# Having Trouble Understanding SD: F > ~G, ~F > ~H & More

• gabwind
In summary, the conversation discusses the difficulty in proving the inconsistency of the statements {F > ~ G, ~ F > ~H, (~ F v G) & H} in SD, and also the method of deriving ~ H from the statements { (R v ~ H), (~ R v ~ H) }. The conversation suggests making provisional assumptions and showing how they lead to an inconsistency or validation in the proof. The conversation also provides steps to get started with the proofs.
gabwind
Hi, I am having trouble showing that
{F > ~ G. ~ F > ~H, (~ F v G) & H}
is inconsistent in SD.

Also I don't understand how one can derive: ~ H
from: {(R v ~ H), (~ R v ~ H)}

I would be grateful to anyone who can help me understand these problems.

For both you need to make provisional assumptions and show that either the assumption yields an inconsistency in the proof (introduced by the assumption) or validates it. For #2, I would make the assumption R.

If you still have trouble, here's a few step to get you started:
1. F -> ~G
2. ~F -> ~H
3. (~F v G) & H
4. ~F v G [3, Simplification]
5. ~F v ~G [1, Implication]
6. (~F v G) & (~F v ~G) [4, 5, Conjunction]
7. ~F v (G & ~G) [6, Distribution]
8. ... assumption time

1. R v ~H
2. ~R v ~H
3. (R v ~H) & (~R v ~H) [1, 2, Conjunction]
4. ... does 3 look familiar?

## 1. What is SD in this context?

SD stands for "sufficient condition and necessary condition." In logic, it is used to describe the relationship between two events or conditions, where one event or condition is necessary for the other to occur, and both events or conditions are sufficient for each other to occur.

## 2. What do F, G, and H stand for?

In this context, F, G, and H are placeholders for different conditions or events. They do not have specific meanings and can be replaced by any other variables to represent different conditions or events.

## 3. How do you read the statement "F > ~G"?

The statement "F > ~G" can be read as "If F is true, then G is not true." This is an example of the sufficient condition and necessary condition relationship, where F is the sufficient condition and ~G (not G) is the necessary condition.

## 4. What does the "~" symbol mean in "F > ~H"?

The "~" symbol in "F > ~H" represents the logical operator "not." It means that if F is true, then H is not true. This is another example of the sufficient condition and necessary condition relationship.

## 5. How do I solve problems involving SD?

Solving problems involving SD requires understanding the relationships between different conditions or events and using logical reasoning to determine their implications. It is important to carefully read and interpret the statements and identify the sufficient and necessary conditions. Drawing diagrams or creating truth tables can also be helpful in solving these types of problems.

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