Type-juggling and (strict) greater/lesser-than comparisons in PHP

Question

PHP is famous for its type-juggling. I must admit it puzzles me, and I'm having a hard time to find out basic logical/fundamental things in comparisons.

For example: If $a > $b is true and $b > $c is true, must it mean that $a > $c is always true too?

Following basic logic, I would say yes however I'm that puzzled I do not really trust PHP in this. Maybe someone can provide an example where this is not the case?

Also I'm wondering with the strict lesser-than and strict greater-than operators (as their meaning is described as strictly which I only knew in the past from the equality comparisons) if it makes any difference if left and right operands are swapped with strictly unequal values:

# Precondition:
if ($a === $b) {
    throw new Exception(
       'Both are strictly equal - can not compare strictly for greater or smaller'
    );
}

($a > $b) !== ($b > $a)

For most of all type comparison combinations these greater / lesser comparison operators are not documented, so reading the manual was not really helpful in this case.

Answer 1

PHP's comparison operators deviate from the computer-scientific definitions in several ways:

In order to constitute an equivalence relation == has to be reflexive, symmetric and transitive:

• PHP's == operator is not reflexive, i.e. $a == $a is not always true:

  var_dump(NAN == NAN); // bool(false)
  

  Note: The fact that any comparison involving NAN is always false is not specific to PHP. It is mandated by the IEEE 754 Standard for Floating-Point Arithmetic (more info).

• PHP's == operator is symmetric, i.e. $a == $b and $b == $a are always the same.

• PHP's == operator is not transitive, i.e. from $a == $b and $b == $c does not follows $a == $c:

  var_dump(true == "a"); // bool(true)
  var_dump("a" == 0);    // bool(true)
  var_dump(true == 0);   // bool(false)
  

In order to constitute a partial order <=/>= has to be reflexive, anti-symmetric and transitive:

• PHP's <= operator is not reflexive, i.e. $a <= $a is not always true (Example same as for ==).

• PHP's <= operator is not anti-symmetric, i.e. from $a <= $b and $b <= $a does not follow $a == $b:

  var_dump(NAN <= "foo"); // bool(true)
  var_dump("foo" <= NAN); // bool(true)
  var_dump(NAN == "foo"); // bool(false)
  

• PHP's <= operator is not transitive, i.e. from $a <= $b and $b <= $c does not follow $a <= $c (Example same as for ==).

• Extra: PHP's <= operator is not total, i.e. both $a <= $b and $b <= $a can be false:

  var_dump(new stdClass <= new DateTime); // bool(false)
  var_dump(new DateTime <= new stdClass); // bool(false)
  

In order to constitute a strict partial order </> has to be irreflexive, asymmetric and transitive:

• PHP's < operator is irreflexive, i.e. $a < $a is never true. Note that this is true only as of PHP 5.4. Previously INF < INF evaluated to true.

• PHP's < operator is not asymmetric, i.e. from $a < $b does not follow !($b < $a) (Example same as for <= not being anti-symmetric).

• PHP's < operator is not transitive, i.e. from $a < $b and $b < $c does not follow $a < $c:

  var_dump(-INF < 0);    // bool(true)
  var_dump(0 < TRUE);    // bool(true)
  var_dump(-INF < TRUE); // bool(false)
  

• Extra: PHP's < operator is not trichotomous, i.e. all of $a < $b, $b < $a and $a == $b can be false (Example same as for <= not being total).

• Extra: PHP's < operator can be circular, i.e. it is possible that $a < $b, $b < $c and $c < $a:

  var_dump(INF < []);           // bool(true)
  var_dump([] < new stdClass);  // bool(true)
  var_dump(new stdClass < INF); // bool(true)
  

  Note: The above example throws a "Object of class stdClass could not be converted to double" notice.

You can find a few nice graphs for PHP's comparison operators on PHP Sadness 52 - Comparison operators.

As a last note, I want to point out that there are two equalities that PHP does guarantee (unlike pretty much everything else). These two always hold, simply because the compiler reduces one to the other:

($a > $b) == ($b < $a)
($a >= $b) == ($b <= $a)