independence-friendly logic (if logic; proposed by jaakko hintikka and gabriel sandu in 1989) is an extension of classical first-order logic (fol) by means of slashed quantifiers of the form formula_1 and formula_2 (formula_3 being a finite set of variables). the intended reading of formula_1 is "there is a formula_5 which is functionally independent from the variables in formula_3". if logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. this greater level of generality leads to an actual increase in expressive power; the set of if sentences can characterize the same classes of structures as existential second-order logic (formula_7). for example, it can express branching quantifier sentences, such as the formula formula_8 which expresses infinity in the empty signature; this cannot be done in fol. therefore, first-order logic cannot, in general, express this pattern of dependency, in which formula_9 depends "only" on formula_10 and formula_11, and formula_12 depends "only" on formula_13 and formula_11. if logic is more general than branching quantifiers, for example in that it can express dependencies that are not transitive, such as in the quantifier prefix formula_15 (formula_9 depends on formula_10, and formula_13 depends on formula_9, but formula_13 does not depend on formula_10).