Help with Symbolic Logic SD+ Question

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In summary, Marie couldn't figure out how to solve the problem and was grateful to both people who responded.
  • #1
Marie120
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Good evening. Would anybody in this room be able to help me with an SD+ question?
My question is as follows:
Show that the following set of sentences is inconsistent in SD or SD+:

{(~C v (E & P)) (triple bar) B, ~E > ~C, ~(P & B) & ~(~P & ~B), B > C}
 
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  • #2
Marie120 said:
Good evening. Would anybody in this room be able to help me with an SD+ question?
My question is as follows:
Show that the following set of sentences is inconsistent in SD or SD+:

{1.(~C v (E & P)) (triple bar) B, 2.~E > ~C, 3.~(P & B) & ~(~P & ~B), 4.B > C}

Don't know that this is quite what you're looking for, but consider this:

[~C v (E & P)] > C (by substituting for B in 4, from the equivalence given in 1)

[(~C v E) & (~C v P)] > C (By distribution)

That line right there is the same as the argument:

1. ~C v E
2. ~C v P
Therefore, C

which can pretty easily be shown to be invalid. It's a roundabout method, but it should work. I'll leave it to you to write a rigorous proof of this.
 
  • #3
Loseyourname, you're only working with assumptions. Assumptions do not have to be tautologies. Like if I assume X -> Y, I am not claiming that X, therefore Y, is logically valid for every substitution of X and Y. Such a claim is false but the assumption X -> Y is certainly not inconsistent with itself.

Marie, I don't know about the terms SP or SP+, so if there are special rules I am unaware of then this reply may not be right. But I have worked it through like this:

(your premises)
1. (~C v (E & P)) <--> B
2. ~E --> ~C
3. ~(P & B) & ~(~P & ~B)
4. B --> C

5. B <--> ~P (line 3)
6. (~C v (E & P)) <--> ~P (lines 5, 1)
7. E & P --> ~P
8. ~P v ~(E & P)
9. ~P v ~E v ~P
10. ~P v ~E
11. ~C --> B (line 1)
12. ~C --> C (lines 4, 11)
13. C (line 12)

This is the main part. You can finish it from here. Of course, there may be a simpler way to do it than how I did it, and I didn't formally go into several steps, particularly 5, 7, and 13.
 
  • #4
Thank you!

Hi

Sorry for my late reply, but I just wanted to say thank you to both Loseyourname and Bartholomew for taking the time to muse over my question. Though I don't have the time right now to apply your solutions to my problem, I definitely will soon.

Have a great New Year! :smile:
 

1. What is symbolic logic and why is it important?

Symbolic logic is a formal system of reasoning that uses symbols to represent statements and logical relationships. It is important because it allows us to analyze complex arguments and identify any flaws in their reasoning.

2. What is the difference between SD and SD+ in symbolic logic?

SD (sentential logic or propositional logic) is a basic form of symbolic logic that deals with simple statements and logical operators, while SD+ (predicate logic or first-order logic) allows for more complex statements and the use of quantifiers.

3. How do I solve a symbolic logic SD+ question?

To solve a symbolic logic SD+ question, you should first identify the premises and conclusion of the argument. Then, use the rules of inference and replacement to derive the conclusion from the premises. Finally, check if the conclusion logically follows from the premises.

4. What are some common mistakes to avoid in symbolic logic SD+ questions?

Some common mistakes to avoid in symbolic logic SD+ questions include not properly translating statements into symbolic form, applying incorrect rules of inference, and making assumptions that are not explicitly stated in the premises.

5. Is there a specific order in which I should apply the rules of inference in symbolic logic SD+ questions?

No, there is no specific order in which the rules of inference should be applied. However, it is important to carefully consider which rule is most appropriate for the given premises and conclusion in order to arrive at a valid conclusion.

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