Given (p ∧ q), use the Fitch system to prove (q ∨ r)

Question

I'm having trouble solving a Fitch Style Proof and I was hoping someone would be able to help me.

Premises: p ∧ q

Goal: q ∨ r

Would be an assumption (p ∧ q => q ∨ r) in first step a correct one?

Then, q ∨ r Implication Elimination: 2, 1

Why is it not correct?

Thank you very much for any help!

No; that assumption amounts to assuming what you have to prove, i.e.that $q \lor r$ follows from $p \land q$. — Mauro ALLEGRANZA, Apr 02 '17 at 11:57
Please, format the question properly, at least using consistently the same symbol for "or". — Mauro ALLEGRANZA, Apr 02 '17 at 11:57
You have to use $\land$-Elim form the premise $p \land q$ to "unpack" it and then derive $q \lor r$. — Mauro ALLEGRANZA, Apr 02 '17 at 11:58
Thank you Mauro for your quick answer! I've corrected my question. But how do I derive q ∨ r? — pramort, Apr 02 '17 at 12:15
From $p \land q$ derive $q$ by $\land$-Elim and then derive $q \lor r$ from $q$ by $\lor$-Intro. — Mauro ALLEGRANZA, Apr 02 '17 at 12:28

score 2 · Answer 1 · answered Apr 02 '17 at 12:26

2

HINT

Your premise says $p \land q$ ... what does this mean in plain English? It means that both $p$ and $q$ are true. Now: are you interested in $p$ being true? Would $p$ help you to get to your goal $q \lor r$? How about $q$?

answered Apr 02 '17 at 12:26

Bram28

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@pramort You're welcome! – Bram28 Apr 02 '17 at 21:22

score 2 · Answer 2 · answered Jul 20 '17 at 04:20

Assumptions should always be for one of three purposes:

1) It's actually a premise. For example, $p \wedge q$ is a legal assumption in this case.

2) It's the beginning of a proof by contradiction (which I think in Fitch is "$\neg$-introduction"), in which case you are later going to "eliminate" the assumption.

3) It's the beginning of a subproof for proving an implication ($\rightarrow$-introduction), in which case you are later going to "eliminate" the assumption.

Whenever you make an assumption, you should think "if this is true, then..." You want to prove "if $p \wedge q$ then $q \vee r$", so you can assume $p \wedge q$. If you also assume $(p \wedge q) \rightarrow (q \vee r)$, then what you're proving is "if $p \wedge q$ and also $(p \wedge q) \rightarrow (q \vee r)$, then $q \vee r$."

The rules of Fitch, other than assumption, are all "Introduction" or "Elimination" of something. Think about which rules you have that can do things to $\wedge$, and which rules you have that can create a $\vee$. For example, from $p \wedge q$ I can derive $p$.

Also, think through why $q \vee r$ might follow from $p \wedge q$. Logic doesn't come from nowhere - it's just a math-y version of how you already deduce things. $p$, $q$, and $r$ all stand for sentences; try out an option. Say $p$ is the sentence "Bob has a hat", $q$ is the sentence "Jill is wearing shoes", and $r$ is the sentence "Joe is tall". Then $p \wedge q$ is the sentence "Bob has a hat AND Jill is wearing shoes". $q \vee r$ is the sentence "Either Jill is wearing shoes, OR Joe is tall. If I know that "Bob has a hat and Jill is wearing shoes", how do I know that "Either Jill is wearing shoes, or Joe is tall"? Well, it's because I know that Jill is wearing shoes, so it doesn't matter whether Joe is tall. In other words, the reason we can get $q \vee r$ from $p \wedge q$ has something to do with $q$, not $r$ or $p$. That might give you a clue on how to start.

score -1 · Answer 3 · answered Aug 18 '19 at 05:39

Here is a proof using a Fitch-style proof checker:

The first line is the premise. The second line is derived from the premise use conjunction elimination ($\land$E). The last line is derived from the second using disjunction introduction ($\lor$I).

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

Given (p ∧ q), use the Fitch system to prove (q ∨ r)

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