You're right, there isn't a "1 but false" ordinarily. You can make one though:

my $one_but_false = OneButFalse->new();
print "value = $one_but_false\n";
print "truth = ", ($one_but_false ? "true\n" : "false\n");

package OneButFalse;
use overload '0+'   => sub {1};
use overload 'bool' => sub {0};
use overload fallback => 1;

sub new {
    my ($pkg) = @_;
    bless {}, $pkg;
}
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See What is truth? for more info.