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Theorem empty-surprise 44250
Description: Demonstrate that when using restricted "for all" over a class the expression can be both always true and always false if the class is empty.

Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. It is important to note that 𝑥𝐴𝜑 is simply an abbreviation for 𝑥(𝑥𝐴𝜑) (per df-ral 3087). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 44248).

If you want the expression 𝑥𝐴𝜑 to not be vacuously true, you need to ensure that set 𝐴 is inhabited (e.g., 𝑥𝐴). (Technical note: You can also assert that 𝐴 ≠ ∅; this is an equivalent claim in classical logic as proven in n0 4190, but in intuitionistic logic the statement 𝐴 ≠ ∅ is a weaker claim than 𝑥𝐴.)

Some materials on logic (particularly those that discuss "syllogisms") are based on the much older work by Aristotle, but Aristotle expressly excluded empty sets from his system. Aristotle had a specific goal; he was trying to develop a "companion-logic" for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature... This is why he leaves no room for such nonexistent entities in his logic." (Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy, While this made sense for his purposes, it is less flexible than modern (classical) logic which does permit empty sets. If you wish to make claims that require a nonempty set, you must expressly include that requirement, e.g., by stating 𝑥𝜑. Examples of proofs that do this include barbari 2703, celaront 2705, and cesaro 2712.

For another "surprise" for new users of classical logic, see alimp-surprise 44248 and eximp-surprise 44252. (Contributed by David A. Wheeler, 20-Oct-2018.)

Ref Expression
empty-surprise.1 ¬ ∃𝑥 𝑥𝐴
Ref Expression
empty-surprise 𝑥𝐴 𝜑

Proof of Theorem empty-surprise
StepHypRef Expression
1 empty-surprise.1 . . . 4 ¬ ∃𝑥 𝑥𝐴
21alimp-surprise 44248 . . 3 (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
32simpli 476 . 2 𝑥(𝑥𝐴𝜑)
4 df-ral 3087 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
53, 4mpbir 223 1 𝑥𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1505  wex 1742  wcel 2050  wral 3082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-ral 3087
This theorem is referenced by:  empty-surprise2  44251
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