Natural Deductions: Strategies to Reason Asymmetry

[kodachrome22]

I am having trouble with some homework on natural deductions. I just don't know how to approach the problems. For example, one problem is to show that irreflexivity and transitivity imply asymmetry. Any strategies on how to reason the deductions would be great (the rules are simple enough, knowing how to decide how to do the problem is where I have trouble).

Thanks, and apologies if this belongs in the HW forum. I only saw physics problems there so I thought I'd get a better response here. Thanks!

 

[Stevo]

I like picturing these sorts of things.

So, imagine you have two points a and b, and an arrow between them which represents a relation R. Assume that R is irreflexive and transitive. Now, because R is transitive, we can go from a to b via R and then back from b to a via R, and this is equivalent to going from a to a via R. However, because R is irreflexive, there is no closed R curve from a to a. We've missed something.

In going from a to b via R and then from b to a via R, we're clearly assuming that R is also symmetric. So, it's possible that we've made two mistakes. Either R is not transitive, in which case not all "broken journeys" can be collapsed to a "shortcut", or R is asymmetric.

So we've killed two birds with one stone. An irreflexive symmetric relation must be intransitive, and an irreflexive transitive relation must be asymmetric.

 

[R0man]

Natural Deductions is a powerful tool used in logic to prove the validity of arguments. However, it can be challenging to know how to approach certain problems. When dealing with asymmetry, there are a few strategies that can help guide your reasoning in Natural Deductions.

Firstly, it is important to understand the definitions of the terms involved in the problem. In this case, irreflexivity means that a relation cannot hold between an object and itself, while transitivity means that if a relation holds between two objects, and another relation holds between those two objects and a third object, then the first relation also holds between the first and third objects. Understanding these definitions will help you identify the key premises and conclusion of the argument.

Next, you can try to break down the problem into smaller steps. This can be done by identifying any sub-arguments or assumptions that need to be made in order to prove the main conclusion. By breaking down the problem into smaller steps, it can become more manageable and easier to approach.

Another strategy is to use proof by contradiction. This involves assuming the opposite of what you are trying to prove, and then showing that it leads to a contradiction. This can be a useful approach when dealing with asymmetry, as it allows you to rule out any other possible explanations for the relationship between the objects.

Additionally, it can be helpful to look for patterns or similarities in the premises and conclusion. This can give you an idea of which rules of Natural Deductions to apply in each step of the proof. For example, in this problem, the premises involve the relations of irreflexivity and transitivity, so you may want to look for rules that involve these types of relations.

Lastly, practice and familiarity with Natural Deductions will also help in reasoning asymmetry. As you work through more problems, you will become more comfortable with identifying key premises, making logical connections, and applying the rules correctly.

Overall, approaching problems in Natural Deductions requires a combination of understanding the definitions, breaking down the problem into smaller steps, using proof by contradiction, and looking for patterns and similarities. With practice and these strategies, you will become more confident in tackling problems involving asymmetry. Best of luck with your homework!