Please use this thread to post your questions about propositional logic.

Note to Glemkat: this is not about you, I just wanted to add a general question thread for use in the future and your question was a perfect starting question. Sorry about the move.

Hi Glemkat!

I am not sure what you are referring to when you say sub-proofs. I have heard of sub-formulas from predicate logic, but I think it may just be a specific rule that you are using that goes by a different name than what I know of. But I am sure I can help you with your question.

Could you elaborate some more on the issue you are having?

Hello VideCorSpoon,

Thank you for your prompt reply and your note about posting general questions.

A subproof in my book is a proof within a proof. I am attaching a screenshot of the problem that I have been working on (at least I hope I did). Know that the proof is incorrect. I was just going through and seeing what I can prove to get to the conclusion.

I would assume that the subproof is the same as what you stated in your previous message about predicate logic. What I would do is take one of the premises and form another proof underneath the premises. I hope that I am making sense. I am thoroughly confused....

glemkat

No problem at all. First I should note is that we are in a sense doing mathematics, but in different languages. Case in point, we are both doing propositional logic, but under different understood systems, so it may take a little bit to understand the system you are using since I am used to a different type of symbolic logic. What are the rules (full names) you can use at this point?

In the mean time, when you are talking about "sub-proofs," I am interpreting this as inference proofs. I use the Herrick system primarily, so the two inference proofs are Indirect proof and conditional proof. Indirect proofs states that to prove P, assume ~P and derive a contradiction and then assert P. A direct proof on the other hand requires a conditional, assuming the antecedent, derive the conditional, and assert the conditional. Yours may be different though, so there needs to be more clarification.

Sorry for the delay in getting back to you on the clarification.

I looked up indirect proofs and conditional proofs online. The conditional proof looks more like what I am currently doing. I think that I did find in a search where you talked about conditional proof here. I did read that but I am not understanding where to begin on more challenging problems. This problem would have three conditional proofs because of three premises. That is what I was thinking anyway. I am not sure. I hope that helps. Thanks for being patient with me.

What is the exact name of your classification of logic? The type of propositional logic that I am using here is set propositional logic. This also goes by other names like sentential logic and SD logic, depending on whose system you are using. Something tells me you are using an arithmetic grounded logic like Boolean or recursion because from what you have said so far, you use a much different set of rules. Then again, you may be doing formal proofs judging from the example you sent. So I am thinking it is an issue in proof structure and translation.

In the propositional logic system I use for example, the basis for conditional proofs is determined by the possibilities of inference, not the basis of premises. You can of course have more than one conditional proof in the same proof (i.e. nested proofs) but it is in a much different format than yours. Same rules, different configuration.

Here is an example of a proof in my system with a single inferred conditional.



I'm trying to piece together your system and mine so that we can move on.

if the comma is the determinate in the statement, how would something like this be symbolised?

it's cold, but it's not windy or foggy

i was thinking (C ~(W v F)). am i close at all? is it appropriate to have a basic proposition followed by a negated compound?

C: it's cold W: it's windy F: it's foggy by the way

HALP

What is the value of translating English into logic?

On a general level, the main reason one would want to translate English into logic is so that that English sentence (compound or otherwise) will fit into the formal propositional logic (in this case) system. You need to in order to utilize the system. But I suppose you are probably asking "why logic?"

Suppose I said this to you; "If Alan is at home, the Bob is at home. If Bob is at home, then Charlie is at home. Thus, if Alan is at home, then Charlie is at home." In the broader range of critical thinking, this may seem like a complex scenario, and thinking about it may take someone a little bit of time to comprehend the "value" of the compound statement. But if you utilize a formal logic system, it makes perfect sense and just a glance infers a truth functional rule, which is that of a hypothetical syllogism.

A hypothetical syllogism breaks down like this (at least to ma at any rate.)

Compound syntactical structure;
"If Alan is at home, the Bob is at home. If Bob is at home, then Charlie is at home. Thus, if Alan is at home, then Charlie is at home."

Breakdown;
If Alan is at home, the Bob is at home. = A-->B
If Bob is at home, then Charlie is at home. = B -->C
Thus, if Alan is at home, then Charlie is at home. = |- A-->C

Propositional formula for compound statement;
A-->B,B-->C, |- A-->C

From this equation, I can infer, deduce, etc. in a variety of symmetric (or even asymmetric) ways to approach an argument. When you get into logic, translating the sentenced you read for an argument into logical syntax turns out to be a lot easier than just looking at the superficial statement.

The reason for translating English into logic is in a sense a short cut for deductive argumentation and so on. You could very well do without the finer points made in propositional, predicate, or any other type of logic, but then you end up coming around full circle again. You would come up with shortcuts (deductions) which would necessitate the very system one would try to avoid. There are many reasons to study logic, such as an increased ability to identify logical structures in arguments given to you, a more refined way to evaluate arguments and the ability to find out what , and just the simple ability to construct and present good arguments for yourself. (Herrick regurgitation) And this is just a broad rationalization for the system, because this is not even factoring in truth-functional implications, necessary or sufficient conditions, and all the other stuff inherent in the knowledge of the system. As to the "value" of translating English to logic, what better way to find the "value" than within a truth functional system like logic. But in my mind, the real value in all of this comes from the part when logic is then translated into English. When you get to that point, you sweat half as much in a task twice as hard as the normal deductive processes we would normally use.

To build on VideCorSpoon's point of the purpose of translating english into logic say you have the very hairy looking argument made by a lawyer in a court room:

"If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard did not see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, ladies and gentlemen of the jury, my client is innocent." *

Although for some the previous argument at first glance would make perfect sense, but for the rest of us it hurts to read. However, logic comes in to save the day. By logically translating the english statements into logical ones you can systematically prove that the previous argument is indeed sound.

So lets go through it and translate and prove that it is sound. First we need to assign variables.
G = Client is guilty
K = Knife was in the drawer
J = Jason Pritchard saw the knife
O = Knife was there on October 10
H = Hammer was in the barn

Now that we have variables to represent everything we can now translate:

[(G -> K) AND (K' OR J) AND (O' -> J') AND (O -> (K AND H)) AND (H')] -> G'

Ok so this is the logic sentence of the lawyers argument. Using Equivalence and Inference rules of logic we can now check to see if his argument is sound. First lets write down what we know (hypothesis).

1. G -> K (hyp)
2. K' OR J (hyp)
3. O' -> J' (hyp)
4. O -> (K AND H) (hyp)
5. H' (hyp)

And we are trying to prove G'

So lets do it.

6. O' OR (K AND H) (Implication 4)
7. (O' OR K) AND (O' OR H) (Distribution 6)
8. (O' OR K), (O' OR H) (Simplification 7)
9. O -> H (Implication 8)
10. K -> J (Implication 2)
11. G -> J (Hypothetical Syllogism 1,10)
12. J -> O (Contraposition 3)
13. G -> O (Hypothetical Syllogism 11,12)
14. G -> H (Hypothetical Syllogism 9,13)
15. G' (Modus Tollens 5,14)

And there we are. We just logically stepped through and proved the lawyer's argument to be true.

*this argument was taken from Mathematical Structures for Computer Science: A Modern Approach to Discrete Mathematics, 6E by Judith L. Gersting.

Translating English into logic to check argument soundness sounds like a good reason. Nonetheless, translation looks to be troublesome when you can arrive at a conclusion mentally without such effort. I didn't need translation to see that the lawyer's argument was sound. I guess the purpose of translation is to use the tools of logic to check argument soundness, when in doubt.




This logic sentence without the original argument in sight and what the variables represent is just gibberish, right?

I think you might be using the same book I used when I took symbolic logic.

Ok. I'm gonna get a sheet of paper and see if I can do it first (I should be able to do it pretty easily). Then, once I finish, I'll give the equivalent here:

Ok. Here's what I got:

1. Dodec(c)
2. Not Small(e)
3. Not Small(e) or Dodec(f) or Small(e)
4. Assume Not Dodec(e) or Dodec(f) (Subproof begins and ends on this line)
5. Assume Small(e) (Subproof begins on this line)
6. Contradiction (Contradiction intro 1,9)
7. Not Dodec(e) or Dodec(f) (Contradiction elim. 6) (Subproof ends on this line)
8. Not Dodec(e) or Dodec(f) (Or elim. 3, 4, 5-7)
9. Assume Not Dodec(e) (Subproof begins on this line)
10. Contradiction (Contradiction intro 1,9)
11. Dodec(f) (Contradiction elim. 10) (subproof ends on this line)
12. Assume Dodec(f) (Subproof begins on this line and ends on this line)
13. Dodec(f) (Or elim. 8, 9-11, 12)

A contradiction means that we have said A and not A. Anything follows from a contradiction.

Terminology such as "sound" and "true" are fairly strictly and consistently defined in logic. "Valid" is the term that should have been used in the three instances above. The argument is valid and the proof demonstrates as much. Arguments aren't generally described as true or false in logic. Also, except in the most trivial of cases, logic won't help with the determination of actual truth or soundness.

Silly me I forgot to mention that the hypothesis were assumed to be true. Given that the hypothesis are true and the proof is valid, the argument is therefore sound.

But how did you make the move from assuming the hypothesis to be true, to stating that "Given that the hypothesis are true"?

That is the distinction I was making. Logic is of no help in the move from assuming the hypothesis to be true and saying that they are true. For hypothetical examples such as these, it is difficult to say that the argument is sound because soundness requires actually true premises.

If this question were on an exam and it was asked "is this argument sound?", how would it be demonstrated that it is sound? If you say the argument is sound if the premises are true - that begs the question; are the premises true? It cannot be said that they are true (for the purpose of demonstrating soundness) merely because they have been assumed to be true.

Hi,

I would like to have some help on how to know on what criteria an ASSUMPTION must be made in a proof ? ie in a proof that needs an ASSUMPTION to solve how should I tackle it ?

thanks
ron

I suppose as part of a RAA proof.

I would learn to know what and how to ASSUME in a proof.

im a begginer !

Well, one thing you can do is to assume the negation of what you want to prove, and then deduce a contradiction from that assumption. You can then argue that since the negation of the assumption you assume cannot be true (since it implies a contradiction) then what you wanted to prove is true.

So:

1. you assume the negation of what you want to prove.
2. you deduce a contradiction from that assumption.
3. You concluded that what you want to prove is true.

This is an RAA proof. (Look up "conditional proof" in Google. Or "indirect proof") RAA is one kind of indirect proof.