Prove validity of a cononclusion

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In summary, the first and third statements are valid arguments, as they are tautologies. However, the second statement is not a valid argument, as it is not a tautology and can be false.
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Movingon
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Prove the validity of the following:

1. It rains, Ali is sick. Ali was not sick. ⊢ It didn't rain.

2. I like maths, I study. I study or don't make an exam. ⊢ I don't make an exam, I do not like Maths.

3. I study, I do not fail in maths. I don't play soccer, I study. I failed in maths. ⊢Therefore I played soccer.

My attempts at solutions so far:

1. ((p → q) Λ ¬q) → ¬p This statement is a tautology so this conclusion is true?

2. Slightly trickier but this was my attempt. ((p → q) Λ (¬q V ¬r)) → (¬r → ¬p) This is not a tautology but has only one place that is false so is the argument true or not?

3. ((p → ¬q) Λ (¬r → p) Λ q) → r This is also a tautology so this argument is valid?
 
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  • #2
Just a question on number 2. If q is the proposition, I study, and r is the proposition, i make an exam. Then why is "I study or don't make an exam", [tex]\neg p \vee \neg r[/tex]?
 
  • #3
daveyinaz said:
Just a question on number 2. If q is the proposition, I study, and r is the proposition, i make an exam. Then why is "I study or don't make an exam", [tex]\neg p \vee \neg r[/tex]?

That was my mistake. Thanks for the correction. It should be [tex]\ p \vee \neg r[/tex]?

So are my attempts at solutions correct? Since 1 and 3 are a tautology, they are right. The 2 is false, because it is not a tautology?
 
Last edited:

Related to Prove validity of a cononclusion

1. How do you prove the validity of a conclusion?

To prove the validity of a conclusion, you must use deductive reasoning. This means starting with a premise or set of premises that are assumed to be true, and using logical steps to arrive at the conclusion. If the premises are true and the reasoning is valid, then the conclusion must also be true.

2. What is the difference between validity and soundness?

Validity refers to the logical structure of an argument, while soundness refers to the truthfulness of the premises and the validity of the reasoning. A valid argument can have false premises and still be considered valid, but a sound argument must have true premises and valid reasoning.

3. Can you prove a conclusion to be true in all cases?

No, it is not possible to prove a conclusion to be true in all cases. This is because there may be exceptions or counterexamples that could disprove the conclusion. However, if the premises are true and the reasoning is valid, then the conclusion is considered to be true in most cases.

4. How can you determine if a conclusion is logically valid?

To determine if a conclusion is logically valid, you can use deductive reasoning and logical rules to evaluate the argument. If the argument follows the rules of logic and the conclusion logically follows from the premises, then the conclusion can be considered valid.

5. What are some common logical fallacies that can make a conclusion invalid?

Some common logical fallacies that can make a conclusion invalid include circular reasoning, false cause and effect, and ad hominem attacks. These fallacies can distort the logical structure of an argument and make the conclusion invalid despite appearing to be logically sound.

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