# are these statements valid argument?

I learned logic they taught:

``````if A then B
not B therefore not A
``````

Here are statements:

``````If something exists then it must adhere to a standard
If it does not adhere to a standard, it will cause suffering.
If it will cause suffering, then it should not exist.
``````

Explain:

``````A: something exists
B: it must adhere to a standard
Not B: it does not adhere to a standard
C: it will cause suffering
Not A: it should not exist

If A, then B
If Not B, then C
If C then, then Not A
``````

We trim off C, then it will be

``````If A, then B
If Not B, then Not A--> why there still `If` here, can we replace `If` to `Therefore`?
``````

If you're asking whether this is a valid argument, with the third line understood as the conclusion,

• If something exists then it must adhere to a standard
• If it does not adhere to a standard, it will cause suffering.
• If it will cause suffering, then it should not exist.

No, it's not. The antecedent of the third line affirms the consequent of the second, and so doesn't have any implications concerning the first line.

The argument is not valid.

From : "if A, then B" and "if not B, then C", you cannot conclude in propositional logic, using hypothetical syllogism (called also transitivity of implication, i.e. "from P => Q and Q => R, infer P => R"), that "if C, then not A".

In your "scenario, valid arguments are :

(a) from "if A, then B" and "if B, then C", you can conclude "if A, then C".

(b) from "if A, then B" you may have "if not B, then not A"; in this case, the second premise must be "if not A, then C", in order to conclude : "if not B, then C".

• Sure you can (conclude something). Those sentences entail, for example, "B v (~A v C)." Feb 12, 2014 at 2:38
• @ChristopherE - you are right; I will amend it. Feb 12, 2014 at 10:35

'Therefore' is the indicator for the conclusion of an argument. What follows is a proof that should hopefully answer your question.

``````  (1)  A --> B                 (Hypotheses)
(2)  ¬B                      (Hypotheses)
(3)  ¬A                      1,2 MTT
(4)  ¬B --> ¬A               2,3 C
Therefore
(5)  (A-->B) --> (¬B --> ¬A) 1,4 C
``````

(5) is a theorem. It says that (A-->B) implies (¬B --> ¬A). To get back to your question, you can write the following true statement:

`If A, then B. Therefore: If Not B, then Not A`

• All very true except he was asking about something other than proof of MT. Feb 12, 2014 at 3:54

If you are translating natural language into formal logic, then it is important to be completely consistent when labeling a statement as "A", "B", etc.

As it stands, you're easing your way from statements of fact ("something exists") into statements of value ("something should exist").

A related problem is that your original statement "If something exists, it must adhere to a standard" is ambiguous. Do you mean "it does adhere to a standard" or "it should adhere to a standard"? These are two different statements, and they have two different negations (does not versus should not).

Your argument may have value, and it's not a bad idea to base it loosely on the structures of formal logic, but you're overreaching if you want to claim it has formal logical validity. Strictly speaking, recasting a statement in the contrapositive can never add content --it's just another way of expressing the original statement.