Talk:Entailment
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I think where the contrast is drawn in this entry between entailment and implication, this should really be a constrast between entailment and Grice's more specific term "implicature".
The things said about implication are possibly true of some uses of that word, but not of its use in mathematical logic.
I suppose in fact the same kind of point can be made about the entry as a whole. The principal uses of the term "entailment" in mathematical logic are not associated with pragmatics but with truth conditional semantics. In that context A entails B if the truth conditions for B are a subset of those for A.
I am not myself well versed in pragmatics, and am not familiar with its use in pragmatics.
Roger Jones (rbj at rbjones.com)
Can anyone explain what "model of" is supposed to mean? How do models apply to logic?
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[edit] Examples
This entry is woefully in need of examples. There seems to be a contextual nuance that differentiates semantic from logical entailment, as well as from the meaning in pragmatism. This nuance isn't conveyed when the same words are used in all of the definitions. —BozoTheScary 16:11, 2 May 2006 (UTC)
- I have some questions regarding this page, and I believe it has to do with your asking for more examples. First of all, what is a frikin “model”?? this thing is so vague and mysterious. And the “model theory” page doesn't help at all. People only talk about high-order logics and that damn “samantics”... Can't we have an explanation of all this in predicate logic?... Can't we frist explain things in a way that people can uderstand, and just then talk in that beautiful way that is so abstract and general?...
- And regarding that two examples given, the first is so silly and meaningless. It's just a tiny syllogism. And it's damn small, There is no way for us to imagine when it would NOT hold. And then the second example requires the readers to understand what the heck is an “empty model”, and I could not, since I don't even imagine what a model is. -- 200.185.249.203 (talk) 23:43, 15 March 2008 (UTC)
[edit] Implication question
What is the general consensus? Premise: A implies B. Conclusion: Not A implies not B. (Or would it be "Not A does not imply not B"?) And from that first conclusion, can it then be said "Not B implies not A" or even "B implies A"? -- 24.153.226.102 18:10, 28 February 2006 (UTC)
- I'm not sure that this is the right place for this particular discussion, but your conclusions are incorrect.
- Given the premise "A implies B", we may validly conclude that "Not B implies not A". However, it does not generally follow that "Not A implies not B"; nor does it follow that "Not A does not imply not B", which is equivalent to "Not (Not A implies not B)".
- One way to understand these assertions is to take as example propositions: A="Charlie's age is in {2,3}"; B="Charlie's age is in {2,3,4,5}". Then your premise: "A implies B" holds, since if Charlie's age is in {2,3}, it is also in {2,3,4,5}. Your required conclusion, "Not A implies not B", is equivalent to "Charlie's age is not in {2,3} implies it is not in {2,3,4,5}", which is clearly false if Charlie is aged 4 or 5.
- You can use the same example to see that the valid conclusions are plausible (but not to prove them).
- An easy way to picture these propositions is to draw the Venn diagrams of the sets of ages for which A and B are true, and also mark the sets not A and not B. In such pictures, we can identify proposition A with its truth conditions, that is, the set of conditions under which A is true, in this case the set of ages for Charlie {2,3}. The Venn diagram for "A implies B" is then equivalent to the Venn diagram for "A is a subset of B".yoyo 11:27, 25 September 2006 (UTC)
- A implies B
The modus ponens rule is: If A is true then B is true.
The modus tollens rule is: If B is false then A is false.
Those are the only two valid logical inferences of which I am aware in Propositional Logic. capitalist 03:39, 1 March 2006 (UTC)
[edit] Completeness and Godel's Theorem
I don't believe that the reference to completeness and Godel's theorem is correct. Godel's theorem can be applied to logics that satisfy a completeness theorem (e.g. axioms of arithmetic on first order logic). The incompleteness theorem does provide a proposition that can't be proven or disproven but is satisfied by the "natural" model. But there are other models for which the proposition does not hold.
Maybe a better example for an incomplete logic would be the second order logic. Second order logic does not satisfy a compactness theorem. I think that this means that a proof system based on finite proofs will not work.
Turtle59 06:31, 13 March 2006 (UTC)
[edit] Diagram
In the diagram, shouldn't B be a subset of A and not vice versa which is what the diagram is? --Eok20 01:49, 1 July 2006 (UTC)
You're perfectly right and I allowed myself to add a subtext to the diagram, as well as correct the conclusion drawn out of it, saying now: "every B has to be an A" instead of "every A is a B", as before. - Frank.Lenzer@uni-jena.de 12 October 2006
If this diagram is showing what I think it is then it is wrong according to the "correction" underneath. Assuming the diagram is showing models of A and B then it shows A entails B. The definition of entailment says A entails B if all models of A are also models of B. Therefore A has less models than B and also A's models are a subset of B's models. Assuming the diagram is showing models then this should be cleared up because it refers to "A" and "B" whereas further up the page these are defined as sets of sentences. - b 16th November 2006
[edit] Terms for the arguments of an entailment
Are there specific words for the two arguments of an entailment? If we have that 
, is A the "entailer" or the "antecedent" or the "conditional" or what? Similarly, what is B called?
[edit] What is a "deducibility relation"?
Also, what is the difference between the |= and |- symbols in implication? I'm having a hard time understanding this section... 169.232.78.24 (talk) —Preceding comment was added at 04:42, 11 June 2008 (UTC)
