Ok, I looked at the code up, down and sideways, then started refactoring common code and realised that half the logic vanishes if the two sets of statements are stored in one structure. I also noticed a bunch of make work creating a transient numbered set of data so I eliminated that. Then the code seemed to simplify down to the point where I could see what was going on. Here's the result:

use strict;
use warnings;

if (!@ARGV) {
    print <<HELP;
    Hi. This is the Lee-Hardy Conclusion Calculator.
    Given a list of statements on the command line using the syntax "p
+ -> q" I
    will deduce the correct conclusion to be made using basic logic an
+d the
    chain rule.
    
    - may be used for negation
    q's can not be duplicated
HELP
    exit;
}

LHCC(parse(@ARGV));


sub LHCC {
    my (%all) = @_;
    my @Conclusion;
    my @alternate;

    foreach my $testKey (keys %all) {
        my @candidate = ($testKey);
        my $nextKey = $testKey;

        while ($nextKey) {
            $nextKey = $all{$nextKey};
            push @candidate, $nextKey;
            
            last if ! exists $all{$nextKey};
        }

        if (scalar @candidate > scalar @Conclusion) {
            @Conclusion = @candidate;
        } elsif (scalar @candidate == scalar @Conclusion) {
            @alternate = @candidate;
        }
    }

    print join ' => ', @Conclusion;
    print "\nor\n";
    print join ' => ', @alternate;
    print <<RESULT;

In other words: $Conclusion[0] => $Conclusion[-1]
            or: $alternate[0] => $alternate[-1]
RESULT
}


sub parse {
    my %all;

    for my $statement (@_) {
        my ($p, $q) = $statement =~ /(\S+)\s*->\s*(\S+)/;
        $all{$p} = $q;
        $_ = /^-(.*)/ ? $1 : "-$_" for $p, $q;
        $all{$q} = $p;
    }

    return %all;
}
[download]

Given "-c->-f" "g->b" "p->f" "c->-b" on the command line it prints:

p => f => c => -b => -g
or
g => b => -c => -f => -p
In other words: p => -g
            or: g => -p
[download]

The code is not robust against bad data, but it does at least generate what seems to be the right result with the sample data.