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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?

sunRise
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  • 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.](https://philosophy.stackexchange.com/questions/54175/given-p-%e2%87%92-q-and-m-%e2%87%92-p-%e2%88%a8-q-use-the-fitch-system-to-prove-m-%e2%87%92-q?rq=1). – J D Nov 25 '19 at 00:45
  • See also [this post](https://philosophy.stackexchange.com/questions/37405/fitch-proof-lpl-exercise-8-17) – Mauro ALLEGRANZA Nov 25 '19 at 10:58
  • what is an idea? sorry, you're using language in a highly imprecise way here –  Dec 27 '19 at 03:31

3 Answers3

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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.

J D
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????

(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)

Eodnhoj7
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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.

Graham Kemp
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  • How is H not also an assumption? – Eodnhoj7 Nov 27 '19 at 00:11
  • Because you are *not* assuming `H`, you are aiming to *derive* it from the assumption of `M`. **However** you can do that by making another assumption – Graham Kemp Nov 27 '19 at 00:13
  • And we are deriving M from an assumption of H: H ---> M. It ends up being recursive assumptions at best. – Eodnhoj7 Nov 27 '19 at 00:51
  • No we are *not* deriving `M` from any prior statement; we *are* raising it as an assumption in a new context. – Graham Kemp Nov 27 '19 at 00:54
  • Premise: H --> M. – Eodnhoj7 Nov 27 '19 at 00:58
  • Yes, but we are not deriving `M` from that (indeed we cannot). – Graham Kemp Nov 27 '19 at 00:59
  • M is the result of H, where --> means "tends towards", "therefore", or however you want to word it. Yeah, M ---> H is observing M as an assumption, but it goes both ways. – Eodnhoj7 Nov 27 '19 at 01:00
  • `H→ M` says "`M` is a consequence of `H`", so we *cannot* derive `M` *unless* we also have `H`, which we do not. Thus we can only *assume* it, make derivations *in context* of that assumption, and close the context by properly *discharging* the assumption (in this case with the rule of *Conditional Introduction*, thereby *deriving* `M→H`). – Graham Kemp Nov 27 '19 at 01:03
  • Yet H as a propostion is also assumed. H if and only if M, necessitate H as an element of M where M simulataneously exists if and only if H considering the identity of M contains H. H as a propostion is assumed, hence H is assumed. – Eodnhoj7 Nov 27 '19 at 01:06
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    We are **not** assuming `H` anywhere! No subproof has been raised with an assumption of `H`. – Graham Kemp Nov 27 '19 at 01:06
  • Tell me how H is not assumed if that is how the premise begins as H --> M. I am not trying to disagree for the sake of disagreeing, but H results in M, but H is a proposition. Take for example: H --> M requires (H-->H) implicitly as an identity property. The statement requires an assumed identity. – Eodnhoj7 Nov 27 '19 at 01:09
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    `H` is the **antecedent** of the conditional statement `H → M`. The words are not synonyms. – Graham Kemp Nov 27 '19 at 01:14
  • Antecedent: a thing or event which exists prior. Yes, but as an antecedent it begins as an assumed proposition. I don't know which words you are referring too when you state "synonyms". – Eodnhoj7 Nov 27 '19 at 01:22
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    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. – Graham Kemp Nov 27 '19 at 02:39
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    Then to start a context, we are left with the antecedent being an assumption...unless you claim it does not start the context. – Eodnhoj7 Nov 27 '19 at 02:49