3

I was looking into the below problem found at this site and that question is whether the given propositional logic is valid or not.

enter image description here

I got few doubts

  • why the You don't write a paper,however you get an extra credit. sentence has been split into two propositional sentences (2)~Q & (3)P and not ~QΛP ?
  • There is an explanation given for the solution which I understood. Now, does that explanation can be summed up saying I wasn't able to apply any propositional laws to the sentences to derive the conclusion and hence it is invalid? I mean is that how you generally solve the valid/invalid question? by trying to apply the laws to the sentences and see if it gives you the conclusion?
  • and if possible can someone explain about converse and inverse error in detail when it comes to prove the validity?

2 Answers2

1
  • why the You don't write a paper,however you get an extra credit. sentence has been split into two propositional sentences (2)~Q & (3)P and not ~QΛP ?

Because assuming $ A \land B $ is equivalent to assuming $ A $ and assuming $ B $, it's in almost any set of rules of inferences, hard to imagine that rule of inference not being present. The author just didn't make this step explicit because it's not important.

  • There is an explanation given for the solution which I understood. Now, does that explanation can be summed up saying I wasn't able to apply any propositional laws to the sentences to derive the conclusion and hence it is invalid? I mean is that how you generally solve the valid/invalid question? by trying to apply the laws to the sentences and see if it gives you the conclusion?

The validity of a proof and the validity of it's conclusion are 2 different things. The validity of a proof depends on every step being correct; the final statement being true is not sufficient for a proof to be correct.

A conclusion is invalid if it's negation is consistent with your assumptions. In the above example, the negation of the conclusion is that you didn't solve the test questions. Does that contract any assumptions? No, you could have gotten extra credit for being the teacher's special friend. No one said that the only way to get extra credit was the 2 possibilities listed.

  • and if possible can someone explain about converse and inverse error in detail when it comes to prove the validity?

Modus ponens is the statement that, if you assume $A \implies B$ and $A$, then you can conclude $B$.

The above proof starts with the following 3 assumptions.
(1) $(Q \land R) \implies P$
(2) $\lnot Q$
(3) $P$

The proof then tries to apply modus ponens to (1) and (3). However that would only work if (1) was $(Q \land R) \leftarrow P$. Since (1) isn't a reverse implication, the application of modus ponens is incorrect.

This is a common error to make in English. If you said "You will get extra credit if you write a paper or if you solve the test questions", how many students do you think would assume that those are the only way? If your mother said "if you don't eat your vegetables, then you can't have dinner", wouldn't you assume that you can have dinner if you eat your vegetables? This is logically incorrect, and it's the error the author is trying to present.

DanielV
  • 23,556
  • Thanks for the answer. So, basically I negate the conclusion and see if the assumptions still hold? If they do, it is invalid. –  Oct 03 '13 at 04:37
  • @rps Negate the conclusion and see if it contradicts the assumptions. Or as the other poster said, see if it is "possible" for the conclusion to be false when the assumptions are true. – DanielV Oct 03 '13 at 05:57
  • Also I dont understand, why its not $(R\rightarrow P)\vee(Q\to P)$, what difference it will make from $(R\vee Q)\to P$. It will be "You will get extra credit if you solve test problem or you will get extra credit, if you write a paper." But isnt it the same? Can you please clarify? I understand by truth table $(R\rightarrow P)\vee(Q\to P)$ and $(R\vee Q)\to P$ are different, but I dont understand why when considering verbal interpretation, they dont feel different. – Mahesha999 Dec 28 '15 at 21:20
  • $(R \lor Q) \implies P$ is the same as $(R \implies P) \land (Q \implies P)$. In common english that is : "If it is raining or if it cold then I will wear a coat" is the same as "If it is raining then I will wear a coat, and if it is cold then I will wear a coat." – DanielV Dec 28 '15 at 22:39
0

"why the You don't write a paper,however you get an extra credit. sentence has been split into two propositional sentences (2)~Q & (3)P and not ~QΛP ?"

I don't know the author's motivation, but it doesn't really affect the problem.

"There is an explanation given for the solution which I understood. Now, does that explanation can be summed up saying I wasn't able to apply any propositional laws to the sentences to derive the conclusion and hence it is invalid?"

Not really. You could have an incomplete set of laws and not come as able to derive the conclusion. Also, you might not know how to apply the laws to derive the conclusion. For example, I'm currently working on a propositional calculus proof which will have over 300 lines. If taken as an argument like this one (which it can in some sense), plenty of people might not know how to apply the laws to get to the conclusion. However, the argument still holds as valid.

"I mean is that how you generally solve the valid/invalid question?"

In general you show an argument invalid by showing that all the premises can come as true, but the conclusion can also come as false. To show an argument valid you either produce a formal proof, or you make a sufficiently rigorous argument such that we can know that a formal proof could get written.

"by trying to apply the laws to the sentences and see if it gives you the conclusion?"

That often works, because people find such arguments sufficiently rigorous such that they know (or at least believe they know) that a formal proof could get written.

"and if possible can someone explain about converse and inverse error in detail when it comes to prove the validity?"

The converse error referenced refers to the fact that from the consequent of a conditional as true, as well as the conditional holding as true, you can't conclude the antecedent of the conditional true also. That is,

(p $\rightarrow$ q)

q

p

is not a valid argument, where the parts above the line indicate the premises of the argument, and the part below the line indicates the conclusion of the argument. Suppose p false, and q true. Then (p $\rightarrow$ q) holds as true also. So, all the premises hold as true, but the conclusion can qualify as false.

The inverse error has the following form:

(p $\rightarrow$ q)

$\lnot$p

$\lnot$q

First off, I'll remark that if $\rightarrow$ was the biconditional in the first premise instead, this would NOT qualify as an error. But, given that it comes as the material conditional or implication as defined by the referenced site, then suppose q true, and p false. Both premises consequently qualify as true, but the conclusion does not, and consequently the argument ends up invalid.

Doug Spoonwood
  • 11,211