Inductive and deductive arguments and mathematical induction

Question

I started reading Paul Teller's A Modern Formal Logic Primer. In the first chapter, the book presents the inductive and deductive arguments with the following examples:

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The inductive argument:

1. Adam has smiled a lot today.

2. Adam has not frowned at all today.

3. Adam has said many nice things to people today, and no unfriendly things.

   Therefore

4. Adam is happy today.

The deductive argument:

1. Adam just got an 'A' on his logic exam.

2. Anyone who gets an 'A' on an exam is happy.

   Therefore,

3. Adam is happy.

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If we exclude the 2) in the deductive argument, it turns into an inductive argument. Also, 1) and 2) in the deductive argument are a form of mathematical induction. Meaning, the 1) and 2) are something of:

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(i) P(1) is true, i.e., P(n) is true for n = 1. (e.g. Adam = 1)

(ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true. (Anyone = n+1, Adam = n)

Then P(n) is true for all natural numbers n. (Adam is happy.)

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Since I'm really a newbie at logic, is the above correct? Are there any kind of connections between mathematical induction and deductive arguments? If yes, then given that logical deduction is connected with mathematical induction, with what is logical induction connected?

Answer 1

Consider the argument presented in the OP that is claimed to be similar to mathematical induction:

(i) P(1) is true, i.e., P(n) is true for n = 1. (e.g. Adam = 1)

(ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true. (Anyone = n+1, Adam = n)

Then P(n) is true for all natural numbers n. (Adam is happy.)

In mathematical induction n ranges over the domain of natural numbers which have a first element (1) and are connected by an order relationship so that given some natural number, n, we can find the next one. This allows mathematical induction to cover all of the natural numbers once one has shown something about the first one and then given an arbitrary natural number that also has that something shown that the next one has that something as well.

Consider the example. In the first sentence the domain is unclear. Certainly Adam is a member of the domain. In the second sentence there is an "anyone" assuming there are more elements in the domain than Adam. In the conclusion, all we know is that Adam is happy. Are there any others in the domain besides Adam? Perhaps Adam is the only element in the domain.

Also, assuming there are others in the domain besides Adam, say Mary, is there an ordering on the elements in the domain like in the natural numbers so that after Adam we have another element, say Joe, and before Mary there is someone, say, Jane? It is unlikely that this ordering exists.

Finally, the first sentence says that Adam is 1. That means that Adam is not n unless n is identical to 1. For the inductive step in mathematical induction, we take an arbitrary n from the natural numbers, which would correspond to an arbitrary person from the domain of people we are considering. We would not take a specific number like 1 or even 10, but an arbitrary natural number and that is why it is left unspecified as n. So claiming Adam is n would not work unless the domain only includes Adam.

Paul Teller provides a good observation about the difference between induction and deduction after warning, "Don't let anyone tell you that these terms have rigorous definitions" (page 3):

We tend to call an argument 'Deductive' when we claim, or suggest, or just hope that it is deductively valid. And we tend to call an argument 'Inductive' when we want to acknowledge that it is not deductively valid but want its premises to aspire to making the conclusion likely.

Even the first statement that "Adam = 1" implying that Adam is happy is inductive. All we know are some premises about Adam listed earlier that

Adam has smiled a lot today.

Adam has not frowned at all today.

Adam has said many nice things to people today, and no unfriendly things.

The conclusion is not deductive because these three premises might be true but Adam might not really be happy. As Teller mentions (page 3), "...the premises do not rule out the possibility that Adam is merely pretending to be happy."

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Reference

Teller, P. (1989). A modern formal logic primer. Prentice Hall. http://tellerprimer.ucdavis.edu/

Answer 2

The claim in question is that the following deductive argument is an instance of mathematical induction, or MI. (Btw, this is your claim, not Teller’s, right?)

Adam just got an ‘A’.
Anyone who gets an ‘A’ is happy.
∴ Adam is happy.

While I know more about logic than mathematics, I find this a very tenuous claim. The deductive argument has this form:

A
Pa
∀x(Px→Qx)
∴ Qa

By contrast, MI takes this form:

MI
P0
∀x(Px→Px+1)
∴∀xPx.

Right away, we note two differences: (i) A contains two predicates, P and Q, while MI contains only one. (ii) MI’s conclusion is a universal claim, while A’s is not. (MI's central purpose is to show that a property is had by all numbers.)

There are two further, conceptually more important differences: (iii) to even formulate MI, we need addition, or at least the notion of successor, but neither is available in FOL. (iv) MI is an axiom of arithmetic, something we add to logic to get PA, for example. A, by contrast, does not rely for its validity on any (non-logical) axioms. To put it more technically, Robinson Arithmetic (for example) doesn’t include MI, but A is still valid in that theory.

In sum: I don’t think that the deductive argument is an instance of mathematical induction.

Answer 3

You are confusing some of the terminology and concepts. Logical induction is a type of reasoning that is different from deductive reasoning. Mathematical induction is the name of a deductive inference procedure in mathematical logic. You seem to be taking the definition of the term “induction” literally where you see it. This is wrong. You should understand that definitions come from context and not dictionaries. The same word can take on different meanings depending on context. However in this case, the name people gave to the inference rule could have been better applied. Let’s begin with the different contexts.

Inductive reasoning is a reasoning pattern that has conclusions that are not certain. That is, the conclusion of an inductive argument is not guaranteed to be true while the premises are true. The conclusion may be true today and the same conclusion false 10 years later. Other factors might make the conclusion true and not just the premises given. You can also have a true conclusion while the premises are false. Think of the conclusion being likely to be true without certainty. Another way to put it is that inductive reasoning is probable. You have heard of statistics or probability haven’t you? That is the conclusion of an argument has a percentile of being correct: i.e., 1% up to 99% of being true. All of the famous sciences fits in this category.

Deductive reasoning is the pattern of reasoning that has certainty in the conclusions if the premise are true. It would be impossible for one to follow the rules and the conclusion of an argument to be false while the premises are true simultaneously. Inductive reasoning does not have this feature. Induction can be correct but this pattern does not ALWAYS give certainty. Already the distinction is one pattern of reasoning (induction) gives truth some of the time and fails some of the time. The other pattern (Deductive reasoning)has rules similar to mathematics and following the rules make it impossible to get the answer wrong. If one gets a wrong answer at least one of the known rules had to be violated. Mathematical induction is a deductive procedure used in reasoning to derive a conclusion in an argument. Mathematical induction is not a distinguishing type of reasoning as inductive reasoning. The reasoning procedure has rules that if followed derive a conclusion that must be true if the premises (this includes assumptions) are true as well. Wikipedia has a helpful write up on mathematical induction. You can also use the Google search engine to give further information about other web pages that discuss mathematical induction further. Many sites dedicated to math will have a description of it.

The example you gave of a deductive argument does not seem to fit the pattern of mathematical induction but of a categorical syllogism. I would write as a categorical syllogism as this: All persons who receive an “A” on an exam are happy persons. Adam has received an “A” on on an exam. Therefore, Adam is a happy person.

A mathematician would translate that into conditional form: If a person receives an “A” on an exam, then he will be a happy person. Adam has received an “A” on an exam. Therefore, Adam is a happy person.

There is no mathematical induction in either case.