The meaning and purpose of setences of the form “….cannot….without…”? How about other similiar forms such as “can” and “with”

logicproof-explanationproof-writing

I am not familiar with the thinking behind terminology of mathematics.

So I keep trying to improve on that.

I currently teach myself to write proofs by reading a textbook.

This book says

We will say that an argument is valid if the premises cannot all be true without
the conclusion being true as well.

Does the logical form of a valid argument mean ¬(P$\land$¬C) ?

How about others? Are they correct or meaningful?

Such as

1

The premises cannot all be true with the conclusion being true.

¬(P$\land$C)

2

The premises cannot all be true without the conclusion being true.

¬(P$\land$¬C)

3

The premises cannot all be true with the conclusion being false.

¬(P$\land$¬C)

4

The premises cannot all be true without the conclusion being false.

¬(P$\land$¬(¬C))

5

The premises can all be true with the conclusion being true.

P$\land$C

6

The premises can all be true without the conclusion being ture.

(P$\land$¬C)

7

The premises can all be true with the conclusion being false.

P$\land$¬C

8

The premises can all be true without the conclusion being false.

P$\land$¬(¬C)

As above there is only the second sentence which is mentioned in the book.

Other seven senteces are made by me out of curiosity.

Best Answer

We will say that an argument is valid if the premises cannot all be true without the conclusion being true as well.

Does the logical form of a valid argument mean ¬(P$\land$¬C) ?

Yes, since the original verbal sentence is rephrased as "it cannot be the case that the premises are all true in conjunction with the conclusion not being true".

Similarly, Translations 1-4 are all meaningful and accurate.

On the other hand, Translations 5-8 are meaningful but inaccurate: for example, Translation 5 asserts that the premises and conclusion are all true, whereas the original verbal sentence allows for a false premise in conjunction with a true conclusion.

5. The premises can all be true with the conclusion being true.

P$\land$C

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