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Vacuous truth has two types conditional statements (if) and universal statements (all).

I intuitively understand why conditional statements can be vacuous truth but I don't understand why universal statements can be vacuous truth.

Please teach me why universal statements can be vacuous truth.

Is it because "all" includes "zero"? But linguistically nouns refer to at least one? "Zero people in the room are playing soccer" is weird.

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  • Vacuous truth is due to the interaction of universal quantifier with conditional (the truth-table for conditional has True for the rows with False in the antecedent). Jan 15 at 10:09
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    "all" does not include "zero"... The reason is that the statement "Every person in the room is blue" is parsed with "For every person x (if x is in the room, then x is blue)". And thus, in an empty room when there are no person in the room the conditional "if x is in the room, then x is blue" is True for every possible value of the variable x. Jan 15 at 10:14
  • @MauroALLEGRANZA Can "all unicorns can fly" be converted into a conditional?
    – user71046
    Jan 15 at 10:51
  • @MauroALLEGRANZA Or should I add some information like "all unicorns in the yard".
    – user71046
    Jan 15 at 11:01

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Apparently a vacuous truth is a conditional statement P ⇒ Q where it is known a priori that P can NEVER be true.

This obviously includes conditionals, but also a universal statement that can be converted to a conditional statement.

That Wikipedia article also lists some examples of that:

∀x:P(x) ⇒ Q(x), where ∀ x: ¬P(x)
∀x: x∈ A:Q(x), where the set A is empty

And from the German Wiki article:

Let Q be a property, then  ∀a  a ∈ ∅  possess that property
Let  x ∈ ℝ: then x < x²  ⇒ x = 42
Let  x ∈ ℝ: then x < 0 ∧ x > 0  ⇒ x = ∞
All even prime numbers p > 2 are dividable by 9

Or in every language: sentences that start with "when pigs fly" or "when hell freezes over".

TL;DR you need to convert the universal statement to a conditional.

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Von Neumann's quote comes to mind: "Young man, in mathematics you don't understand things. You just get used to them." Here is an argument that I hope may get you used to the idea of a vacuous answer...

Suppose we have the equation C.Y = C.X where C is a constant. We have X and we wish to solve for Y. We would normally divide both sides of the equation by C, yielding Y = X.

However, there is also the numerically vacuous solution where C = 0, and Y can take any value.

Formal logic reduces logic to a set of lexical rules in much the same way we simplify equations. Like solving for numbers, logic also can have vacuous statements that cannot reliably be evaluated as True or False.

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How can I understand vacuous truth?

The notion of vacuous truth is a recent invention of mathematicians. It came as a consequence of the fact that mathematicians and philosophers working on formal logic following George Boole's 1847 book on the subject eventually decided that the disjunction ¬α ∨ β was the logical equivalent of the implication α → β. The problem is that the disjunction ¬α ∨ β is of course true if ¬α is true, i.e. if α is false. So, they had to postulate that α → β is true when α is false. Somewhat later, they decided to call the postulated truth of α → β in this case "vacuous truth", presumably to suggest the idea that this case was not logically significant.

There is of course no logical reason for the implication α → β to be true if α is false. And the disjunction ¬α ∨ β is also not the logical equivalent of the implication α → β.

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    What is your logical reason that there is no logical reason for α → β to be true when α is false? What about α must be false?
    – PW_246
    Jan 16 at 16:59
  • @PW_246 "What is your logical reason that there is no logical reason for α → β to be true when α is false?" I didn't say there is one, but, presumably if there was one we would know by now! - 2. "What about α must be false?" Sorry, is that your proposal for a logical reason?! Jan 16 at 17:27
  • We do already know. The only sensible way to get rid of that ¬A⊨A→B is to use a paraconsistent logic. Also, if A is unsatisfiable, then clearly whenever it’s satisfied, B is also satisfied. That’s all I meant.
    – PW_246
    Jan 16 at 18:06
  • @PW_246 Sorry, no, obviously, for ¬A is clearly not a good reason, a logical reason, for A → B to be true. Jan 17 at 10:28
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A few points worth noting. Gracias for the question.

A vacuous truth is a conditional whose antecedent is false, but/and because a conditional is true despite that, the conditional is a vacuous truth.

It seems related to the following: Inverse/denying the antecedent fallacy p ⇒ q ¬p Ergo ... ??? To conclude ¬q is to commit the fallacy.

As for the example given in Wiki: All cellphones in the room are turned off (when there are no cellphones in the room) is true. If so the contradictory some cellphones in the room are not turned on must be false. In other words, everything in the room is either not a cellphone or is turned off.

It is obvious why, from this, universal statements with an empty subject class/category must be deemed true. If all cellphones in this room are turned off were treated as false (empty subject class), the contradictory some cellphones in the room not turned on must be true. Since particular negative statements have existential import, it means there exists at least 1 cellphone (in the room), but ... there are no cellphones in the room.

I wonder why logicians would call empty-subject-class universal statements vacuous. For the above reasons, it seems imperative that we treat such statements as true or else ... all hell will break loose.

I can however see a glimmer of justification. A universal statement establishes a logical connection between a subject class and a predicate class. If the subject class is empty, we aren't really saying anything at all. If I say unicorns are flying creatures, I mean NOTHING are flying creatures (the subject class is empty), but I definitely don't mean NO THING(s) are flying creatures (the predicate class is empty). Phew!

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This is a statement of formal logic, so translating it into informal language can be misleading or counterintuitive. You also need to be clear WHICH system of formal logic you're in.

In addition, and in contrast to tautologies, which can be determined structurally, a vacuous truth requires knowing your universe of discourse and what exists in it.

In propositional logic, a statement of the form P -> Q is vacuously true in the case that P is false. Since a false statement implies anything, you can write "P -> Q" for any Q whatsoever, if you have already established P is false.

In first-order logic, the concept can be extended to universal statements about objects that do not exist in the universe of discourse. For All x, [statement Y] will be vacuously true for any statement Y in the event that no x exists.

To borrow your example, For all people-in-the-room, people in the room are playing soccer is vacuously true if there are no people in the room. Again, this follows from the rules of formal first-order-logic, it is not necessarily a good analog for how people talk in natural language.

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