Strategies for Identity (=)
The Language Proof and Logic textbook groups the presentation of the
inference or derivation rules of the FITCH system. First it considers formal
proofs using just the identity rules (LPL, sec 2.3), then the truth‐functional
opertor rules together with identity (LPL, sec 6.1‐6.6 and), and finally proofs
using the quantifier rules as well as the previous two sets of rules (LPL, sec?). A
more common practice is to leave the identity rules until last. Here we’ll follow
the textbook and present strategies for the identity rules first, then the truth‐
functional operator rules, and lastly the quantifiers.
Here are the strategies for identity alone:
1. IF there are any unjustified lines in the proof THEN set the
formula in the unjustified line with the lowest line number as the
current goal G AND continue with the remaining strategy rules.
2. REIT: IF the current goal G is any formula AND G occurs on an
accessible line THEN use Reit AND return to strategy rule 1.
3. =INTRO BACKWARDS: IF the current goal G is any formula n1=n1
THEN use =Intro to obtain G AND return to strategy rule 1.
4. =ELIM: IF the current goal G is any formula P(...n1...) in which n1
occurs one or more times AND n1=n2 or n2=n1 is a formula on an
accessible line AND P(...n2...) is a formula on an accessible line,
where P(...n2...) is the result of replacing one or more occurrences
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of n1 in P(...n1...) with n2, THEN use =ELIM to obtain G AND return
to strategy rule 1.
5A. ADD P(...n2...) TO PROVE: IF the current goal G is any formula
P(...n1...) in which n1 occurs one or more times AND n1=n2 or
n2=n1 is a formula on an accessible line THEN add P(...n2...), where
P(...n2...) is the result of replacing one or more occurrences of n1
in P(...n1...) with n2, as a formula on a new line in the proof to be
proved AND return to strategy rule 1. Applying this tactic or rule
may mean adding some unjustified lines to the proof earlier than
the line containing G.
5B. ADD n2=n1 OR n1=n2 TO PROVE: IF the current goal G is any
formula P(...n1...) in which n1 occurs one or more times AND n1=n2
or n2=n1 is a proper positive subformula of a formula on accessible
lines AND P(...n2...) is a positive subformula of a formula on an
accessible line, where P(...n2...) is the result of replacing one or
more occurrences of n1 in P(...n1...) with n2, THEN add n1=n2 or
n2=n1 as a formula on a new line in the proof to be proved AND
return to strategy rule 1. Applying this tactic or rule may mean
adding some unjustified lines to the proof earlier than the line
containing G.
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6. SYMMETRY TACTIC: IF the current goal G is a formula n1=n2
AND n2=n1 is on an available line THEN use the Symmetry tactic
backwards to get G AND return to strategy rule 1. Applying this
tactic or rule may mean adding some unjustified lines to the proof
earlier than the line containing G.
7. ADD n2=n1 TO PROVE: IF the current goal G is a formula n1=n2
AND n2=n1 is a proper positive subformula of a formula on an
available line THEN add n2=n1 as a formula on a new line in the
proof to be proved AND return to strategy rule 1. Applying this
tactic or rule may mean adding some unjustified lines to the proof
earlier than the line containing G.
In common with many other parts of the strategies to be presented
later, one of the strategy rules given here says to use a tactic. Tactics are fixed
sequences of steps, sometimes involving just inference rule applications, that
can be used to derive or select a formula of a particular form or ‐‐ in the case of
a tactic explained later ‐‐ a pair of formulas. In this case the tactic is the
Symmetry tactic. The tactic provides a sequence of FITCH derivation rule
applications that can be used to prove a formula of the form n1=n2. This tactic
goes as follows:
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i.
j.
k.
i,j
Figure 4. The Symmetry tactic
There are not many different kinds of examples of proof problems to
illustrate these identity strategies on. After all there is only one logic operator,
identity. Moreover the derivation problems are not very challenging. So the
primary point of these illustrations will be to show the basics of how the
strategies work.
The first example is Exercise 2.17 from the Language Proof and Logic
textbook:
Figure 5.
The strategies are ordered. We always start with the lowest numbered
strategy and work our way down. So we start at step 1. According to strategy
rule 1, the current goal formula or the formula we are currently trying to derive
is the one in line 4, SameCol(a,d). As the proof gets constructed this goal will
change. We now look at rule 2 REIT. Clearly this won’t apply. Nor will 3 =INTRO.
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Strategy rule 4 =ELIM almost applies. SameCol(a,d) is of the form P(...n1...),
where n1 is d and an identity formula c=d exists on accessible line in the proof,
line 3. But no formula of the form P(...n2...) is on an accessible line. In this case
that would be the formula SameCol(a,c) and it is nowhere to be found. So
strategy 4 does not quite apply. We next try strategy 5A ADD P(...n1...). This
does apply. SameCol(a,d) is of the form P(...n1...), where n1 is d. An identity
formula c=d exists on accessible line in the proof, line 3. That strategy says to
add to the proof SameCol(a,c) as a line to be proved and return to the
beginning of the strategies.
The partly completed proof now looks like this:
Figure 6.
There are two unjustified lines. But the strategies at step 1 tell us to make the
formula in line 4 our current goal. We go down the strategies until one applies.
In this case strategy 4 =ELIM applies. It tells us to apply =elim to lines 1 and 2 to
justify the formula in line 4. So we do that and return to the beginning. The
partly completed proof looks like this:
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Figure 7.
The strategies at step 1 tell us to make the formula in line 5 our current
goal. We go down the strategies until one applies. In this case strategy 4
=ELIM again applies. It tells us to apply =elim to lines 4 and 3 to justify the
formula in line 5. We do that and the proof is c0mpleted. There are no more
lines to be justified in the proof.
Figure 8.
The second example is Exercise 2.18 from the Language Proof and Logic
textbook:
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Figure 9
Strategy rule 1 sets the current goal as proving Between(c,d,e) in line 4. The
first applicable strategy after this is 5A. ADD P(...n2..) TO PROVE. Here n1 could
be c with n2 being a or n1 could be e with n2 being b. The strategy doesn’t
choose between these. We’ll pick the first. The result is that the proof now
looks like this:
Figure 10
we now go back to the top of the strategies where strategy rule 1 assigns
Between(a,d,e) as the current goal formula. The first strategy rule to apply
after rule 1 is rule 4. =ELIM. If what it directs is done we get:
Figure 11
This completes the proof.