5

I'm a newbie to logic so please forgive me if this is a basic question. I've searched the web and stack exchange but can't seem to find an answer.

The book I'm reading on predicate logic (forallx) says that although the argument '${\rm A}$ therefore ${\rm B}$' may be valid, this doesn't mean that ${\rm A}$ necessarily entails ${\rm B}$.

To me, validity and entailment both seem to have the same definition, namely that ${\rm A}$ entails ${\rm B}$ (or an argument is valid) if it is not possible for the premise to be true and the conclusion false.

Clearly my understanding of the above definition is wrong if entailment and validity are in fact different things, but I can't see the difference between them.

Can anyone enlighten me please? And are the definitions different in propositional logic versus predicate logic?

Many thanks

Max

Max
  • 51
  • 2
    I suspect that validity is being used for syntactic consequence and entailment for semantic consequence. In first-order logic these are the same thing, but that's a highly nontrivial fact. Someone who knows more logic can explain it better. – eyeballfrog Mar 05 '21 at 12:16
  • 1
    Thanks. That would make sense since it is in the section on interpretability. I wonder whether the word "therefore" is synonymous with "implies"? – Max Mar 05 '21 at 12:39
  • 2
    It is very unusual for validity not to be a semantic concept. So maybe it could be the other way round from the interpretation suggested by @eyeballfrog. Can you tell us what the book is. – Rob Arthan Mar 05 '21 at 13:44
  • 3
    A sentence is valid. An argument is valid. In a valid argument, the premises entail the conclusion. – Mauro ALLEGRANZA Mar 05 '21 at 17:29
  • 1
    @RobArthan Thank you.you Rob. It's the example on p. 281 of forallx re foxes and vixens: http://forallx.openlogicproject.org/forallxyyc.pdf – Max Mar 06 '21 at 06:22
  • 1
    @MauroALLEGRANZA Thank you. So are the terms 'valid' and 'entail' synonymous/interchangeable? – Max Mar 06 '21 at 06:25
  • 1
    Thanks for providing the link. The book does seem to reserve the term validity as a property of arguments and definitely defines entailment semantically, so @eyeballfrog's suspicions seem to be correct. Be warned that other authors use these terms differently. – Rob Arthan Mar 06 '21 at 15:07
  • I found a link to support @MauroALLEGRANZA statements. https://logiccourse.com/textbook/sample-chapter-1/chapter/1-let-the-adventure-begin/section/1-4-entailment-validity-demo/ – tbhaxor Mar 16 '24 at 14:23

2 Answers2

2

  • OP: "It's the example regarding foxes and vixens on p. 281 of forallx.openlogicproject.org/forallxyyc.pdf " enter image description here

    The author is ostensibly cautioning about the relationship between the validity of an argument and FOL-entailment (first-order-logic entailment), yet their given example is making a different point, which is that one must be careful to retain all pertinent information when symbolising an argument (in the example, the original argument contains an implicit/unstated premise "Every vixen is a fox" that became lost in translation).

  • As for what the author says the relationship between argument validity and FOL-entailment is, from a quick keyword-search of the book, I believe it is this: "In FOL, an argument is valid iff its conclusion is a FOL-entailment of its premises."

ryang
  • 38,879
  • 14
  • 81
  • 179
1

Consider the following instance:

 (0 != 0) therefore (God exists)

Since 0 = 0 the premise is false, hence the implication is valid. Does this prove God exists?

[note: If you happen to believe in God, change the above conclusion to 'God does not exist'. The argument still works]

PMar
  • 11
  • 4
    It means anything follows, including (God exists) $\land$ (God does not exist), and including "aliens from Mars of declared war on Earthlings. So, I do not understand your question. The implication is true, no matter what, but the conclusion need not be true, even though the implication is true. – amWhy Mar 05 '21 at 14:46
  • 1
    @amWhy Newbie question: Why is it useful in logic to have a true implication with a conclusion that may not be true? – Max Mar 06 '21 at 06:33