logical connectives ∧, ∨, →, and ↚, as long as the other operand is not affected; that is: Conversely, for the logical connectives ↑, ↓, ↛, and ←, the quantifiers flip: Rules of inference A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the universal quantifier. Universal instantiation concludes that, if the propositional function is known to be universally true, then it must be true for any arbitrary element of the universe of discourse. Symbolically, this is represented as where c is a completely arbitrary element of the universe of discourse. Universal generalization concludes the propositional function must be universally true if it is true for any arbitrary element of the universe of discourse. Symbolically, for an arbitrary c, The element c must be completely arbitrary; else, the logic does not follow: if c is not arbitrary, and is instead a specific element of the universe of discourse, then P(c) only implies an existential quantification of the propositional function. The empty set By convention, the formula is always true, regardless of the formula P(x); see vacuous truth. Universal closure The universal closure of a formula φ is the formula with no free variables obtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of is . As adjoint In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint. For a set , let denote its powerset. For any function between sets and , there is an inverse image functor between powersets, that takes subsets of the codomain of f back to subsets of its domain. The left adjoint of this functor is the existential quantifier and the right adjoint is the universal quantifier . That is, is a functor that, for each subset , gives the subset given by those in the image of under . Similarly, the universal quantifier is a functor that, for each subset , gives the subset given by those whose preimage under is contained in . The more familiar form of the quantifiers as used in first-order logic is obtained by taking the function f to be the unique function so that