Symbolic Logic

6.4

This one took me 7 steps (including the premise).

Keep in mind that in order to apply a rule, you need an *exact*
instance of the rule. That is, you must already have sentences which exactly fit
the pattern of what the rule says you need as justification, and the new
sentence you write down must be an exact instance of what the rule allows you to
derive. (Occasionally Fitch will let you be a little sloppy, but
not often; when in doubt, make sure the pattern in your proof is exactly the same as the
pattern in the rule.)

In particular, when you use vElim, each subproof must show
*exactly* the sentence you want to conclude. For instance, if you need to
prove C ⋁
B, it isn't close enough to prove B
⋁
C. The two sentences are tautologically equivalent, true, but they're syntactically
distinct, so they don't exactly fit the pattern you need.

6.5, 6.6

6.5 took me 10 steps (9 non-premise steps). 6.6 took me 12 steps (11
non-premise steps).

The homework problems for this time would
be easier if the Distribution equivalences were rules in the formal system! In
fact 6.5 and 6.6 amount to proving both directions of one of those equivalences,
so *if *you could use them as rules the proofs would only take one
line!

However, the formal system does *not* have the distribution
laws as rules. Instead it has (so far) only the more basic rules we have
discussed: the introduction and elimination rules for =, v, and &. So the
proofs need to be constructed using only those rules.

Note that ⋁Elim
requires a subproof for each disjunct. The other rules don't require subproofs,
so you'll want to construct subproofs only if you're using ⋁Elim.
(At this point -- later we'll add other rules that use subproofs.) Also it may
help to keep in mind that the premises of the argument and anything derived from
them in the main proof is available for use in the subproofs.

6.9

Took me 6 steps (4 non-premise steps).

If you approach the problem with the right strategic mindset I don't think it
should take that long. Here's how to apply our general strategy:

First step: look at the premises and see whether any of the elimination rules
apply. (Answer: no. One premise is an atomic sentence, and the other is a
negation.)

Next step: look at the conclusion and think about the introduction rules. The
conclusion, ~Cube(c), is a negation, so the rule to think about is negation
introduction. Look up the rule if necessary to see how it works.

When you look up negation intro, you see that you assume the opposite of what
you want to prove in a subproof, derive a contradiction, and can then get the
negation of your starting assumption back out in the main proof.

That seems promising, so give it a try. Now you're in a subproof in which you've
assumed Cube(c). You want to derive a contradiction. To do that you will need to
use the _|_Intro rule, and in order to use that rule, you need some
sentence or other on one line, and the negation of that very same sentence on
another line. At this point
in the proof you have three sentences you can work with: Cube(c), Cube(b), and ~(Cube(c) &
Cube(b)). Can you see a way to use two of these sentences to derive the opposite
of the third? If so, then there's your contradiction!

6.12

Similar to You Try It #3.

6.24

Took me 10 steps (9 non-premise steps). It's easiest to prove ~A and ~B separately, then use &Intro to get the final conclusion.

6.25

Took me 10 steps (9 non-premise steps). I needed to use subproofs inside a subproof.

Curtis Brown | Symbolic Logic | Philosophy Department | Trinity University

cbrown@trinity.edu