Questions about an apparently simple argument.

[ughaibu]

If an argument has premises that are all true propositions, then its conclusion must be true. So, if "a" and "b" are collections of true propositions and at least one of them is non-empty, in the following expression "P" must be true: ((a∧b)→P). By contraposition, if the same argument has a conclusion that is not true, then at least one of the propositions in at least one of "a" or "b" must be not true. This licenses the assertion ~Ǝ a,b((a∧b)→~P). Given this, it seems to me that we can write an argument like this:
1. ~Ǝ a,b((a∧b)→~P)
2. P∨~P
3. ((P∨~P)∧~~P)→P
4. ((((P∨~P)∧~P)→~P)∧~Ǝ a,b((a∧b)→~P))→P
5. (~Ǝ a,b((a∧b)→~P))→P
6. (Ǝ a,b((a∧b)→~P))∨P.
However, the person I'm discussing this with states that they will reject any argument of this form because line 4 includes a "circular proof". But I don't understand how this objection can work, as lines 3 and 4 are disjunctive syllogisms that together produce a disjunction elimination. Can anyone explain to me why the move in lines 3 and 4 is unacceptable, please.
Also, I suggested that if we allow "P" to be the proposition "some arguments are valid", the left disjunction in line 6 would be inconsistent and the above argument would prove that there is at least one valid argument. The objection here was that "P" cannot be interpreted in this way because validity is a feature of the assertion that includes "P". Again, I don't understand this objection because the above argument is independent of any particular interpretation of "P" and "some arguments are valid" seems to me to take a truth value in propositional logic. Again, can somebody explain this objection, please.

 

[JeffM]

I admit that I am stuck on your first step. Your premise is that there exist a and b AND A valid argument that a and b jointly entail P. How again do you deduce your first step that a and b do not exist and that there exists a valid argument that the nonexistence of a and b entails not P. It seems to me that what you can say from your initial premise is that not P implies a does not exist, or b does not exist, or there is no valid argument that a and b jointly entail P. You say it is "licensed." How?

 

[ughaibu]

It seems to me that what you can say from your initial premise is that not P implies a does not exist, or b does not exist, or there is no valid argument that a and b jointly entail P.

Essentially, that's what I was trying to say. A valid argument consists of a collection of statements that are either assumptions or are entailed by assumptions stated further up. We can order these in two collections "a" and "b", with one of these possibly being empty, and charcterise all valid arguments as having the form ((a∧b)→P) and if all the assumptions in "a" and "b" are true, then "P" must be true

You say it is "licensed." How?

If any argument is valid, then if its conclusion is not true, at least one of the assumptions must be not true, so there must be an untrue statement in either "a" or "b" in the case of the conclusion "~P".
What I mean by "this licenses the assertion ~Ǝ a,b((a∧b)→~P)" is just that if the argument is valid and the conclusion is not true, then there must be a proposition in either "a" or "b" that is not true, there doesn't exist a collection of true propositions "a" and "b" such that "a" and "b" jointly entail a conclusion that isn't true.
I haven't any formal training in mathematics and in any case I'm not the world's best communicator, so I apologise for any eccentricities of terminology.

 

[JeffM]

My point is this, which may be your friend's point too.

"If a is true and b is true, then P" does indeed entail that

"If P is false, then a is false or b is false."

But that does not support the statement that "If a is false or b is false, then P is false." It may be that c makes P true. In any case, the conditional statement "if a is false or b is false," does not warrant the statement that a or b is false.

Let's take an example.

"If China is more populous than India and India is more populous than Japan, then China is more populous than Japan" is a valid argument.

Because it is a valid argument so is

"If China is not more populous than Japan, then China is not more populous than India or India is not more populous than Japan" is also a valid argument.

But that does not make the assertion that "China is not more populous than India and India is not more populous than Japan" a valid argument.

 

[pka]

If an argument has premises that are all true propositions, then its conclusion must be true. So, if "a" and "b" are collections of true propositions and at least one of them is non-empty, in the following expression "P" must be true: ((a∧b)→P). By contraposition, if the same argument has a conclusion that is not true, then at least one of the propositions in at least one of "a" or "b" must be not true. This licenses the assertion ~Ǝ a,b((a∧b)→~P).

Like your friend and Jeff, I too do not follow you. We teach that arguments are valid or invalid not true or false.
Moreover you are using symbols that are not standard, such as ~Ǝ. You may mean something like \(\neg(\exists x)[P(x)\vee Q(x)]\)??? But your use lacks any context or range.
As for propositions Copi was famous for his saying: Any statement implies a true statement and any statement is implied by a false statement.
If you are serious then find a good mathematics library and a copy of Irving Copi's Symbolic Logic. Unless we are using the same terms this is a pointless exercise.

 

[ughaibu]

that does not support the statement that "If a is false or b is false, then P is false."

But I haven't made that assertion, at least not intentionally.

 

[ughaibu]

Moreover you are using symbols that are not standard, such as ~Ǝ. You may mean something like \(\neg(\exists x)[P(x)\vee Q(x)]\)??? But your use lacks any context or range.

Okay, I'll try rewriting it.

If you are serious then find a good mathematics library and a copy of Irving Copi's Symbolic Logic. Unless we are using the same terms this is a pointless exercise.

Well, I live in Japan, but I guess I might be able to find a copy.

 

[ughaibu]

If I remove the existence quantifier and write line 1. like this: ((Q∧R)→~P)→(~Q∨~R) have I captured my meaning, that if the conclusion of a valid argument is not true, then at least one of it's premises is not true?
If so, can I then proceed like this:
1. ((Q∧R)→~P)→(~Q∨~R)
2. P∨~P
3. ((P∨~P)∧~~P)→P
4. ((((P∨~P)∧~P)→~P)∧(((Q∧R)→~P)→(~Q∨~R)))→P
5. (((Q∧R)→~P)→(~Q∨~R))→P
6. ~(((Q∧R)→~P)→(~Q∨~R))∨P.
Is this comprehensible and unambiguous?

 

[JeffM]

You can treat symbolic logic as a set of rules for manipulating strings, but in fact the rules are designed to match the rules of valid logic.

So let's try your first step with an example.

(1) If the number q is a power of 3 and the number r is a power of 5, then the sum of the numbers q and r is not even, which entails that either q is not a power of 3 or r is not a power of 5.

But that is absolute nonsense. And what you have posited as a rule for manipulating strings is not a rule.

In fact, however, what you are writing as step 1 seems not to be your first step. In your mind, your ACTUAL first step is

(Actual 1) If the number q is a power of 3 and the number r is a power of 5, then the sum of q and r is even.

And you are saying that actual 1 "licenses" labelled 1. It does not. Instead, it licenses

(2) If the sum of the numbers q and r is not even, then either the number q is not a power of 3 or the number r is not a power of 5.

As pka has said, symbolic logic is formally about the logical validity of an argument rather than its truth. But a valid argument applied to true premises reaches a true conclusion. Instead of trying to determine general validity of a type of argument by manipulating strings of abstract symbols according to rules that you do not know well, try constructing a true example of the type of argument you believe to be valid. We shall understand that you are not going to claim that a single example proves general validity, but at least we shall understand what kind of argument you are talking about.

 

[ughaibu]

(1) If the number q is a power of 3 and the number r is a power of 5, then the sum of the numbers q and r is not even, which entails that either q is not a power of 3 or r is not a power of 5.

But that is absolute nonsense. And what you have posited as a rule for manipulating strings is not a rule.

Thanks for the reply, but I don't understand what you're telling me. If Q and R are powers of odd numbers then their total must be an even number, so if P is "is an odd number" then (Q∧R)→~P. Do I need to assert P before line 1 or am I still missing the point?

[ETA: I don't know what I was trying to get at with that example. I'll think about it and try to clearly express what I want to say. Thanks for the replies.]

 

[JeffM]

Thanks for the reply, but I don't understand what you're telling me. If Q and R are powers of odd numbers then their total must be an even number, so if P is "is an odd number" then (Q∧R)→~P. Do I need to assert P before line 1 or am I still missing the point?

[ETA: I don't know what I was trying to get at with that example. I'll think about it and try to clearly express what I want to say. Thanks for the replies.]

Pka will probably scold me because I am conflating truth and validity. But yes, it is a true statement that if q and r are powers of 3 and 5 respectively, then their sum is an even number. And it is a false statement if q and p are powers of 3 and 5 respectively, then their sum is not even. So do you think an argument that produces a false statement from a true statement can be a valid argument?

The previous paragraph assumes that my guess that your step 1 is the result of some previous mistake, that step 1 is not the actual first step. But perhaps I am wrong. So let's start with a true negative implication

We agree that the following is a true statement: if the numbers q and r are powers of respectively 3 and 4, their sum is NOT odd. You then say as your explicit step 1, that NECESSARILY the preceding true statement implies that q is not a power of 3 or r is not a power of 5. But that conclusion does not necessarily follow. If I have three dollars in one pocket and five dollars in my other pocket and say truthfully the sum of the dollars in my pockets is not odd and you say that therefore I can't have three dollars in one pocket or five dollars in the other pocket, I can prove you to be wrong by emptying my pockets. Once again, a valid argument applied to true premises cannot produce a false conclusion. Your argument is not valid.

 

[ughaibu]

Once again, a valid argument applied to true premises cannot produce a false conclusion.

An argument of the form ((Q∧R)→P) is only invalid in the case that Q or R (or both are not true and P is true. So, can I get what I mean with this: ~((Q∧R)→P)→((~Q∨~R)∧P)?

(I don't see how I can talk about what I want to talk about without mentioning truth, so I've got a strong suspicion that it's not a mathematical dispute that I'm involved in. Anyway, please indulge me a little longer.)

 

[JeffM]

An argument of the form ((Q∧R)→P) is only invalid in the case that Q or R (or both are not true and P is true.

This is simply wrong.

Validity refers to the structure of an argument. Truth refers to the propositions that go into an argument. So here is a valid form of argument.

Structure 1:

Major premise: All A are in B.
Minor premise: All C are in A.
Conclusion: Therefore, all C are in B.

Check it out with a Venn diagram.

OK. Let's apply your test of validity.

Major premise: All snakes are mammals.
Minor premise: All monkeys are snakes.
Conclusion: Therefore, all monkeys are mammals

Here we have an example of Structure 1 where both premises are false yet the conclusion is true. So you would conclude that structure 1 is invalid. No one will buy that.

What we can say is that an argument is invalid if we can find an example that has true premises and a false conclusion. (Just the opposite of what you were saying.)

Let's look at Structure 2.

Major premise: If D, then E.
Minor premise: E.
Conclusion: D.

Can we find an example of Structure 2 with true premises and a false conclusion?

Major premise: If X is president of the U.S., then X may live in the White House.
Minor premise: Melanie Trump may live in the White House.
Conclusion: Melanie Trump is president of the U.S.

The premises are true, but the conclusion is false. We conclude that Structure 2 is invalid.

Here is a an article with a very annoying mess of advertisements but also with what I think is a relatively clear explanation of the complex relationship between validity and truth.

What is the difference between Truth and Validity?

Truth and validity are two different notions. Truth is predicated of propositions whereas validity is predicated of arguments. Propositions are either true or false.

www.preservearticles.com

(I don't see how I can talk about what I want to talk about without mentioning truth, so I've got a strong suspicion that it's not a mathematical dispute that I'm involved in. Anyway, please indulge me a little longer.)

You seem to be thinking about some specific argument that you are having with your friend. But you are correct that you cannot determine the truth of some specific argument by debating the abstract validity of some chain of reasoning. The conclusion of your specific argument may be true even though the general structure of your argument is invalid. Monkeys are mammals no matter how idiotic the argument that gets you there. Or the conclusion of your specific argument may be false even though the argument is universally valid because one or more of your specific premises is false. Despite recognizing that to determine whether an argument is factually true depends on facts, you and your friend seem to be caught up in discussing something that is fact free, namely what arguments are valid, which means arguments that will necessarily arrive at true conclusions if the premises are true. A valid argument does not say anything at all about the truth of a conclusion if one or more of the premises inserted into the argument is false.

I welcome anyone to correct or supplement this response.

 

[ughaibu]

An argument of the form ((Q∧R)→P) is only invalid in the case that Q or R (or both) are not true and P is true.

This is simply wrong.

Okay, so all arguments with the form ((Q∧R)→P) are valid, full stop(?)
Is there any argument that cannot be phrased as ((Q∧R)→P)?

 

[JeffM]

An argument of the form ((Q∧R)→P) is only invalid in the case that Q or R (or both) are not true and P is true.

This is wrong as I said before. The correct formulation is

[MATH]q \land r \longrightarrow p[/MATH]
is shown to be invalid by an example such that q and r are both true and p is false. You have it backwards.

Okay, so all arguments with the form ((Q∧R)→P) are valid, full stop(?)
Is there any argument that cannot be phrased as ((Q∧R)→P)?

I think you misunderstand what

[MATH](q \land r) \longrightarrow p[/MATH]
means. It represents a statement of the form "q and r together entail p."

So you can indeed phrase any argument that way.

If all snakes are Russians and all kangaroos are French, then the orbit of the earth lies outside that of Jupiter.

There. I just put an argument into that format. I can put arguments that are true and valid in that format. I can put arguments that are true but invalid into that format. I can put arguments that are false but valid into that format. And I can put arguments that are false and invalid into that format.

It is true that when we are thinking about the structure of arguments, we usually make the assumption that the arrow symbol represents an argument previously demonstrated to be valid. But, apparently, you and your friend are arguing about whether a particular form of argument is indeed valid. If you assume that the arrow represents a valid argument when the validity of that argument is the subject of debate, you are indeed arguing in a circle and begging the question. You don't prove an argument valid by drawing an arrow.

This whole thread is getting silly because you seem unwilling to grasp the distinction between validity and truth and you never turn any abstract symbols into the actual specifics that you and your friend are arguing about. In fact, I do not know whether you and your friend are arguing about whether some conclusion is true or whether some chain of reasoning is valid. Do you two know?

 

[ughaibu]

you can indeed phrase any argument that way.

So, all arguments are valid.

 

[pka]

An argument of the form ((Q∧R)→P) is only invalid in the case that Q or R (or both) are not true and P is true. Okay, so all arguments with the form ((Q∧R)→P) are valid, full stop(?)
Is there any argument that cannot be phrased as ((Q∧R)→P)?

Look at its TRUTH TABLE!

 

[JeffM]

So, all arguments are valid.

Just because you can say something does not make it valid or true. Stop being ridiculous.

 

[ughaibu]

Look at its TRUTH TABLE!

In your first response on this thread you said "arguments are valid or invalid not true or false" now you're capitalising "truth table".
Let's cut to the chase, is it possible to express the notion of validity in the terms of propositional logic? If not, why not?

 

[ughaibu]

Stop being ridiculous.

If any argument can be expressed as (Q∧R)→P and all such arguments are valid by virtue of their structure, then as all arguments of the form (Q∧R)→P are valid, all arguments are valid.