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I am learning to derive proofs of some sentences based on logical axiom schemes and inference rules. But there is a lot of unclear moments, like getting hypothesis. The one such example would be $A \Rightarrow B$ and I take $A$ and will use DT. Than if I have $A, B \mid- X$, i can take A, B. Is it possible from $A \vee B \Rightarrow X$ to take $A$ or $B$ ? And the same for $A \wedge B$. Can I take $A$ and $B$ as individual hypothesis ?

Thanks for any description.

Hippalectryon
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Timotei
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2 Answers2

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Let's say that your premises are $A$, $A \vee B \Rightarrow X$.

Now you can use a rule of inference called addition and infer that $A \vee B$ is true. (This is not a hypothesis).

Now using Modus Ponens, you can infer that $X$ is true.

The formal proof would look like this:

Premises

1)$\quad A$
2)$\quad A\vee B \Rightarrow X$

Proof

3)$\quad A \vee B\quad\quad\quad \quad\quad \text{Addition(1)}$

4)$\quad X\quad\quad\quad \quad\quad \text{Modus Ponens(2, 3)} $

You have proved that $X$ is true.

Now let's say that your premises are $A \wedge B$, $A \Rightarrow X$. You want to prove that $X$ is true.

Now you can use a rule of inference called Simplification (or Conjunction Elimination) and infer that $A$ is true. (This is also not a hypothesis).

Using Modus Ponens, you can infer that $X$ is true.

The formal proof would look like this:

Premises

1)$\quad A \wedge B$
2)$\quad A \Rightarrow X$

Proof

3)$\quad A\quad\quad\quad \quad\quad \text{Simplification(1)}$

4)$\quad X\quad\quad\quad \quad\quad \text{Modus Ponens(2, 3)} $

Paul
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  • But what if I don't have A. I have just A∨B⇒X. Than is it correct to say that I have A hypothesis. As in this dummy sample: A∨B⇒X, than to prove that X, I take premise A∨B. And than I don't know what to do, I take another one hypothesis A. No matter if it's good or not. I am interesting if I can take hypothesis from conjunction and disjunction ? – Timotei Nov 11 '15 at 13:02
  • If you only have A∨B⇒X, then you cannot prove X. The only thing that you can prove is that A⇒X. You would take A as a hypothesis, then use Addition to get A∨B and Modus Ponens to get X. But you have to remember that if you make an hypothesis, then you have not proved anything inside the hypothesis. You have only proven (Hypothesis⇒X) – Paul Nov 11 '15 at 13:07
  • The same goes for $A \wedge B$ – Paul Nov 11 '15 at 13:11
  • Ok, but if I don't know that A∧B is true. Can I use A alone as hypothesis to work around proof of A⇒X. – Timotei Nov 11 '15 at 13:39
  • If you only have A⇒X as a premise, then you cannot prove that X is true.(Not even with a hypothesis). What a hypothesis does is, it lets you deduce something with the assumption that something else is true. So what you are left with is (Hypothesis⇒Whatever you have deduced from that hypothesis). – Paul Nov 11 '15 at 16:48
  • If you have A and A⇒X as premises, then you could prove that X is true. – Paul Nov 11 '15 at 16:49
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Oky, maybe there is some kind of good learning materials. I have to understand the things I can do to prove something. The basic strategies if any. For example, I am looking to this one and just don't know the things i can do.

(Xv!X)->(X->Z)v(Z->X).

I usually try to create some basic hypothesis and try to find some good axiom scheme, but it works for 2 line proofs. Is it really so creative process to create the proof ?

Timotei
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