# Logic terminology question **URL:** \ **Category:** Factual Questions\ **Created:** [April 17, 2012, 6:32pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068 "2012-04-17T18:32:56Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [April 17, 2012, 6:32pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/1 "2012-04-17T18:32:56Z")
In a thread in GD (I think GD at least) **colonial** links to a [page](http://instruct.westvalley.edu/lafave/goodbadlogic.htm) about the terminology of logic, but one of its statements is confusing. (The argument in the thread itself is irrelevant) > [@](#): > > Inductive arguments with good logic are those in which the premises make the conclusion likely. Such arguments are usually called **strong**. Inductive arguments with bad logic are called **weak**. Weak arguments are those in which the arguer claims the premises make the conclusion likely, but the arguer is mistaken: the conclusion isn’t really likely. I thought strong induction was when P(k), P(k+1), P(k+2)… P(n) are true, the P(n+1) can be shown to be true, whereas weak induction just says if P(n) is true, P(n+1) can be shown to be true. Rr as my instructor at the time said, if you can get from any arbitrary stair in the staircase to the next, it’s weak induction. If you can only get to the next stair after travelling (at least some if not all of) the previous ones, it’s strong. The page, on the other hand, seems to be using “weak induction” to mean “unsound.” A few google searches for “weak vs strong induction” confirm what I was taught. Whence the disparity? Is it a difference between, say, the logic terminology of philosophy and mathematical logic? Is the page just wrong? ---
**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:** [April 17, 2012, 6:48pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/2 "2012-04-17T18:48:15Z")
> [@Jragon](#): > > I thought strong induction was when P(k), P(k+1), P(k+2)… P(n) are true, the P(n+1) can be shown to be true, whereas weak induction just says if P(n) is true, P(n+1) can be shown to be true. You’re thinking of **mathematical induction** , a mathematical proof technique, which is not the same thing as inductive reasoning. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [April 17, 2012, 7:03pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/3 "2012-04-17T19:03:02Z")
> [@Thudlow\_Boink](#): > > You’re thinking of **mathematical induction** , a mathematical proof technique, which is not the same thing as inductive reasoning. But they stem from the same type of reasoning. Why use such wildly different definitions for the same terms? After all, deductive proofs use “valid” and “invalid” (and soundness etc) the same way. ---
**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:** [April 17, 2012, 7:37pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/4 "2012-04-17T19:37:28Z")
> [@Jragon](#): > > But they stem from the same type of reasoning. No, they’re not really the same type of reasoning. Inductive reasoning: I’ve seen a lot of crows, and all the crows I’ve observed have been black. Therefore I conclude that (it is likely that) all crows are black. Mathematical induction-style proof: I’ve seen one or more crows, which have been black. Furthermore, I can demonstrate that all other crows are gentically related to the ones I’ve observed, in such a way that they must have the same color. Therefore I conclude that (I know for sure) all crows are black. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [April 17, 2012, 7:39pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/5 "2012-04-17T19:39:55Z")
Confusingly, mathematical induction is an example of _deductive_ reasoning. ---
**Author:** ![cynyc](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@cynyc](https://boards.straightdope.com/u/cynyc)\ **Post date:** [April 17, 2012, 7:41pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/6 "2012-04-17T19:41:45Z")
> [@Thudlow\_Boink](#): > > No, they’re not really the same type of reasoning. > > Inductive reasoning: I’ve seen a lot of crows, and all the crows I’ve observed have been black. Therefore I conclude that (it is likely that) all crows are black. > > Mathematical induction-style proof: I’ve seen one or more crows, which have been black. Furthermore, I can demonstrate that all other crows are gentically related to the ones I’ve observed, in such a way that they must have the same color. Therefore I conclude that (I know for sure) all crows are black. Um. Could this be dumbed down to the term _hypothesis_? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [April 17, 2012, 7:43pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/7 "2012-04-17T19:43:56Z")
> [@Thudlow\_Boink](#): > > No, they’re not really the same type of reasoning. > > Inductive reasoning: I’ve seen a lot of crows, and all the crows I’ve observed have been black. Therefore I conclude that (it is likely that) all crows are black. > > Mathematical induction-style proof: I’ve seen one or more crows, which have been black. Furthermore, I can demonstrate that all other crows are gentically related to the ones I’ve observed, in such a way that they must have the same color. Therefore I conclude that (I know for sure) all crows are black. I don’t think that’s really a mathematical induction style of proof. It’d have to be something like: Put all the crows in order. The first crow is black. If any crow is black, the next one is black as well. Therefore, all the crows are black. The argument you gave may be construed as working towards a demonstration of the third line above. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [April 17, 2012, 7:46pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/8 "2012-04-17T19:46:11Z")
> [@cynyc](#): > > Um. Could this be dumbed down to the term _hypothesis_? Hypotheses are produced by a kind of inductive reasoning (namely, “inference to the best explanation”) so in that sense, yes. “Inductive reasoning” is a very broad term, encompassing many different kinds of arguments. The only thing they all have in common is that the premises are intended to give the conclusion merely probable support. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [April 17, 2012, 7:47pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/9 "2012-04-17T19:47:33Z")
How _did_ mathematical induction come to be called mathematical induction? I assume that “induction” in the broader sense came first, and “mathematical induction” was constructed from that concept somehow. But is that not right? ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [April 17, 2012, 7:54pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/10 "2012-04-17T19:54:59Z")
> [@Jragon](#): > > I thought strong induction was when P(k), P(k+1), P(k+2)… P(n) are true, the P(n+1) can be shown to be true, whereas weak induction just says if P(n) is true, P(n+1) can be shown to be true. Why aren’t “strong” and “weak” the other way around? I would have expected to be able to say something like “I couldn’t prove that P(N) implies P(N+1). I could only prove the weaker statement that P(N-1) & P(N) together imply P(N+1)”. ---
**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:** [April 17, 2012, 7:56pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/11 "2012-04-17T19:56:49Z")
> [@Frylock](#): > > I don’t think that’s really a mathematical induction style of proof. > > It’d have to be something like: > > Put all the crows in order. > The first crow is black. > If any crow is black, the next one is black as well. > Therefore, all the crows are black. Well, yeah, your example is closer to the way (“weak”) mathematical induction actually works. It’s typically used to prove a statement about an arbitrary positive integer, such as “For any positive integer _n_, the _n_th crow is black.” In my example, I was trying to highlight the difference between an inductive argument and a proof that worked similarly to the way mathematical induction works. ---
**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:** [April 17, 2012, 8:05pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/12 "2012-04-17T20:05:20Z")
> [@Frylock](#): > > How _did_ mathematical induction come to be called mathematical induction? I assume that “induction” in the broader sense came first, and “mathematical induction” was constructed from that concept somehow. But is that not right? I don’t know for sure, but I think so. Following up a footnote in the Wikipedia article led me to [this page](http://www.earlham.edu/~peters/courses/logsys/math-ind.htm), which the OP may wish to consult if he’s still confused about the distinction between “ordinary” induction and mathematical induction. It says > [@](#): > > “Mathematical induction” is unfortunately named, for it is unambiguously a form of deduction. However, it has certain similarities to induction which very likely inspired its name. It is like induction in that it generalizes to a whole class from a smaller sample. In fact, the sample is usually a sample of one, and the class is usually infinite. Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class. Hence the conclusion of a mathematical induction does not contain more information than was latent in the premises. Mathematical inductions therefore conclude with deductive certainty. Anybody know the origin(s) of the term “mathematical induction”? ---
**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:** [April 17, 2012, 8:12pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/13 "2012-04-17T20:12:17Z")
> [@ZenBeam](#): > > Why aren’t “strong” and “weak” the other way around? > > I would have expected to be able to say something like “I couldn’t prove that P(N) implies P(N+1). I could only prove the weaker statement that P(N-1) & P(N) together imply P(N+1)”. IIUC it’s because “P(k), P(k+1), P(k+2)… P(n) are true” is a stronger statement than just “P(n) is true.” The stronger hypothesis does, as you point out, make the conclusion P(n+1) easier to deduce. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 17, 2012, 8:25pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/14 "2012-04-17T20:25:14Z")
For what it’s worth, I think the etymology in both cases is something like “induction” as marching from claims about claims about particular instances\* to claims about all instances (“All…”, “Every…”, “Whenever…”). Which both mathematical and non-mathematical induction do. \*: (Albeit, in the mathematical case, this may mean _arbitrary_ particular instances) ETA: Oh, damn, I started writing that 20 minutes ago, then went off and came back, and didn’t realize new posts had been made in the meanwhile. Well, [here](http://pballew.blogspot.com/2009/09/mathematical-induction-brief-history-of.html). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 17, 2012, 9:41pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/15 "2012-04-17T21:41:29Z")
> [@Frylock](#): > > > [@Thudlow\_Boink](#): > > > > No, they’re not really the same type of reasoning. > > > > Inductive reasoning: I’ve seen a lot of crows, and all the crows I’ve observed have been black. Therefore I conclude that (it is likely that) all crows are black. > > > > Mathematical induction-style proof: I’ve seen one or more crows, which have been black. Furthermore, I can demonstrate that all other crows are gentically related to the ones I’ve observed, in such a way that they must have the same color. Therefore I conclude that (I know for sure) all crows are black. > > I don’t think that’s really a mathematical induction style of proof. > > It’d have to be something like: > > Put all the crows in order. > The first crow is black. > If any crow is black, the next one is black as well. > Therefore, all the crows are black. > > The argument you gave may be construed as working towards a demonstration of the third line above. Think of it as induction on the length of the path from the first crow to any other crow on the crow family tree. > [@ZenBeam](#): > > Why aren’t “strong” and “weak” the other way around? > > I would have expected to be able to say something like “I couldn’t prove that P(N) implies P(N+1). I could only prove the weaker statement that P(N-1) & P(N) together imply P(N+1)”. Strong induction makes a stronger assumption than weak induction. ---
**Author:** ![Kimmy\_Gibbler](https://avatars.discourse-cdn.com/v4/letter/k/bbe5ce/32.png) [@Kimmy\_Gibbler](https://boards.straightdope.com/u/Kimmy_Gibbler)\ **Post date:** [April 17, 2012, 9:46pm UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/16 "2012-04-17T21:46:49Z")
> [@ZenBeam](#): > > Why aren’t “strong” and “weak” the other way around? > > I would have expected to be able to say something like “I couldn’t prove that P(N) implies P(N+1). I could only prove the weaker statement that P(N-1) & P(N) together imply P(N+1)”. Strong induction IF for every natural number _n_ it is true that … if all natural numbers less than _n_ are in _S_, then _n_ is also in _S_ THEN _S_ = **N** Weak induction IF if any natural number _n_ is in _S_, then _n+1_ is also in _S_ and 0 is in _S_ THEN _S_ = **N** Strong induction is called strong because you just have to prove the general rule that for every natural, the assumption that all smaller naturals are in _S_ entails that _n_ is in _S_. If you can do that, you have shown that _S_ is **N**. For weak induction, you have to show the general rule that the if _n_ is in _S_ then _n+1_ is in _S_. Then you have to prove an initial case (usually 0, sometimes 1). Only then have you shown that _S_ is **N**. Mathematicians and logicians think having to show that something holds for a particular case makes you less of a man, and therefore call that method weak induction. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 18, 2012, 1:00am UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/17 "2012-04-18T01:00:49Z")
> [@Kimmy\_Gibbler](#): > > Strong induction is called strong because you just have to prove the general rule that for every natural, the assumption that all smaller naturals are in _S_ entails that _n_ is in _S_. If you can do that, you have shown that _S_ is **N**. Nope. The inductive step holds vacuously for any set whose membership criteria are never satisfied, so you need to establish a base case here as well. > [@](#): > > Mathematicians and logicians think having to show that something holds for a particular case makes you less of a man, and therefore call that method weak induction. This is nonsense and bullshit. ---
**Author:** ![Kimmy\_Gibbler](https://avatars.discourse-cdn.com/v4/letter/k/bbe5ce/32.png) [@Kimmy\_Gibbler](https://boards.straightdope.com/u/Kimmy_Gibbler)\ **Post date:** [April 18, 2012, 1:19am UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/18 "2012-04-18T01:19:41Z")
> [@ultrafilter](#): > > Nope. The inductive step holds vacuously for any set whose membership criteria are never satisfied, so you need to establish a base case here as well. I’m not sure that is so. Do you agree that this is the principle of strong induction (where _A_ is the universal quantifier, _e_ is membership, and _=\>_ is the conditional)? (An)[(Am)(m\ meA) =\> neA] Therefore: A = **N** Then, taking the part before the “Therefore:” as true, one implication is (Am)(m\<0 =\> meA) =\> 0eA Since, under your hypothesis, the membership criteria of A are never satisfied, the consequent is false. To preserve the truth of the conditional, the antecedent must be false. But, (Am)(m\<0 =\> meA) cannot be false, because there is no natural less than 0. Accordingly, the part above the “Therefore:”, the inductive step, is inconsistent with A being the null set. > [@](#): > > This is nonsense and bullshit. Ligthen up, Francis. 'Twas a joke. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 18, 2012, 6:41am UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/19 "2012-04-18T06:41:01Z")
> [@ultrafilter](#): > > Nope. The inductive step holds vacuously for any set whose membership criteria are never satisfied, so you need to establish a base case here as well. Huh? What **Kimmy Gibbler** said is correct; one doesn’t need a separate base case. Show P(0) is exactly the same as showing True -\> P(0), which is exactly the same as showing (For all x \< 0, P(x)) -\> P(0), which is already handled as just a special instance of the general pattern **Kimmy Gibbler** set out. The induction principle “(For all y, (For all x \< y, P(x)) -\> P(y)) -\> for all z, P(z)”, with no special handling of base cases, holds true in any well-founded order; indeed, it serves as one common constructive definition of well-foundedness. ---
**Author:** ![TATG](https://avatars.discourse-cdn.com/v4/letter/t/50afbb/32.png) [@TATG](https://boards.straightdope.com/u/TATG)\ **Post date:** [April 18, 2012, 11:41am UTC](https://boards.straightdope.com/t/logic-terminology-question/619068/20 "2012-04-18T11:41:50Z")
> [@Indistinguishable](#): > > ETA: Oh, damn, I started writing that 20 minutes ago, then went off and came back, and didn’t realize new posts had been made in the meanwhile. Well, [here](http://pballew.blogspot.com/2009/09/mathematical-induction-brief-history-of.html). One of the articles used for that blog entry is avaliable at [JSTOR](http://www.jstor.org/stable/2972638), it is short but thorough, and not behind a pay-wall. It is well worth a look. On the terms “strong” and “weak” in this context, given two logics, if, for every set of premises, one of the logics entails all the other does, and for some set of premises it entails more than the other, it is said to be the stronger logic. Or, in short, a stronger logic is one that entails more. [Next page](https://boards.straightdope.com/t/logic-terminology-question/619068.md?page=2)