Invalid form of an argument (Propositional Logic)

[Afive]

Hi all,

the modus ponens, that is (A -> B) and A, therefore B, is a valid argument.

If we use the form (1) (A -> B) and B, then A, the argument is no longer valid because
for the assignment A = False, B = True we have the premises (A -> B) and B both True but the conclusion A is false.

It is said that propositional logic must hold only for the form of the arguments, not for the meaning of the propositions.

But try using A = "Today is Saturday", B = "Tomorrow is Sunday"

The assignment A = False, B = True which makes the argument invalid is not possible because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it it is fundamental.

Thanks for the replies

 

[lex]

If we use the form (1) (A -> B) and B, then A, the argument is no longer valid because...

But try using A = "Today is Saturday", B = "Tomorrow is Sunday"

The assignment A = False, B = True which makes the argument invalid is not possible because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it it is fundamental.

No. Take the propositions: A= "Today is Saturday", B="Tomorrow is Sunday"

Consider the argument:
If "Today is Saturday" then "Tomorrow is Sunday"
"Tomorrow is Sunday"
Therefore "Today is Saturday"

Is this a valid argument?
To ask this question, is to ask about the form of the argument. So the question you have just asked is:
Is the argument -
[MATH]A\rightarrow B\\ B\\ \therefore A\\ \text{a valid argument.}[/MATH]In other words: "is it the case that irrespective of what A and B are, when [MATH]A\rightarrow B[/MATH] and [MATH]B[/MATH] are true, does [MATH]A[/MATH] necessarily have to be true"?
The answer to this question is No.

 

[Zermelo]

Hi all,

the modus ponens, that is (A -> B) and A, therefore B, is a valid argument.

If we use the form (1) (A -> B) and B, then A, the argument is no longer valid because
for the assignment A = False, B = True we have the premises (A -> B) and B both True but the conclusion A is false.

It is said that propositional logic must hold only for the form of the arguments, not for the meaning of the propositions.

But try using A = "Today is Saturday", B = "Tomorrow is Sunday"

The assignment A = False, B = True which makes the argument invalid is not possible because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it it is fundamental.

Thanks for the replies

That’s why we use apstract letters to represent statements. The main problem in this is; maybe there is some other day such that Sunday comes after it. This maybe doesn’t make sense at first, but you used A->B, which only means if today is Saturday that tomorrow is Sunday, it doesn’t tell us that “if tomorrow is Sunday, then today is definitely Saturday”. In order to pass this information on, we would use A<=>B, saying that those are equivalent statements, today is Saturday if and only if tomorrow is Sunday, and that we can “go backwards with the deduction”, hence the left arrow A<=B. Try this out with a better example, A:it’s raining. B:the streets are wet

 

[Dr.Peterson]

Hi all,

the modus ponens, that is (A -> B) and A, therefore B, is a valid argument.

If we use the form (1) (A -> B) and B, then A, the argument is no longer valid because
for the assignment A = False, B = True we have the premises (A -> B) and B both True but the conclusion A is false.

It is said that propositional logic must hold only for the form of the arguments, not for the meaning of the propositions.

But try using A = "Today is Saturday", B = "Tomorrow is Sunday"

The assignment A = False, B = True which makes the argument invalid is not possible because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it it is fundamental.

Thanks for the replies

It is possible for an invalid argument (with true premises) to yield a true conclusion! You have not shown that the argument is valid, only that, in this instance, A is true.

In fact, what you are saying is in itself an example showing that "(A -> B) and B, then A" is invalid! Your A is "the argument '(A -> B) and B, then A' is valid", and your B is "the conclusion is true". It is true that if the argument is valid, then the conclusion is true; and it is true that your conclusion (that today is Saturday) is true. But that does not imply that the argument is valid, because it is not!

This is called the fallacy of the converse, or "affirming the consequent".

 

[lex]

It is true that if the argument is valid, then the conclusion is true;

A valid argument can have a false conclusion.
E.g.
If 4 is a prime number then 4 divides 27
4 is a prime number
Therefore 4 divides 27

This is a valid argument.
Again, the point is that the validity of an argument refers to the form of the argument.
The above argument has the form:
[MATH]A \rightarrow B\\ A\\ \therefore B[/MATH]which is valid, since if the premisses are true, the conclusion is necessarily true.

The error being made by the OP in thinking their proposed argument is apparently valid, is not that the conclusion is true alone, but that the particular premisses cannot recreate the truth values which show that an argument form is invalid. But they don't have to. Again it is the form of the argument we are looking at, and seeing if we can assign truth values to the statements in the premisses, such that the premisses are true and the conclusion is false, and we can. Therefore the argument is invalid.

The proposed argument is:
If "Today is Saturday" then "Tomorrow is Sunday"
"Tomorrow is Sunday"
Therefore "Today is Saturday"

The form is:
[MATH]A \rightarrow B\\ B\\ \therefore A[/MATH]Assign False to A and True to B, then Premiss 1 is True and Premiss 2 is True, but the conclusion is False.
(Therefore the proposed argument is invalid).

 

[Afive]

Thanks to lex and Zermelo and Dr.Peterson.

Obviously I agree with the fact that the argument is invalid and remains invalid
(lex kindly wanted to propose it in the traditional form of the arguments, although I prefer to speak of an argument as "a conjunction of premises implying a conclusion"
so that I can express the whole argument in question with a single proposition ((A → B) ∧ B) → A
so as to be able to check with the truth tables whether the proposition is or is not a tautology (tautology that corresponds to the validity of the argument),
and this proposition turns out not to be a tautology)

To Zermelo :

It's raining so the street is wet
The street is wet
So it's raining

it is obviously invalid because there can be many reasons for which the road is wet but it does not rain.
"The street is wet" and "it's raining" are generally indipendent, for them there are possible four assignable truth values
and there are four possible cases in the reality.

but in my invalid argument,

If "Today is Saturday", "Tomorrow is Sunday"
"Tomorrow is Sunday"
Therefore "Today is Saturday"

"Tomorrow is Sunday" and "Today is Saturday" are inextricably related to each other in a sort of one-to-one functional dependence,
for them there are possible four assignable truth values
but there are only two possible cases in the reality, when both are true or when both are false, so your argument is not equivalent to mine.

To Dr.Peterson:
Your conclusion is that when from wanting to know the validity of the formalism ((A → B) ∧ B) → A one passes to wanting to know the correctness (or correspondence in reality , aka "to be true")
of the argument must conclude that B = "Tomorrow is Sunday" and A = "Today is Saturday", taken together, give the possibility of having a formally invalid argument which is also a correct argument. This is terrible.
I can accept a valid reasoning that has no correspondence in reality ("If the sky is blue then moon is made of cheese " and "the sky is blue" therefore "the moon is made of cheese")
but I cannot accept that an invalid reasoning has correspondence in reality
because the purpose of knowing whether an argument is valid or not is to exclude that an invalid reasoning has a true conclusion in reality.
If having an invalid reasoning is not enough to exclude that a conclusion
is true, my conclusion is that knowing whether an argument is valid or not is useless!

 

[Dr.Peterson]

but in my invalid argument,

If "Today is Saturday", "Tomorrow is Sunday"
"Tomorrow is Sunday"
Therefore "Today is Saturday"

"Tomorrow is Sunday" and "Today is Saturday" are inextricably related to each other in a sort of one-to-one functional dependence,
for them there are possible four assignable truth values
but there are only two possible cases in the reality, when both are true or when both are false, so your argument is not equivalent to mine.

Do you understand that the validity of an argument is entirely a matter of the form of the argument, and has nothing to do with the meaning of the statements? Relationships not expressed in the argument are relevant to reality, but not to claims about the validity of the argument itself.

To Dr.Peterson:
Your conclusion is that when from wanting to know the validity of the formalism ((A → B) ∧ B) → A one passes to wanting to know the correctness (or correspondence in reality , aka "to be true")
of the argument must conclude that B = "Tomorrow is Sunday" and A = "Today is Saturday", taken together, give the possibility of having a formally invalid argument which is also a correct argument. This is terrible.

No, a formally invalid argument is not a correct argument. It just happens to have a correct conclusion. Surely you know the difference.

I can accept a valid reasoning that has no correspondence in reality ("If the sky is blue then moon is made of cheese " and "the sky is blue" therefore "the moon is made of cheese")
but I cannot accept that an invalid reasoning has correspondence in reality
because the purpose of knowing whether an argument is valid or not is to exclude that an invalid reasoning has a true conclusion in reality.

The validity of an argument is only part of what you need to consider in reasoning. You have to also consider whether the premises are true, for instance. This is the difference between validity and soundness.

But you should certainly see that a particular true conclusion can be arrived at from true premises by invalid means. That happens all the time in politics, for example. See section 2.5 (p12-13) here, for example.

If having an invalid reasoning is not enough to exclude that a conclusion
is true, my conclusion is that knowing whether an argument is valid or not is useless!

Nonsense. Are you saying that if anyone has ever given an invalid argument that, say, the sky is blue, then the sky must not be blue? An invalid argument absolutely does not imply its conclusion is false.

Validity is not everything, but it is not useless. It is just part of how we determine truth.

 

[lex]

give the possibility of having a formally invalid argument which is also a correct argument. This is terrible.
...
the purpose of knowing whether an argument is valid or not is to exclude that an invalid reasoning has a true conclusion in reality

Perhaps what is causing you unease here is the use of the word 'conclusion'; the thought that somehow an invalid argument has logically 'produced' a conclusion which is necessarily true? I know to say that an invalid argument has a true conclusion is a bit unnerving.
In an invalid argument, the conclusion does not follow logically from the premisses. The 'conclusion' is simply an assertion, which may happen to be true or false.

E.g.
If I am mortal then the moon is made of cheese
The moon is made of cheese
Therefore I am mortal

or
E.g.
If I am mortal then the moon is made of cheese
The moon is made of cheese
Therefore I am over 21

The 'conclusion' (assertion) happens to be true. But the argument is useless as there is no logical connection between the truth of the premisses and the truth of the 'conclusion'. (It is an invalid argument; the conclusion does not follow from the preceding argument, whether or not the 'conclusion' is true). The final line still has the form of a conclusion even though it is in no way implied by what went before.

 

[JeffM]

Hi all,

the modus ponens, that is (A -> B) and A, therefore B, is a valid argument.

If we use the form (1) (A -> B) and B, then A, the argument is no longer valid because
for the assignment A = False, B = True we have the premises (A -> B) and B both True but the conclusion A is false.

It is said that propositional logic must hold only for the form of the arguments, not for the meaning of the propositions.

But try using A = "Today is Saturday", B = "Tomorrow is Sunday"

The assignment A = False, B = True which makes the argument invalid is not possible because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it it is fundamental.

Thanks for the replies

You say "but the conclusion A is false."

A is not a conclusion. It is a premise. There is the root of your confusion.

Let's go to the truth table associated with implication [MATH]U \implies V.[/MATH]
U true, V true: implication true
U true, V false: implication false
U false, V true: implication true
U false, V false: implication true

Proposition C is [MATH]A \implies B.[/MATH]
Let's look at the truth table associated with logical and as in [MATH]P \land Q[/MATH]
P true, Q true: conjunction true
P true, Q false: conjunction false
P false, Q true: conjunction false
P false, Q false: conjunction false

Putting those together for

[MATH]((A \implies B) \land A) \implies B \text { is equivalent to } (C \land A) \implies B[/MATH]
Now you have created an example where C is always true by definition. So you have only two cases, A true or A false.

C true, A true therefore we get B true.
C true, A false therefore B false

Both of those are true statements.

Turning those into English

C true, A true means that the compound statement in English "If the statements that 'Today is Saturday means tomorrow is Sunday' and 'Today is Saturday' are both true, then the statement that 'Tomorrow is Sunday'" is true.

C true, A false means that the compound statement in English "If the statements that 'Today is Saturday means tomorrow is Sunday' and 'Today is not Saturday' are both true, meaning the statement 'Today is Saturday' is false, then the statement that 'Tomorrow is Sunday is false'" is true.

We always get a true statement out of it.

 

[lex]

You say "but the conclusion A is false."

A is not a conclusion. It is a premise. There is the root of your confusion.

Afive's argument is: [MATH]((A \rightarrow B) \land B) \rightarrow A)[/MATH]A is the conclusion.

 

[pka]

the modus ponens, that is (A -> B) and A, therefore B, is a valid argument.
The assignment A = False, B = True which makes the argument invalid is not possible because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it it is fundamental.

\(\begin{array}{*{20}{c}} A&B&|&{A \Rightarrow B} \\ \hline T&T&|&T \\ T&F&|&F \\ F&T&|&T \\ F&F&|&T \end{array}\)
It may be just me, but I found the thread to most confusing. The above is the truth-table that defines implication (A implies B).
This is the common practice of the vast majority of symbolic logic textbooks. The following is taken from Copi's.
There is just one case in which the implication is false: A is true and B is false. In all other cases implication is true.
If \(1+2=4\) then \(3+3=6\). is a true statement. If \(1+2=4\) then \(3+3=7\). is a true statement. If \(1+2=3\) then \(3+3=6\). is a true statement
If \(1+2=3\) then \(3+3=7\). is a false statement. The only one of the four that is false.
Here is Copi's summery of that: A false state implies any statement is true. Rows III & IV
While a true statement is implied by any statement. Rows I & III

 

[lex]

@pka

It may be just me, but I found the thread to most confusing.

It's definitely not just you.

Yes, everyone here is in agreement on the truth table for implication, at least.
Btw for greater clarity I think I would put those statements in brackets: (If 1+2=4 then 3+3=7) is a true statement.

 

[JeffM]

Afive's argument is: [MATH]((A \rightarrow B) \land B) \rightarrow A)[/MATH]A is the conclusion.

You are correct.

Zermelo hit the nail on the head.

I’ll ask Subhotosh Khan to delete my post as irrelevant, but right now it is dinner time.

 

[lex]

Zermelo hasn't hit the point. The implication in the opposite direction is not part of the argument, therefore is irrelevant to whether or not the argument is valid. The argument simply is invalid for the reasons stated before.
Enjoy your dinner. Things always look brighter after a bit of food!

 

[JeffM]

Actually, I think that was what I was getting at: A was not a conclusion in the original syllogism, so treating A as a conclusion is irrelevant. Moreover, I read zermelo as showing why you cannot treat A entails B as being equivalent to B entails A. I am not so sure now whether my response needs to be deleted. The OP asked about A true or false and B true or false. And that I believe I addressed that.

Delete or not?

 

[Afive]

Perhaps what is causing you unease here is the use of the word 'conclusion'; the thought that somehow an invalid argument has logically 'produced' a conclusion which is necessarily true? I know to say that an invalid argument has a true conclusion is a bit unnerving.
In an invalid argument, the conclusion does not follow logically from the premisses. The 'conclusion' is simply an assertion, which may happen to be true or false.

E.g.
If I am mortal then the moon is made of cheese
The moon is made of cheese
Therefore I am mortal

I think you are very close to the essence of my question: in general when we have an invalid argument we cannot talk about one
conclusion but only of a proposition that can be true or false.

In the particular invalid argument ((A → B) ∧B) → A) we cannot speak of A as a
conclusion but only as a a proposition that can be true or false. That is, A does not follow from both premises,
but at the same time we have that when one goes to investigate the meaning of individual propositions he/she finds that this proposition A is necessarily true when proposition B is true (and vice versa)
and A is necessarily false when proposition B is false (and vice versa). In other words, it is not possible (in reality) that A and B have different truth values. So, A seems " to follow" from just B, in a very broad sense.

Reality is a little different from propositional letters written on a piece of paper
I just ask for another clarification: what happens if, always for the same argument , A and B are instead formulas, in particular tautologies?
In this case the premises and conclusion are always true, and in a certain sense the conclusion seems to follow from the premises but do you think it is still right to talk about the validity or invalidity of the argument?

 

[Zermelo]

To Zermelo :

It's raining so the street is wet
The street is wet
So it's raining

it is obviously invalid because there can be many reasons for which the road is wet but it does not rain.
"The street is wet" and "it's raining" are generally indipendent, for them there are possible four assignable truth values
and there are four possible cases in the reality.

but in my invalid argument,

If "Today is Saturday", "Tomorrow is Sunday"
"Tomorrow is Sunday"
Therefore "Today is Saturday"

"Tomorrow is Sunday" and "Today is Saturday" are inextricably related to each other in a sort of one-to-one functional dependence,
for them there are possible four assignable truth values
but there are only two possible cases in the reality, when both are true or when both are false, so your argument is not equivalent to mine.

Yes, but note, imagine that I was an alien that knew nothing about saturdays or sundays or whatever. You only gave me information that A=>B, if today is saturday that tommorow is sunday, I know nothing else! In “the real world”, those statements are equivalent without a doubt, but in the language of logic, you need to clearly express their equivalence. That’s why math is beautiful and works and is true for any statement you could imagine, if we would go around drawing up conclusions from our experiences, then math wouldn’t be a universal language, and the use of math would be heavily dependent on the past experience of the reader. I repeat once again, that’s why we use A and B, because A and B can be any statements. Again, this may be confusing when thinking about Saturdays and Sundays, but imagine working with some really complicated high level math, there is no room for ambiguity and error! Love that you are thinking about this and defending your argument. Dr Peterson and others gave you a clear mathematical answer to why this doesn’t work, this is just my interpretation ?

 

[Zermelo]

Zermelo hasn't hit the point. The implication in the opposite direction is not part of the argument, therefore is irrelevant to whether or not the argument is valid. The argument simply is invalid for the reasons stated before.
Enjoy your dinner. Things always look brighter after a bit of food!

Yes, the explanation I gave was a little crude, but the only reason that there is any confusion here is that the OP is using the fact that those facts are equivalent, hence he can make it work (he can use the fact that Saturday is the only day before Sunday, thus, he is unknowingly using (A<=>B)^A=>B). When I gave him a second set of statements to try his logic out on it, the argument failed, because the statements weren’t equivalent. That is basically a counterexample saying that an invalid argument sometimes reaches a true conclusion, depending on the premises. I’m not an English native speaker so he is better off reading other answers for the mathematical clarity, but this is just me giving a simpler outlook. If you read the OPs answer to my post, he says that there is a “one on one functional dependance in his statements, that there are just 2 possible cases: when they are both right or both wrong”, which is literally equivalence, and that is causing the confusion here. A and B are not some random statements such that the conclusion ends up being true, they are equivalent statements, and the OP sees that and heavily relies on that.

 

[lex]

@Zermelo
Your point is expressed very clearly here.
I see you are expressing the source of the OP's confusion. You have correctly pointed out that the only reason that there is any confusion with this particular case is that the two statements "Today is Saturday" and "Tomorrow is Sunday" are logically equivalent.
This certainly adds light to the situation and is, I think, a most valuable contribution to the discussion.

I focussed on the OP's logical argument which, as a fact, doesn't use "Today is Saturday" [MATH]\leftrightarrow[/MATH] "Tomorrow is Sunday":

Their logical argument again is:
"Today is Saturday" [MATH]\rightarrow[/MATH] "Tomorrow is Sunday"
"Tomorrow is Sunday"
[MATH]\therefore[/MATH] "Today is Saturday"

and I focussed on their fundamental misunderstanding of the concept of validity of an argument, which is a question of the Form not the Content of the argument. So even the argument with particular statements:

"Today is Saturday" [MATH]\rightarrow[/MATH] "Tomorrow is Sunday"
"Tomorrow is Sunday"
[MATH]\therefore[/MATH] "Today is Saturday"

is not valid, since the form of the argument:

[MATH]\mathscr{A} \rightarrow \mathscr{B}\\ \mathscr{B}\\ \therefore \mathscr{A}[/MATH]can have the truth assignments: [MATH]\mathscr{A}[/MATH] false, [MATH]\mathscr{B}[/MATH] true: so that the premisses are true and the conclusion is false. No more thought is needed. As you know, we adhere rigidly to definitions in mathematics and simply test against the definition, to determine whether or not it applies.

I do appreciate that your input has given great clarity to the hidden assumption in their initial thinking, the thinking which prompted them to propose their logical argument as being valid.

 

[lex]

In other words, it is not possible (in reality) that A and B have different truth values. So, A seems " to follow" from just B, in a very broad sense.

In that case, as Zermelo says very succinctly in #18, you are actually using (A[MATH]\leftrightarrow[/MATH]B and B). Then A actually does follow from these premisses! (In no mathematico-logical sense does A follow from A[MATH]\rightarrow [/MATH]B, and B).

what happens if, always for the same argument , A and B are instead formulas, in particular tautologies?
In this case the premises and conclusion are always true, and in a certain sense the conclusion seems to follow from the premises but do you think it is still right to talk about the validity or invalidity of the argument?

The nice thing about mathematics is that we define terms rigorously and simply apply the definition rigidly to determine whether the term applies.
The only criterion for calling an argument valid, is a test on the form of the argument.
In examining the form of the argument, with total freedom to assign truth values to any of the variables, if we can thus assign truth values that make the premisses true and the conclusion false, then the argument is invalid, otherwise it is valid.

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