# What is the meaning and purpose of setences of the form "....cannot....without..."? How about other similiar forms such as "can" and "with"?

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.

• The sentence in the book is, indeed, $\neg(P\wedge\neg C)$. The others are all meaningful, but the only ones that correctly describe the validity of arguments are the ones that are logically equivalent to $\neg(P\wedge\neg C)$. Mar 24 at 2:56
• "cannot" says something can't happen. "can" says something might, or might not, happen. It doesn't say something must happen. If I had to interpret, say, $P\wedge C$, I'd interpret it as "premises and conclusion are true," not as "premises may be true when conclusion is true." Mar 24 at 3:34
• It sounds great. So do you suggest that the second half of the logic forms, namely sentences #5-#8, ARE not accurate? I am not familiar with the thinking behind terminology of mathematics. Thanks, STILL. Professor, your comments are valuable to me. Mar 24 at 4:27
• For what it's worth, in my opinion the sentence you are having difficulty with should have been written in a more straightforward way, such as -- For us, an argument is valid means: if all the premises are true, then the conclusion is true. The textbook's sentence mixes an "if ... then" construction (which is actually "if and only if") at the sentence level (i.e. if something is the case, then we say the argument is valid) with an implied "if ... then" construction in the definition of "valid", along with some confusing negation (double-negation?) constructions. Mar 24 at 6:41
• By the way, the author is Daniel J. Velleman whose PhD advisor was Mary Rudin. Ms. Rudin is the wife of Walter Rudin. Mr. Rudin is the author of some famous textbooks on analysis. Though a bit interesting in my view. Mar 24 at 9:03

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

• Wow! you are alway of assistance, as usual. I am still taking baby steps forward. Mar 24 at 8:54
• Put it simply. The original verbal sentence does not exclude the case where a false premice and a true conlcusion. What a fantastic discusion I need! Appreciate it. Mar 24 at 11:30
• @StatsCruncher Based on our previous exchanges about terminologies and logic, I wouldn't have guessed that English isn't your first language! Thanks again for your enthusiasim, and, to repeat my deleted comment, these 8 exercises are less about mathematics and more about interpreting, or even disambiguating, natural language (English). Mar 24 at 12:36
• @StatsCruncher Speaking of doing mathematics in English, it just occurred to me that (East?) Asian languages are known to be higher-context languages than English, yet appear to many to be more literal/logical when it comes to elementary mathematics (or perhaps just counting). Mar 24 at 14:26
• Thanks for sharing your personal experience. Mandarin is quite different from East Asian languages such as Korean and Japanese. For me, when it comes to advanced mathematics, I prefer English to Mandarin. At least thinking and writing proofs in English is more logical than in Mandarin beyond high school maths. Mar 24 at 14:49