Theorem alimp-surprise 42526

Description: Demonstrate that when using "for all" and material implication the consequent can be both always true and always false if there is no case where the antecedent is true.

Those inexperienced with formal notations of classical logic can be surprised with what "for all" and material implication do together when the implication's antecedent is never true. This can happen, for example, when the antecedent is set membership but the set is the empty set (e.g., x e. M and M = (/)).

This is perhaps best explained using an example. The sentence "All Martians are green" would typically be represented formally using the expression A. x ( ph -> ps ). In this expression ph is true iff x is a Martian and ps is true iff x is green. Similarly, "All Martians are not green" would typically be represented as A. x ( ph -> -. ps ). However, if there are no Martians ( -. E. x ph), then both of those expressions are true. That is surprising to the inexperienced, because the two expressions seem to be the opposite of each other. The reason this occurs is because in classical logic the implication ( ph -> ps ) is equivalent to -. ph \/ ps (as proven in imor 428). When ph is always false, -. ph is always true, and an or with true is always true.

Here are a few technical notes. In this notation, ph and ps are predicates that return a true or false value and may depend on x. We only say may because it actually doesn't matter for our proof. In metamath this simply means that we do not require that ph, ps, and x be distinct (so x can be part of ph or ps).

In natural language the term "implies" often presumes that the antecedent can occur in at one least circumstance and that there is some sort of causality. However, exactly what causality means is complex and situation-dependent. Modern logic typically uses material implication instead; this has a rigorous definition, but it is important for new users of formal notation to precisely understand it. There are ways to solve this, e.g., expressly stating that the antecedent exists (see alimp-no-surprise 42527) or using the allsome quantifier (df-alsi 42534) .

For other "surprises" for new users of classical logic, see empty-surprise 42528 and eximp-surprise 42530. (Contributed by David A. Wheeler, 17-Oct-2018.)

Hypothesis

Ref	Expression
alimp-surprise.1	|- -. E. x ph

Assertion

Ref	Expression
alimp-surprise	|- ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) )

Proof of Theorem alimp-surprise

Step	Hyp	Ref	Expression
1		imor 428	. . . 4 |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
2	1	albii 1747	. . 3 |- ( A. x ( ph -> ps ) <-> A. x ( -. ph \/ ps ) )
3		alimp-surprise.1	. . . . 5 |- -. E. x ph
4	3	nexr 2062	. . . 4 |- -. ph
5	4	orci 405	. . 3 |- ( -. ph \/ ps )
6	2, 5	mpgbir 1726	. 2 |- A. x ( ph -> ps )
7		imor 428	. . . 4 |- ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) )
8	7	albii 1747	. . 3 |- ( A. x ( ph -> -. ps ) <-> A. x ( -. ph \/ -. ps ) )
9	4	orci 405	. . 3 |- ( -. ph \/ -. ps )
10	8, 9	mpgbir 1726	. 2 |- A. x ( ph -> -. ps )
11	6, 10	pm3.2i 471	1 |- ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705

This theorem is referenced by:  empty-surprise  42528