The proposition All unicorns are beautiful is vacuously true in standard logic.
the verdict
SUPPORTED
the evidence backs this
refutedsupported
the weight of evidence
2 sources for · 0 against
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Reference materials and logic discussions establish that universal statements regarding non-existent entities such as unicorns are vacuously true in standard logic because the antecedent cannot be satisfied.
Vacuous truth
In mathematics and logic, a vacuous truth is a conditional or universal statement(specifically a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. An example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia".
It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" (alternatively said "for all x in this room, if x is a cellphone then x is turned off") will be true when no cell phones are present in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true. The conjunction of the two: "all cell phones in the room are turned on and all cell phones in the room are turned off", can only be true vacuously, and it implies "there are no cell phones in the room".
Vacuous statements are also used as a rhetorical device for creating verbal irony. A common example is the "Queen of England" retort; For example, "I'm a great swimmer". "If you're a great swimmer, then I'm the Queen of England", using a transp
# Is it okay that vacuously true statements can contradict each other?
Tags: logic, propositional-calculus, predicate-logic
- Score: 11
- Views: 1240
- Answers: 4
- Answered: yes
- Asked by: SebastianLinde (455 rep)
- Asked: 2022-01-26
- Site: math
## Question
I am reading How to prove it by Velleman, and something is bothering me. On page 74, he states that "all unicorns are green" is vacuously true and so is "all unicorns are purple." He goes on to state that this two statements do not contradict each other and does not give any reasons why. This does not make sense to me. If both statements are considered to be true, the conclusion is that they contradict each other.
I understand the mathematical reasoning behind the conclusion, but that still does not explain why we should accept that both of the statements that are supposedly true do not contradict each other. Does this signal to a problem in the theory of quantificational logic mainly in how the conditional connector is defined?
## Answers
### Answer by GEdgar (score: 17)
Both statements are true. So of course they do not contradict each other.
Perhaps say it another way: From the two statements "all unicorns are green
Everything we examined (2)
This check searched the claim as stated. It did not run a separate search for evidence against it.