, cheaper, or safer than determining the truth of the conclusion itself

Will a bridge designed in this way collapse in a high wind when bearing any load that does not exceed the maximum specified load? There are two options. You could build such a bridge and see what happens. Practical, real-world people, however, choose the other option of testing the validity of the design.

 

[lex]

To put it a different way, if an argument is valid, all you need to do to determine whether its conclusion is true is to determine the truth of iits premises, a process that is often quicker, cheaper, or safer than determining the truth of the conclusion itself

Not quite.

[MATH]A \rightarrow B\\ A\\ \therefore B[/MATH]is a valid argument.
If one of the premisses is False, we can't say whether its conclusion is true.

If 3 is an even number then 2 is a prime number
3 is an even number
Therefore 2 is a prime number

If 3 is an even number then 4 is an odd number
3 is an even number
Therefore 4 is an odd number

 

[JeffM]

Not quite.

[MATH]A \rightarrow B\\ A\\ \therefore B[/MATH]is a valid argument.
If one of the premisses is False, we can't say whether its conclusion is true.

If 3 is an even number then 2 is a prime number
3 is an even number
Therefore 2 is a prime number

If 3 is an even number then 4 is an odd number
3 is an even number
Therefore 4 is an odd number

@lex

Yes, my language was inexact in one sense, but I think you missed what I was trying to say.

I did not say or imply , “If an argument is logically valid and at least one of its premises is false, you know its conclusion is false.”

I said "If an argument is logically valid and its premises are true ..., you know that the conclusion is certainly true."

I doubt you disagree with that statement.

I also said, "If an argument is logically invalid, you do not know whether the conclusion is true or false..."

I doubt you disagree with that statement.

Nor did I say “If an argument is valid and at least one of its premises is false, you do not know whether the conclusion is true or false.“ Obviously I agree with that. I did not, however, say anything about logically valid arguments and false premises because the argument I was rebutting is about the purported "uselessness" of assessing validity. That argument does not purport to say that determining truth or falsity is useless. So in that context false premises are not relevant.

Where I think I was unclear was in a hidden assumption I was making in the paragraph that you quoted. I was no longer trying to make an argument in pure logic but about real world practice. In the huge majority of cases, we can determine whether a statement is true or false directly. We do not rely on logic at all. We rely on logic when directly determining the truth of a statement is time consuming, difficult, or dangerous. We often have two options: direct determination or indirect determination. So what I should have said is something like “All you have to do to determine whether a conclusion is true is to determine it directly or to determine that an argument reaching that conclusion is both logically valid and has true premises.” I was assuming that direct determination was being excluded as impractical as in my bridge example.

So to restate exactly what I was trying to say

If we are relying on indirect determination of the truth of a conclusion in the real world, we know that the conclusion is true if it is supported by an argument that is logically valid and relies on premises that are all true, but we do not know whether the conclusion is true or false if the argument is logically invalid or relies on premises that are not all true. Knowledge versus ignorance. And to obtain knowledge indirectly, the conjunction of validity of argument and truth of premises is sufficient. And that is the usefulness of a valid argument, which arises in those cases where direct examination of the truthfulness of a conclusion is impractical.

 

[lex]

@JeffM
The statement I disagree with is the one I quoted:
"if an argument is valid, all you need to do to determine whether its conclusion is true is to determine the truth of its premises".
I think you have clarified what you meant to say and what you believe!

 

[Afive]

1.
[MATH]\mathscr{A}\rightarrow \mathscr{B}\\ \mathscr{B}\\ \therefore \mathscr{A}[/MATH]is not a valid argument

since the form:
[MATH]p\rightarrow q\\ q\\ \therefore p[/MATH]is not valid

2.
[MATH](\mathscr{A}\rightarrow \mathscr{A})\rightarrow (\mathscr{B}\rightarrow \mathscr{B})\\ (\mathscr{B}\rightarrow \mathscr{B})\\ \therefore (\mathscr{A}\rightarrow \mathscr{A})[/MATH]has a different form from 1.
and the argument is valid

since the form:
[MATH](p\rightarrow p)\rightarrow (q\rightarrow q)\\ (q\rightarrow q)\\ \therefore (p\rightarrow p)[/MATH]is valid


equivalently
1.
[MATH](\mathscr{A}\rightarrow \mathscr{B})\land \mathscr{B})\rightarrow \mathscr{A}[/MATH] is not a tautology
since the form:
[MATH]((p\rightarrow q) \land q) \rightarrow p[/MATH] is not a tautology

2.
[MATH](((\mathscr{A}\rightarrow \mathscr{A})\rightarrow (\mathscr{B}\rightarrow \mathscr{B})) \land (\mathscr{B}\rightarrow \mathscr{B})) \rightarrow (\mathscr{A}\rightarrow \mathscr{A})[/MATH] is a tautology
since the form:
[MATH](((p\rightarrow p)\rightarrow (q\rightarrow q)) \land (q\rightarrow q)) \rightarrow (p\rightarrow p)[/MATH] is a tautology

Thank you for the effort to have formalized the passage from an invalid argument to a valid one
by replacing the atomic formulas in the argument with tautologies.
My point, however, was to highlight that, in addition to the obvious change from invalid to valid,
also to make it clear that when you want to build
an argument based on tautologies no longer you have the flow of information that makes you conclude that the models of the premises are included in the models of the conclusion,.
In some books it is reported as a property of tautology that of being derivable from an empty set of premises,
the hidden meaning of these words is precisely that of the lack of new information
in reasoning made with a conclusion that is a tautology. So turning the argument into valid with tautologies doesn't help.

 

[JeffM]

when you want to build
an argument based on tautologies no longer you have the flow of information that makes you conclude that the models of the premises are included in the models of the conclusion.

This sentence is entirely incoherent. What does “models of premises” mean? What does “models of the conclusion“ mean? What kind of “flow of information” makes the models of the premises be included in the models of the conclusion? This is word salad. Unless you define your terms, it is just gibberish.

“In some books” is a red herring. Quote what you are discussing in full and exactly and give a citation to text and page so that what you quote can be viewed in context.

 

[lex]

Thank you for the effort to have formalized the passage from an invalid argument to a valid one
by replacing the atomic formulas in the argument with tautologies.
My point, however, was to highlight that, in addition to the obvious change from invalid to valid,
also to make it clear that when you want to build
an argument based on tautologies no longer you have the flow of information that makes you conclude that the models of the premises are included in the models of the conclusion,.
In some books it is reported as a property of tautology that of being derivable from an empty set of premises,
the hidden meaning of these words is precisely that of the lack of new information
in reasoning made with a conclusion that is a tautology. So turning the argument into valid with tautologies doesn't help.

This is all a bit too philosophical for me. These couple of pages (though on induction) might lead to some further reading if you are interested! (I notice our example of 'affirming the consequent' is mentioned).

 

[Afive]

This sentence is entirely incoherent. What does “models of premises” mean? What does “models of the conclusion“ mean? What kind of “flow of information” makes the models of the premises be included in the models of the conclusion? This is word salad. Unless you define your terms, it is just gibberish.

“In some books” is a red herring. Quote what you are discussing in full and exactly and give a citation to text and page so that what you quote can be viewed in context.

In the attached file you can find the simplest explanation of what formula models are. (If you get lost, consider alpha as the premises, beta as the conclusion, M (alpha) as the set of models of the premises, M (beta) as the set of models of the conclusion.) The book is by Russell and Norvig

 

[JeffM]

First off, you could have said from the get-go that you were trying to make sense of propositional calculus in terms of notions and vocabulary from computer science. In particular, you expressed the view that validity in the sense that it is used in logic is "useless."

Second, there is a very important sentence that you seem to have ignored: We have described a reasoning process whose conclusions are guaranteed to be true in any world in which the premises are true; in particular, if KB is true in the real world, then any sentence α derived from KB by a sound inference procedure is also true in the real world.

You started this thread by talking about validity and were bothered by

[MATH]\{(A \implies B) \land B\} \implies A[/MATH] is not a valid argument.

You provided an example (a model in the words of your text) such that, A does entail B and, if B is true then A is necessarily true.

Zermelo provided an example (a model) such that A does entail B and, if B is assumed to be true, then A is not necessarily true.

Your text says an inference procedure is a sound inference procedure if it works for every conceivable model. But Zermelo demonstrated that the procedure you were asking about did not work for at least one model. "Sound inference procedure" is simply a different way of saying "logically valid."

Third, could you please tell me where "flow of information" is mentioned, and its relevance to "sound inference procedures" is explained? I must have missed it.

 

[Afive]

Third, could you please tell me where "flow of information" is mentioned, and its relevance to "sound inference procedures" is explained? I must have missed it.

Regarding the flow of information, without referring to the Latin etymology, we can think of it in general as the knowledge of something
which goes from A to B (where B can also be A), hopefully without uncertainty (entropy).
When I said that there is no more information flow I was referring to an argument in which all formulas are tautologies, I explain it
very quickly:

1) invalid argument ((P → Q) ∧ Q) → P

2) Modus ponens ((P → Q) ∧ P) → Q

You can build truth tables for 1) and 2) and notice that they are different.

Now, I always replace each letter with the same formula and each formula is a tautology, I get:

1 *) ex invalid argument (((P → P) → (Q → Q)) ∧ (Q → Q)) → (P → P)

2 *) Modus ponens (((P → P) → (Q → Q)) ∧ (P → P)) → (Q → Q)

If you go and check with the truth tables can you distinguish 1 *) from 2 *) ? Not me, I have lost knowledge of something compared to before.
I conclude that tautologies have disrupted the flow of information.

With this I thank everyone again for the discussion, hoping to have more opportunity in future.