# small arguments

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If we are presented with the following arguments :

1) If 1<0 ,then 1-1< -1 => 0<-1. But since $\neg(1<0)$ ,then $\neg(0<-1)$.

2) if 1>0 then 1+1>0+1 => 2>1 . And since 2>1 ,then 1>0

What true facts can we use so that we can decide whether the above arguments are valid or non valid??

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Well (1) is denying the antecedent (2) is affirming the consequent so from inspection they're invalid.

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Well (1) is denying the antecedent (2) is affirming the consequent so from inspection they're invalid.

So $\neg(0<-1)$ and 1>0 are not true since the arguments are invalid??

Besides $\neg(0<-1)$ and 1>0 not only are true ,but can be logically concluded in the following way:

2>1 => 2-1>1-1 => 1>0

$\neg(1<0)\Longrightarrow 1\geq 0\Longrightarrow 1-1\geq -1\Longrightarrow 0\geq -1\Longrightarrow\neg(0<-1)$

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If an argument is invalid, the conclusion can still be true, but it does not follow from the premises.

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If an argument is invalid, the conclusion can still be true, but it does not follow from the premises.

Is that a general statement ??

Yes.

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Yes.

Based on what axiom theorem or definition??

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Very well. Proof by contradiction:

Let us suppose that an invalid argument's conclusion is necessarily incorrect. Thus, let us make this argument:

1. If an animal is a reptile, it lays eggs.
2. Lizards lay eggs.
3. Therefore, lizards are reptiles.

However, this commits the fallacy of affirming the consequent. (Platypuses lay eggs too, and they're mammals.) Therefore, the conclusion must be incorrect, and lizards are not reptiles.

This is a contradiction, since lizards are, in fact, reptiles. Therefore an argument can be invalid while the conclusion is correct.

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1) An example or examples cannot support a general statement .

2) In my post #3 i showed that the conclusions of both arguments are logically implied

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My point is that one single example of an invalid argument's conclusion being correct proves that it is at least possible. My example was a proof by contradiction.

The conclusions are indeed implied, and true, but they are not implied by the arguments in post #1.

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1) An example or examples cannot support a general statement .
Is that a general statement?

But really, if you didn't already grasp that an invalid argument can have a true conclusion then you might want to start completely from scratch.

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Well (1) is denying the antecedent (2) is affirming the consequent so from inspection they're invalid.

Why when "denying the antecedent"or when "affirming the consequent" as you say the arguments are invalid?

When is an argument invalid ??

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When the conclusions do not logically follow from the premises. The conclusions in the arguments the tree talked about do not follow logically, because they deny the antecedent or affirm the consequent. Neither is a valid logical operation.

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O.K then ,why when denying the antecedent or affirming the consequent the conclusions of an argument do not logically follow??

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http://en.wikipedia.org/wiki/Affirming_the_consequent should answer that. I don't have my logic textbook handy.

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Here is an example where we deny the antecedent and the argument is still valid :

we say : If i was rich i would have bought Kennedy Airport ,but since i am not rich i cannot bye Kennedy Airport.

Is it not that argument valid??

There might be other cases that we do not know where although we deny the antecedent or affirm the consequent are valid

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Here is an example where we deny the antecedent and the argument is still valid :

we say : If i was rich i would have bought Kennedy Airport ,but since i am not rich i cannot bye Kennedy Airport.

Is it not that argument valid??

It is neither valid, nor a meaningful English sentence.

There might be other cases that we do not know where although we deny the antecedent or affirm the consequent are valid

Edit: I mixed up "valid" and "sound". Fixed.

No, never. Validity requires true logic; such that true premises guarantee a true conclusion. The conclusion has no effect on the validity of an argument, although one can be guaranteed than an argument is invalid if its conclusion is untrue but its premises true.

A valid but unsound argument:

If the moon is made of green cheese, then astronauts visiting the moon will get plenty of dairy products.

The moon is made of green cheese.

Therefore, astronauts visiting the moon will get plenty of dairy products.

This argument is valid, because if the premises are true the conclusion is guaranteed to be true. However it is unsound, because one of the premises is false. The argument is an instance of one of the simplest of arguments:

If A, then B. A. Therefore B.

You can tell that the argument is valid by looking at this form. However to be sound both the first two statements (the premises) must be true as well.

The following argument is unsound and invalid:

If an animal is a mammal, then it has fur.

A dog has fur.

Therefore, a dog is a mammal.

All of the statements (the two premises and one conclusion) are true. However, the conclusion does not follow from the premise. This is an example of one of the simplest of mistakes:

If A, then B. B. Therefore A.

In real life, you won't find a single example of something that has (real) fur and is not a mammal. This is because the opposite case happens to be true: fur is one of the defining characteristics of mammals. However, that is just a coincidence and does not make up for the faulty logic. Can you come up with an example that shows how this second example (the generic one) can lead to an incorrect conclusion given true premises?

Edited by Mr Skeptic
mixup of "sound" and "valid"

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There might be other cases that we do not know where although we deny the antecedent or affirm the consequent are valid

No, never.

Let logic decide that with its axioms and theorems , non of which you have cited so far

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I think what you will find is that you cannot use the axioms and theorems of logic to prove any argument of the form

1. If P, then Q.
2. Q.
3. Therefore, P.

You cannot prove #3 with the theorems of formal logic. That is why affirming the consequent is a fallacy.

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Let logic decide that with its axioms and theorems , non of which you have cited so far

The definition of "valid" suffices for that. The conclusion does not follow from the premises, ergo the argument is invalid. It's like if you asked my by what axiom or theorem I prove that a nonhuman is not human.

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triclino, here is the truth table for logical implication:

$\begin{array}{ccc} p & q & p\rightarrow q \\ F & F & T \\ F & T & T \\ T & F & F \\ T & T & T \end{array}$

Note that p=>q is true regardless of the value of q when p is false. Not that you can do anything with it, though. If p then q doesn't say a thing about q when p is false, and that is what you did wrong in the original post.

Aside: If p then q says quite a bit about p when q is false.

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triclino, here is the truth table for logical implication:

$\begin{array}{ccc} p & q & p\rightarrow q \\ F & F & T \\ F & T & T \\ T & F & F \\ T & T & T \end{array}$

Note that p=>q is true regardless of the value of q when p is false. Not that you can do anything with it, though. If p then q doesn't say a thing about q when p is false, and that is what you did wrong in the original post.

Aside: If p then q says quite a bit about p when q is false.

This the truth table for the definition for implication and not for logical implication.

There is a great difference between implication and logical implication.

In a simple implication the value of p =>q depends on the values (T,F) of p and q as the above table shows .

In a logical implication the value of P=>Q is always true irrespective of the values of P and Q.

Merged post follows:

Consecutive posts merged
I think what you will find is that you cannot use the axioms and theorems of logic to prove any argument of the form

1. If P, then Q.
2. Q.
3. Therefore, P.

You cannot prove #3 with the theorems of formal logic. That is why affirming the consequent is a fallacy.

oh ,yes you can

You can prove whether an arguments is valid or non valid thru the axioms or theorems of logic, otherwise we would have a situation similar to the one in this thread.

Merged post follows:

Consecutive posts merged
The definition of "valid" suffices for that. The conclusion does not follow from the premises, ergo the argument is invalid. It's like if you asked my by what axiom or theorem I prove that a nonhuman is not human.

Little you know about logic.

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There is a great difference between implication and logical implication.

Depends on who you read. Besides, if p then q is typically construed as material implication, the reason being that not q says a lot about p. (More later).

That you disagreed with Cap'n Refsmmat about the fallacy of affirming the consequent suggests that you are the one who needs to read your logic book again.

Two rules of logic apply here based on material implication, if P then Q. These are modus ponens and modus tollens.

Modus ponens says that given (1) if P then Q and (2) P then one can conclude (3) Q.

Modus tollens says that given (1) if P then Q and (2) not Q then one can conclude (3) not P.

There is nothing one can conclude from if P then Q given not P or given Q.

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If you do not want to learn the difference between implication and logical implication,that is not my problem.

Does not your book of logic have a section explaining logical implication??

M.ponens and M. tollens has nothing to do whether our arguments are valid or non valid.

If an argument does not fit the general form of another valid argument that does not mean that it is invalid

Does not your book of logic have a theorem deciding whether an argument is valid or non valid??

Edited by triclino

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If you'd like to share what your logic book says, that'd be great. Mine's 200 miles away at the moment, and in any case my philosophy professor's definition of "valid" and "invalid" agrees with what has been demonstrated above. (See Philosophical Writing by Martinich, my professor.)

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