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I'm trying to understand propositional logics and the concepts of entailment, but I'm struggling. The concepts don't seem to be difficult in theory, but are very strange-looking when examined. For example, I understand that the formal definition of entailment is that a ⊨ b iff M(a) ⊆ M(b).

However, the first example that my textbook provides is that false ⊨ true but true ⊭ false. I'm struggling to understand how false can entail true.

wadda_wadda
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1 Answers1

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Natural language: $a$ entails $b$ if, whenever $a$ is true, $b$ is true.

Models: $a$ entails $b$ if every model of $a$ is a model of $b$ - that is, if $M(a)\subseteq M(b)$.

So think about "$\perp$" ("false"). When is $\perp$ true? Or, what is the set of models of $\perp$?

It may be easier to think about the following set-theoretic problem first:

Why is $\emptyset\subseteq A$ for every set $A$?

Noah Schweber
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    So stated another way (with regards to the false ⊨ true but true ⊭ false example) -- the absence of evidence doesn't preclude the possibility of occurrence, but evidence of occurrence precludes the impossibility of occurrence? – wadda_wadda Oct 28 '15 at 19:57
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    Um, I don't quite follow that - I think you're trying a bit too hard. Consider this. Suppose your child comes up to you and says, "Can I have dessert before dinner if Santa Claus exists?" You wouldn't have a problem saying "yes," since - because Santa doesn't exist - you would not be hurting your child's nutrition. :P That is, the sentence "If Santa exists, then I can have dessert before dinner" ("false implies true") is true. This isn't an exact parallel, but I think it will make things clearer. – Noah Schweber Oct 28 '15 at 20:00
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    Meanwhile, if you think this is weird, you've got a point: consider the sentence "If I am not the prime minister of Lichtenstein, then this statement is false." Natural language and formal logic don't mix very well (or I've been shirking my duties). – Noah Schweber Oct 28 '15 at 20:02
  • See also http://math.stackexchange.com/questions/1474054/implication-in-mathematics-how-can-a-imply-b-when-a-is-false/1474063#1474063. – Noah Schweber Oct 28 '15 at 20:08
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    @wadda_wadda Basically, a valid argument beginning from a false premise may lead to a true conclusion ($\bot\vDash\top$, or $\bot\vDash\bot$), but a valid argument beginning from true premises can not lead to a false conclusion ($\top\nvDash\bot$). – Graham Kemp Oct 29 '15 at 00:49