An argument is valid if the premises CANNOT all be true without the conclusion being true as well

Question

"An argument is valid if the premises CANNOT all be true without the conclusion being true as well."

Argument_1:

P or Q.

not Q.

Therefore, P.

Argument_1 is valid.

Argument_2:

B or M.

M or C.

Therefore, B or C.

Argument_2 is invalid.

Here are my three questions.

Question_1

What is the meaning of the following two sentences?

"the premises CANNOT all be true without the conclusion being true as well."

and

"the premises CAN all be true without the conclusion being true as well."

Do they share any same characteristic?

such as "ALL premises MUST be true"?

If one premise is true and the other premise is false,

could we say that

"the premises CANNOT all be true without the conclusion being true as well."?

or

could we say that

"the premises CAN all be true without the conclusion being true as well"?

Question_2

Is the following sentence correct?

"the premises CAN all be true without the conclusion being true as well IF AND ONLY IF an argument is invalid."

Question_3

Argument_3:

P.

not P.

Therefore, Q.

Is argument_3 valid?

If yes.

Is the reason that the premises CANNOT all be true without the conclusion being true as well.

Thank you all.

Answer 1

It can be useful to go back to the source of formal logic : Aristotle.

An argument must be valid "by virtue of form alone".

In Aristotle's logic :

A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so [emphasis added]. (Prior Analytics I.2, 24b18-20)

The core of this definition is the notion of “resulting of necessity” . This corresponds to a modern notion of logical consequence: X results of necessity from Y and Z if it would be impossible for X to be false when Y and Z are true. We could therefore take this to be a general definition of “valid argument”.

Aristotle proves invalidity by constructing counterexamples. This is very much in the spirit of modern logical theory: all that it takes to show that a certain form is invalid is a single instance of that form with true premises and a false conclusion. However, Aristotle states his results not by saying that certain premise-conclusion combinations are invalid but by saying that certain premise pairs do not “syllogize”: that is, that, given the pair in question, examples can be constructed in which premises of that form are true and a conclusion of any of the four possible forms is false.

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Consider for simplicity arguments like the above, with two premises : P-1 and P-2 and call C the conclusion.

1)

Saying that :

"the premises CANNOT all be true without the conclusion being true as well"

means :

it is not the case that : P-1 and P-2 are true and that C is false

that is equivalent to saying that :

if we have that P-1 and P-2 are true, then also C is true.

The "it is not the case that ..." means that we cannot find a "situation" where ...

Instead :

"the premises CAN all be true without the conclusion being true as well"

means that :

it is the case that P-1 and P-2 are true and that C is false

i.e. we can find a "situation" where ...

In a "situation" where we have one of the Ps true and the other false, we cannot said nothing about C, i.e. the truth values of the premises do not "force" the truth value of the conclusion.

2)

If :

"the premises CANNOT all be true without the conclusion being true as well"

is the defintion of valid argument, then an invalid argument is defined by the negation of the previous statement.

An argument with

premises P-1 and P-2 and conclusion C is invalid precisely when (i.e. : iff) it is the case that P-1 and P-2 are true and that C is false

i.e.

iff "the premises CAN all be true without the conclusion being true as well".

3)

The argument :

if P and not P, therefore Q

is valid because there is no "situation" where the premises CAN be both true : it is a case of Vacuous truth.

We have to consider the definition of invalidity; in order to assert that the above argument is invalid, we have to find a "situation" where both premises are true and the conclusion is false; but we cannot find such a situation, because in every prossible situation, if one premise is true, the otehr is false.

Thus we cannot assert that the argument is invalid; but if it is not invalid, it must be valid.

Answer 2

Question 1: The first sentence describes a valid argument; the second describes an invalid argument. If you want to describe it a bit more metaphysically, in the former case the truth of the premises necessitates the conclusion, and in the latter case they do not necessitate the conclusion. So, they are opposites in that way. Neither of them share the characteristic "All premises must be true." For example, a valid argument where not all premises are true is:

1. Pigs can fly.
2. If pigs can fly, the moon is green.
3. Therefore, the moon is green.

An example of an invalid argument where not all premises are true is:

1. 2 is even.
2. 3 is even.
3. Therefore, I like cookies.

Question 2: It depends on what you mean by "invalid argument". For example, when talking about a purely formal argument like

1. P
2. P implies Q
3. Therefore, Q

whether or not it is valid or not is a purely syntactic notion about symbols. In any introductory logic textbook you learn the correct syntactic derivations for what counts as a valid argument. However, your question uses modal terms like "can" and "cannot". For much of the intuitive content of what it means for an argument to be invalid, your equivalence is correct. This intuitive consideration is what motivates the formal definitions of what it means to be valid. To get out of metaphysical waters about modality however, logicians have tried to steer away from your formulation. To appreciate the subtlety, consider the argument:

1. I am a human.
2. Therefore, in every possible world I am a human.

Most metaphysicans would probably accept that it cannot be the case that (1) is true but (2) is false. However, this would count as an invalid argument in a logician's sense.

Question 3: Yes, the argument is formally valid. The reason it is formally valid is by convention, but this convention was motivated by your modal considerations. It cannot both be the case that the premises are true and the conclusion false, since the premises can never be true!