Order of Quantifiers

Mathematical Logic · Axiom Academy

Understanding why the sequence of ∀ and ∃ changes meaning 1. Same Type Quantifiers: Order Doesn't Matter When all quantifiers are the same type (all ∀ or all ∃), their order can be swapped without changing the meaning. Example: "For all people x and all people y, x + y = y + x" means the same thing regardless of whether we check x first or y first. 2. Mixed Quantifiers: Order DOES Matter When we mix ∀ and ∃, the order becomes critical. The outer quantifier sets the scope, and the inner quantifier can depend on the outer variable. "For each x, there exists a y (possibly different for each x) such that P(x,y)" The y can change depending on x "There exists one y that works for all x such that P(x,y)" The same y must work for every x 3. Classic Example: "Everyone Loves Someone" Let L(x,y) mean "x loves y". Compare these two statements: Each person loves at least one person (but possibly different people). Alice might love Bob, Bob might love Carol, etc. "Someone is loved by everyone" There exists one specific person that everyone loves. Like a universally beloved celebrity. Key Difference: The first statement (∀x∃y) is much weaker and easier to satisfy - everyone just needs to love somebody . The second (∃y∀x) is much stronger - it requires one special person to be loved by everybody . Let's visualize why ∃y∀x P(x,y) → ∀x∃y P(x,y) is always true, but the reverse is false. Matrix Visualization: Think of P(x,y) as a truth table where rows are x values and columns are y values:

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