l a b o r a t o r y f o r a d va n c e d s t u d i e s

Introduction to Mathematical Induction

Hugo Sereno FerreiraDepartment of Informatics Engineering

Faculty of Engineering, University of Porto

hugo.sereno@fe.up.pt

July 13, 2011

Abstract. It is argued that there may be a case where all gambozinos are white.

1 introduction

Most of us learn that there are two types of reasoning, namely deductive and induc-tive. In deductive reasoning, one usually starts from a general observation (a set ofpremises) and arguments towards a specific conclusion. For example, fromthe classicstatement all men are mortal, along with the observation that Socrates is a man, it fol-lows that Socrates ismortal. In amore abstract sense, we are asserting that, should a setof things have a certain property1, and should something belong to that set, then itmust have that same property, i.e. (i) @xP(x)Ñ Q(x), (ii)P(Socrates)$Q(Socrates).

Inductive reasoning, on the other hand, makes generalizations based on individ-ual instances. Imagine that you go outside in a quest to observe the gambozino, ananimal so rare that it only appears minutes before the sunset in a rainy day. You werelucky to catch three young, white gambozinos, drinking water by the lake. Since it isthe first time you see one, you assume that, probably, most gambozinos are white. Inthe following days you repeat the feat, and you now seem to be confident that allgambozinos are white. Hence, (i) P(a) ^ Q(a), (ii) P(b) ^ Q(b), (iii) P(c) ^ Q(c),(iv) …$ @xP(x)Ñ Q(x)?

Of course it doesn’t seemvalid to assume that, only because youhave seen a dozenof gambozinos, all must have the same characteristics. But what if you had observedhundreds, or even millions? Is it sound to begin an argument based on a probability

1This is formally equivalent to @xPPQ(x), but we are being relaxed.

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(most ), and conclude an universal assertion ( all ), due to the sheer number of obser-vations?

2 proving an universal

It was the skeptic Sextus Empiricus who first questioned the validity of inductivereasoning, by positing that a universal rule could not be established from an incom-plete set of particular instances: “When they propose to establish the universal from theparticulars by means of induction, they will effect this by a review of either all or some of theparticulars. But if they review some, the induction will be insecure, since some of the partic-ulars omitted in the induction may contravene the universal; while if they are to review all,they will be toiling at the impossible, since the particulars are infinite and indefinite.”

According to Sextus Empiricus, only if one had observed all gambozinos could oneconclude the universal statement, since even just a single particular would be suffi-cient to disprove the generalization.Therefore, (i) P(a)^Q(a), (ii) P(b)^␣Q(b)$

@xP(x)Û Q(x).

3 monotonicity of entailment

There are other different ways to show that inductive reasoning isn’t a valid form ofargumentation. For example, let us go back to Socrates and the mortality of men.Suppose Socrates is observed to never die2. We should then reject the conclusion,not because of its form, but because of its premises; either Socrates isn’t a man, or notall men are mortal. Should we still accept the premises, then Socrates will (someday)die, by the sheer force of our logic.

Now suppose that we observe more men that don’t die. We may add this factto our list of premises, but we now seem to be in a position of inconsistency. Somemen die, others don’t, and thus being a man isn’t sufficient to guarantee its mortal-ity: P(x) Û Q(x). We can’t even begin to reject the conclusion since our premisesare contradictory. Otherwise, no matter how many new premises you add, the con-clusion is always a direct consequence of the hypothesis.

This characteristic, i.e., that in a consistent argumentation onemay add premiseswithout affecting the validity of the conclusion, is called the monotonicity principle,and one can see that the inductive reasoning violates it: adding a black gambozino re-jects theprevious conclusion, so (i)P(a)^Q(a), (ii)P(b)^␣Q(b), (iii)…&@xP(x)ÑQ(x).

2Wemay have a practical problem with this premise since, to observe that Socrates never dies, onewould have to (i) wait an infinite amount of time, and (ii) be immortal.

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4 there are infinitely many primes

In inductive reasoning, the premises do not guarantee the conclusion, although theymay give it some probability or plausibility. In order to prove an universal claim onehave toobserve every instanceof that claim, or else assume it as a (potentially refutable)hypothesis.

But, mathematicians keep proving things about numbers without actually ob-serving every one of them. For example, while it is only believed that every evennumber is a sum of two primes3, it is actually known that there is an infinite numberof primes. So, if it is true that mathematical induction involves a sort of generaliza-tion, how can we ensure its validity within a logical framework?

Mathematical induction is actually a very different type of reasoning, and the artgoes as back as 2000 years. Euclid is recognized as probably the first one to haveimplicitly used it for proving that there are infinitely many primes. The reasoningis similar to the following: suppose you were searching for prime numbers, and youhad already collected a very fine list of them, p1,p2, . . . ,pn. Let P be the product ofall the prime numbers in the list, P = p1 p2 ¨ ¨ ¨pn. Let q = P + 1. Then, q is eitherprime or not: (i) if q is prime then there is at least one more prime than is listed, and(ii) if q is not prime then some prime factor p divides q. This factor p is not on ourlist: if it were, then it would divide P (since P is the product of every number on thelist); but as we know, p divides P+1 = q. Then pwould have to divide the differenceof the two numbers, which is (P+ 1)´P or just 1. But no prime number divides 1 sothere would be a contradiction, and therefore p cannot be on the list. This means atleast one more prime number exists beyond those in the list. ‚

5 mathematical induction

The above proof is based on a very particular type of structure inherent to naturalnumbers, and it is precisely that structure that allows us to prove something for everynumber, despite there are infinitely many of them. Let us delve a little bit more onhow a proof by induction works before coming back to logic. Suppose you want toprove that the following statement holds for all natural numbers n:

0+ 1+ 2+ ¨ ¨ ¨+ n =n(n+ 1)

2(1)

3Actually, the Goldbach conjecture is a fine example of how mathematical induction is differentfrom simple induction. It states that every even integer greater than 2 can be written as the sum of twoprimes, e.g. 10 = 7 + 3 . But despite no even number ever found violates this rule, the conjecture remainsmathematically unproven.

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The proof consists of two distinct (but intertwined) steps: first, we show that thestatement holds when n is equal to the lowest value that n is given in the originalstatement, that is when n = 0:

0 =0(0+ 1)

2(2)

Thenweneed to show that, if the statement holds for somen, then the statementalso holds in the subsequent of n, i.e. when n+ 1 is replaced by n:

(0+ 1+ ¨ ¨ ¨+ k) + (k+ 1) =(k+ 1)((k+ 1) + 1)

2(3)

Using 1, we can rewrite the left-hand side, so all that remains is to (algebraically)prove the equality:

k(k+ 1)

2+ (k+ 1) =

(k+ 1)((k+ 1) + 1)

2(4)

which is trivial. Therefore 1 holds. ‚

6 the falling of dominoes

The previous application of induction is based on the fact that every natural numberis connected to every other by a known rule: summing. In fact, if you take the lowest ofthe natural numbers 0, and keep adding 1 to the result, youwill eventually reach everynatural number that it exists. Therefore if you prove that (i) if any arbitrary numberk has a propertyP, then k+1must also have that property, and (ii) the lowest of thosenumbers has P, then it follows that every number n has that property.

And here lies the slight of hand! Mathematical induction is similar to the sequen-tial effect of falling dominoes. Put every one of them in a line, and prove that, if anarbitrary domino falls, the one next to him must fall4. Then push the first one and,voilá: every one of them falls.

7 to induction or not induction

We have argued that a consistent logic cannot support inductive conclusions basedon the observation of instances, so how ismathematical induction reconciliatedwithlogic? The trick is very simple; mathematical induction on natural numbers is actu-ally a form of deductive reasoning, as shown in the following second-order clause:

4This may seem tricky, but you could assume some form of consistent newtonian physics, and afixed distance between pieces.

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@P[P(b)^ (@kPNP(k)Ñ P(k+ 1)]Ñ @nPNP(n) (5)

where P is any proposition, b, k and n are natural numbers, and b assumes the low-est value for which P holds (usually 0 or 1). Remember Socrates and the mortality ofmen? The concept here is very similar. One asserts that, for every proposition P, ifan individual has a certain property, the next individual also has that property. Sincewe know (by observation) that the first individual in a series has the property, thenit follows that every individual has that same property. The universal statement is aconsequence of the established premises, and not a generalization based on individ-ual case analysis. ‚

8 all gambozinos are white

This form of induction does not necessarily involve numbers; one can actually gener-alize it to any type ofwell-founded structure, i.e., any structure whose elements relateto each other in a finite number of ways, essentially creating a “chain”. Back to gam-bozinos, imagine that you are able to assert that (i) if a gambozino is white, its descen-dants will always bewhite, and (ii) the first two gambozinos to exist werewhite5. Then,by the nature of the structure that rules the gambozino ascendency, all gambozinos areproven to be white6.

9 conclusion

Mathematical induction is a powerful tool in deductive reasoning that allows toproveproperties of an infinite number of elements without having to actually observe ev-ery one of them. It works whenever the elements we are dealing with are part of awell-founded relation, andwe are able to assume properties over that relation. Math-ematical induction is thus well beyond inductive reasoning, able to assert the veracityof an argument over its mere plausibility.

5There are actually premises thatwe’ve disregarded for the sake of simplicity, such as (iii) except forthe first two of them, a gambozino can only exist through sexual reproduction, and (iv) the parenthoodof a gambozino is an anti-simetric, anti-reflexive and anti-transitive relation.

6Unless genetic manipulation is allowed, but then you would be attacking the premise, not theconclusion.

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