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Theorem empty-surprise 48138
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 3057). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 48136).

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 4342, 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, http://www.iep.utm.edu/aris-log/ 4342). 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 2659, celaront 2661, and cesaro 2668.

For another "surprise" for new users of classical logic, see alimp-surprise 48136 and eximp-surprise 48140. (Contributed by David A. Wheeler, 20-Oct-2018.)

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

Proof of Theorem empty-surprise
StepHypRef Expression
1 empty-surprise.1 . . . 4 ¬ ∃𝑥 𝑥𝐴
21alimp-surprise 48136 . . 3 (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
32simpli 483 . 2 𝑥(𝑥𝐴𝜑)
4 df-ral 3057 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
53, 4mpbir 230 1 𝑥𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532  wex 1774  wcel 2099  wral 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2164
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-ral 3057
This theorem is referenced by:  empty-surprise2  48139
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