the semantics of if logic thus defined is a conservative extension of first-order semantics, in the following sense. if formula_23 is an if sentence with empty slash sets, associate to it the first-order formula formula_141 which is identical to it, except in that each if quantifier formula_142 is replaced by the corresponding first-order quantifier formula_143. then formula_137 iff formula_145 in the tarskian sense; and formula_139 iff formula_147 in the tarskian sense.