# How to prove H → M ￢H → ￢M prove H↔M?

I'm using the program Fitch and I need to make a formal proof for this:

1. H → M
2. ￢H → ￢M

Prove: H↔M

Any ideas on how to do so?

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– J D
Nov 25, 2019 at 0:42
• Hello, it might be difficult to get homework help unless you shown you have put some effort into it and tried some things. Here's a good example..
– J D
Nov 25, 2019 at 0:45
• what is an idea? sorry, you're using language in a highly imprecise way here
– user38026
Dec 27, 2019 at 3:31

Okay,

Normally I keep my nose out of logic especially, but this one is straightforward, so I'll give you a clue.

Note, that in this argument, you have a conclusion which is the biconditional. The biconditional's definition requires two criteria be met, and ONE of the two premises in the argument satisfies it. The other doesn't, but I'm suggesting you might want to write out the inverse, converse, and contrapositive of both premises and see how they relate to the definition of the biconditional.

Once you find the two premises to satisfy the definition of the biconditional, you should be all set!

Good luck.

• I think it would be best at this point to just give your answer. Nov 27, 2019 at 1:03

????

(H --> H) --> M

H --> M

(H --> -H) <--> (M <--> -H)

(-H --> -H) --> --H

-H --> H

(-H --> -M) <--> (-M <--> H)

(H --> M --> -H --> H --> -M) --> (H <--> M)

• What does "????" mean? Nov 27, 2019 at 1:20
• Thought I am not entirely sure about, as in "question it". Nov 27, 2019 at 1:23

You have `H → M` as one premise, so deriving `M → H` will allow you to introduce the biconditional. So introduce that conditional the usual way (aka via a conditional proof).

``````|  H → M       Premise
|_ ¬H → ¬M     Premise
|  |_ M        Assumption
|  |  :
|  |  H
|  M → H      Conditional Introduction
|  H ↔ M      Biconditional Introduction
``````

The steps between the assumption of M and the derivation of H should not be hard. Looking at the second premise will be helpful.

• How is H not also an assumption? Nov 27, 2019 at 0:11
• We are not assuming `H` anywhere! No subproof has been raised with an assumption of `H`. Nov 27, 2019 at 1:06
• `H` is the antecedent of the conditional statement `H → M`. The words are not synonyms. Nov 27, 2019 at 1:14
• In the Fitch System of Natural Deduction an assumption is a statement raised (without needing to be derived) to start a context of contingent derivations (aka a subproof). That is all that it is. Nov 27, 2019 at 2:39
• Then to start a context, we are left with the antecedent being an assumption...unless you claim it does not start the context. Nov 27, 2019 at 2:49