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No premises given.

In addition, any tips for doing Fitch problems? In particular, how should one begin them? The trickiest thing for me is figuring out exactly where to start. Once I get that figured out, the rest usually comes pretty quickly.

Thanks!

3 Answers 3

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For Fitch proofs in general: typically your goal will give you the 'proof plan'.

In this case, for example, your goal is a conditional, so you'll want to set this up as a conditional proof, i.e. a $\to \: Intro$:

  1. $\qquad P$ Assumption (assumption of subproof, that is)

.

. (skip some lines)

.

n. $\quad Q \to P$ (desired last line of subproof ... we'll worry about how to get it later)

n+1. $P \to (Q \to P)$ $\to Intro$ 1-n

OK, so now we have a new goal: the $Q \to P$ that is the last line of the subproof. Since that is a conditional itself, once again we will set this up with a conditional proof:

  1. $\qquad P$ Assumption (assumption of subproof, that is)

  2. $\qquad \qquad Q$ Assumption (assumption of subproof with subproof)

. (skip some lines)

.

n-1. $\qquad \qquad P$ (desired last line of inside subproof)

n. $\qquad Q \to P$ $\to Intro$ 2 - n-1

n+1. $P \to (Q \to P)$ $\to Intro$ 1-n

OK, so that's the plan .. now you just have to figure out how to get $P$ as the last line of the inside subproof, and you're there!

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    Everything makes sense so far. I'm still a bit confused about how to get P from the Q assumption. Could you perhaps give me a hint (maybe the first step or two from the "skip some lines" part)? EDIT: I've figured it out, literally had to simply reiterate P :) – 2017-02-03
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    @AndrewGuo Hint: When you start a subproof, you don't just have the assumption of that subproof to work with, but you can still use *any* of the earlier statements (as long as they don't occur in a subproof that has since been closed). – 2017-02-03
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With only the two terms in the goal for the proof and no premises, you'll need to assume at least one term.

Maybe not the most elegant solution, but this worked for me:

1. | p ------------- assumption  
2. | | q ----------- assumption  
3. | | q & p ------- and introduction 1,2  
4. | | q ----------- and elimination 3  
5. | | p ----------- and elimination 3  
6. | q => p -------- implication introduction 2,5
7. | p => (q => p) - implication introduction 1,6

Of note, 

1. | p ------------- assumption  
2. | | q ----------- assumption

is insufficient to make an implication introduction of (p => q) but I am not entirely sure why...

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    It is one of the motivating design choices of natural deduction that all expressions in the proof are subformulas of the premises and conclusion. So expression 3 can be normalized out of the proof, it is unnecessary. – 2017-04-22
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    @DanielV thanks for the clarification. I'm just using the Fitch tool with the Stanford Into to Logic course and that's what it accepted for proof of the formula sans premises. I admit I am new to Fitch - can you recommend a resource for further understanding natural deduction? Thanks! – 2017-04-22
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    I don't really know any good sources for natural deduction. Natural deduction is one of those "tip of the iceberg" things, part of a huge theory of types and computable proofs. And I'm not saying your proof is wrong, just that it is mentioning irrelevant information in step 3. – 2017-04-22
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    @DanielV ...much to learn :) – 2017-04-22
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$\def\fitch#1#2{~\begin{array}{|l}#1\\\hline#2\end{array}}$

Well now, $p\to(q\to p)$ effectively states: "If we first assume $p$, then if we subsequently assume $q$, we will find that $p$ is (already assumed) true."   Which is obvious; but this also tells us how the fitch proof is arranged: make two assumptions, restate the first, then make two deductions to discharge those assumptions.

$${\fitch{}{\fitch{p\hspace{15ex}\text{Assumption}}{\fitch{q\hspace{13.5ex}\text{Assumption}}{p\hspace{13.5ex}\text{Restatement}}\\ q\to p\hspace{10.5ex}\text{Deduction (conditional introduction)}}\\ p\to(q\to p)\hspace{5.5ex}\text{Deduction (conditional introduction)}}\\\blacksquare}$$