Can This Logical Argument Prove 't' Given the Premises?

[alexk307]

Homework Statement

For the given set of premises, show the following is a valid argument.

~rAs
q>r
pAs>t
pVq

therefore t

where > is the implication, A is and, and V is or.

Homework Equations

The Attempt at a Solution

I know that I can convert the implications to or statements.
I used ~p>q and q>r with the hypothetical syllogism to make ~p>r

Also used pV~q and pVq to conclude p.

How should these types of problems be done? Should I convert them to and and or statements, or use mostly implications?

 

[mighty2000]

Thank you for posting your question. I am a scientist and would like to help you with your problem. To show that the given argument is valid, we need to use basic logical rules and principles.

First, let's break down the premises and the conclusion into logical statements using the given symbols:

1. ~rAs (not r and s)
2. q>r (q implies r)
3. pAs>t (p and s implies t)
4. pVq (p or q)
5. t (the conclusion)

To prove that this argument is valid, we need to show that the conclusion (t) follows logically from the premises. In other words, we need to show that if the premises are true, then the conclusion must also be true.

To do this, we can use a logical proof. Here is one possible way to prove the given argument:

1. ~rAs (premise)
2. q>r (premise)
3. pAs>t (premise)
4. pVq (premise)
5. ~r (from 1, using simplification)
6. ~rVq (from 5, using addition)
7. r>q (from 2, using contraposition)
8. ~r>p (from 7, using hypothetical syllogism)
9. ~r>pAs (from 8, using addition)
10. ~rAs>t (from 3 and 9, using hypothetical syllogism)
11. ~rAsV~rVq (from 1 and 6, using conjunction)
12. ~rAsVp (from 11 and 4, using disjunctive syllogism)
13. t (from 10 and 12, using disjunctive syllogism)

Therefore, we have shown that if the premises are true, then the conclusion (t) must also be true. This means that the given argument is valid.

In general, when solving logical problems like this, it is helpful to convert implications to equivalent statements using logical rules, such as contraposition and hypothetical syllogism. You can also use other logical rules, such as conjunction, disjunction, and negation, to manipulate the statements and come up with a valid argument.

I hope this helps answer your question. Let me know if you have any further questions or if you would like me to