# Syllogism question **URL:** \ **Category:** Factual Questions\ **Created:** [July 8, 2012, 12:58am UTC](https://boards.straightdope.com/t/syllogism-question/627568 "2012-07-08T00:58:48Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![supery00n](https://avatars.discourse-cdn.com/v4/letter/s/5f9b8f/32.png) [@supery00n](https://boards.straightdope.com/u/supery00n)\ **Post date:** [July 8, 2012, 12:58am UTC](https://boards.straightdope.com/t/syllogism-question/627568/1 "2012-07-08T00:58:48Z")
Hey, I have a question about whether a pair of syllogisms is equivalent to another. The first two are: -if not A, then not X -if not B, then not X Is: if not A or not B, then not X equivalent to the first two syllogisms? Why or why not? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 8, 2012, 1:18am UTC](https://boards.straightdope.com/t/syllogism-question/627568/2 "2012-07-08T01:18:29Z")
Yes, in most systems of logic, “(A implies C) and (B implies C)” is equivalent to “(A or B) implies C”. Why? Because both of these have the exact same meaning: that either an A or a B can be exchanged for a C, so to speak. ---
**Author:** ![tapu](https://avatars.discourse-cdn.com/v4/letter/t/49beb7/32.png) [@tapu](https://boards.straightdope.com/u/tapu)\ **Post date:** [July 8, 2012, 1:22am UTC](https://boards.straightdope.com/t/syllogism-question/627568/3 "2012-07-08T01:22:54Z")
They’re equivalent. It’s not IF A and B, then not X. It’s if A or B, then not X. The two statements in the first set are not dependent on each other, so you only need A **or** B to result in ~X ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 8, 2012, 1:26am UTC](https://boards.straightdope.com/t/syllogism-question/627568/4 "2012-07-08T01:26:27Z")
Entering the expression {(~A \> ~X) & (~B \> ~X)} = {(~A + ~B) \> ~X} into the truth table generator here [http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/](http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/) verifies that they are equivalent. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [July 8, 2012, 1:34am UTC](https://boards.straightdope.com/t/syllogism-question/627568/5 "2012-07-08T01:34:12Z")
I agree with the other posters with one proviso. You should be using the standard non-exclusive or. If your “or” means either A or B but not both, then they aren’t the same. ---
**Author:** ![supery00n](https://avatars.discourse-cdn.com/v4/letter/s/5f9b8f/32.png) [@supery00n](https://boards.straightdope.com/u/supery00n)\ **Post date:** [July 8, 2012, 1:37am UTC](https://boards.straightdope.com/t/syllogism-question/627568/6 "2012-07-08T01:37:52Z")
Cool, thanks. Is there a specific name for the general class of syllogisms that this particular one is an example of? (i.e. De Morgan’s law, commutative law) ---
**Author:** ![supery00n](https://avatars.discourse-cdn.com/v4/letter/s/5f9b8f/32.png) [@supery00n](https://boards.straightdope.com/u/supery00n)\ **Post date:** [July 8, 2012, 2:08am UTC](https://boards.straightdope.com/t/syllogism-question/627568/7 "2012-07-08T02:08:58Z")
Also another logical question: Say that A and B are both necessary conditions for C, and that there are no additional necessary conditions for C other than A and B. Does this mean that A and B together are sufficient conditions for C? ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [July 8, 2012, 2:18am UTC](https://boards.straightdope.com/t/syllogism-question/627568/8 "2012-07-08T02:18:52Z")
> [@supery00n](#): > > Say that A and B are both necessary conditions for C, and that there are no additional necessary conditions for C other than A and B. Does this mean that A and B together are sufficient conditions for C? Depends on exactly what you mean. It might be the case that A is necessary and B is necessary and either X or Y is necessary. And A and B and either X or Y is sufficient. It’s true that A and B together are not sufficient, but it’s not clear if you could say there are no other necessary conditions. It is true that neither X nor Y is necessary, but it is also true that X or Y is necessary, so it depends on how you interpret “there are no additional necessary conditions.” ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 8, 2012, 2:34am UTC](https://boards.straightdope.com/t/syllogism-question/627568/9 "2012-07-08T02:34:08Z")
> [@supery00n](#): > > Also another logical question: > > Say that A and B are both necessary conditions for C, and that there are no additional necessary conditions for C other than A and B. Does this mean that A and B together are sufficient conditions for C? No. You have C -\> A and C -\> B, but that doesn’t allow you to conclude that any combination of A and B will imply C. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 8, 2012, 4:21am UTC](https://boards.straightdope.com/t/syllogism-question/627568/10 "2012-07-08T04:21:53Z")
> [@supery00n](#): > > Also another logical question: > > Say that A and B are both necessary conditions for C, and that there are no additional necessary conditions for C other than A and B. Does this mean that A and B together are sufficient conditions for C? What does it mean to say “there are no additional necessary conditions for C other than A and B”? One way of reading that is as saying “A and B together are equivalent to C”, in which case, the answer to your question is trivially yes. But it’s not clear that that’s what you meant, since that’s so trivial. ---
**Author:** ![TATG](https://avatars.discourse-cdn.com/v4/letter/t/50afbb/32.png) [@TATG](https://boards.straightdope.com/u/TATG)\ **Post date:** [July 8, 2012, 5:38am UTC](https://boards.straightdope.com/t/syllogism-question/627568/11 "2012-07-08T05:38:40Z")
> [@supery00n](#): > > Also another logical question: > > Say that A and B are both necessary conditions for C, and that there are no additional necessary conditions for C other than A and B. Does this mean that A and B together are sufficient conditions for C? What do you think necessary and sufficient mean? If you are just taking them to accord with (material) implication, then among the necessary conditions will be C -\> C; and so among the necessary conditions will be a sufficient condition. > [@Indistinguishable](#): > > What does it mean to say “there are no additional necessary conditions for C other than A and B”? One way of reading that is as saying “A and B together are equivalent to C”, in which case, the answer to your question is trivially yes. But it’s not clear that that’s what you meant, since that’s so trivial. Saying how the later is a reading of the former might not be so trivial, though. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 8, 2012, 6:23am UTC](https://boards.straightdope.com/t/syllogism-question/627568/12 "2012-07-08T06:23:03Z")
Well, I would say it trivially. I might interpret “There are no additional necessary conditions for Y other than X” to directly mean “X is both necessary and sufficient for X”. What else does “There are no additional necessary conditions” mean?, if not “This is everything; this is all it takes; nothing else is required; this is enough. Aka, this is sufficient.”