Is mathematical induction a kind of inductive reasoning?

No, despite the name. Psychologists Lance Rips and Jennifer Asmuth explain that mathematical induction is a deductive technique. You prove a base case, then prove that if a statement holds for any number k it also holds for k + 1, and the conclusion then follows for all natural numbers by strict deduction. The shared name is a loose analogy, since both try to establish a general statement.

Can the same argument be both deductive and inductive?

On some definitions, yes. The Internet Encyclopedia of Philosophy points out that if the type of argument depends on what the person making it intends, then one argument can count as deductive for one person and inductive for another. That's one reason the article argues the distinction is less clear-cut than textbooks suggest.

What is the problem of induction?

It's a puzzle raised by David Hume in A Treatise of Human Nature (1739) and his Enquiry (1748). The Stanford Encyclopedia of Philosophy explains that Hume asked what justifies moving from past observations to beliefs about cases we haven't observed, and argued that neither demonstrative nor probable reasoning can supply that justification.

Who came up with abductive reasoning?

The Stanford Encyclopedia of Philosophy traces the term to the philosopher Charles Sanders Peirce, who used it for reasoning that generates new hypotheses. Today it more often means inference to the best explanation, and the SEP notes that Peirce's own proposal doesn't capture what most people mean by abduction now.

Deductive vs. Inductive Reasoning

Deductive reasoning aims for certainty: if the premises are true, the conclusion has to be true. Inductive reasoning aims for probability: true premises make the conclusion likely, but it could still turn out false. The Internet Encyclopedia of Philosophy (IEP) defines a valid deductive argument as one whose structure guarantees a true conclusion whenever the premises are true. A strong inductive argument is one whose premises make the conclusion likely (IEP). A third type, abductive reasoning, infers the best available explanation for what you observe (Stanford Encyclopedia of Philosophy).

Deductive reasoning

In a deductive argument, the premises, if true, give conclusive evidence for the conclusion (Lander University). The classic example, used by both the IEP and Lander (IEP; Lander University):

All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

Deductive arguments are judged by two standards.

Valid means the argument’s form makes it impossible for the premises to be true and the conclusion false (IEP). Validity is about structure, not about whether the premises are actually true (IEP).
Sound means the argument is valid and all its premises are true. A sound argument always has a true conclusion (IEP).

An argument can be valid and still reach a false conclusion. The IEP’s example: “All toasters are items made of gold. All items made of gold are time-travel devices. Therefore, all toasters are time-travel devices.” The form works, but the premises are false, so the argument is valid but not sound (IEP).

Inductive reasoning

Inductive reasoning moves from what you’ve observed to what you haven’t, or to general laws. The Stanford Encyclopedia of Philosophy (SEP) gives the example of bread. Because bread that looks a certain way has nourished you so far, you expect the next similar piece will too (SEP). The conclusion is reasonable, but it isn’t guaranteed. It’s not a contradiction for the next piece of bread to fail to nourish you (SEP).

The IEP’s example shows the structure:

Most Greeks eat olives. Socrates is a Greek. Therefore, Socrates eats olives.

Inductive arguments have their own pair of standards (IEP):

Strong means the premises make the conclusion likely.
Cogent means the argument is strong and its premises are true.

Strength comes in degrees. The probability of an inductive conclusion can range from very low to very high but never reaches 100 percent, and adding evidence can make it more probable (Lander University). New premises can strengthen or weaken an inductive argument, while they can’t change whether a deductive one is valid (IEP).

Abductive reasoning

Abduction is usually described today as “inference to the best explanation” (SEP). For example, say you come home to wet footprints in the hall and a dripping umbrella by the door. You conclude someone came in from the rain, because that explains what you see better than the alternatives.

Like induction, abduction is ampliative. Its conclusion goes beyond what’s contained in the premises, so it can be wrong even when the premises are true. The difference is that abduction relies on what would explain the evidence, while induction relies on observed frequencies or statistics (SEP).

Comparison table

	Deductive	Inductive	Abductive
If the premises are true…	The conclusion must be true	The conclusion is probably true	The conclusion is the best available explanation
A good argument is called	Valid	Strong	(judged by how well it explains)
Good, with true premises	Sound	Cogent	n/a
Can new evidence change the verdict?	No, validity stays fixed	Yes, strength can rise or fall	Yes, a better explanation can replace it
Example	All men are mortal; Socrates is a man; so Socrates is mortal	Most Greeks eat olives; Socrates is Greek; so he eats olives	Wet footprints and a dripping umbrella; so someone came in from the rain

Sources for the table: IEP, Deductive and Inductive Arguments; IEP, Validity and Soundness; SEP, Abduction.

Where each kind is used

Math

Mathematical proof is deductive. Psychologists Lance Rips and Jennifer Asmuth note that a proven conclusion shares one property with every deductively valid conclusion: it must be true whenever the premises are true (Rips and Asmuth). Induction still plays a part. Students and mathematicians try examples and use rules of thumb to come up with conjectures, and then they prove them deductively. The mathematician Polya called the two “demonstrative reasoning and plausible reasoning” and said they complete each other (Rips and Asmuth).

Science

Science uses all three. Inferences from observations to general laws are inductive (SEP). The hypothetico-deductive method works out what should be observed if a hypothesis is true, then tests it. If the predicted result fails to appear, the hypothesis can be rejected. If it does appear, that supports the hypothesis without proving it (SEP). Abduction shows up too. The SEP describes astronomers proposing an undiscovered planet to explain oddities in the orbit of Uranus. Neptune was found later (SEP).

Everyday life

Everyday arguments are often hard to sort cleanly into deductive or inductive (IEP). Plenty of daily reasoning is inductive (“the bus has been late every morning this week, so I’ll leave early”) or abductive (“the car won’t start and the lights are dim, so it’s probably the battery”). The SEP notes abduction’s role in medical diagnosis, too (SEP).

Common mistakes

Believing that “deduction goes from general to specific, induction from specific to general.” This rule of thumb is common, but Lander University’s logic guide says it is “no longer considered correct in logic” (Lander University). The IEP calls it “much too crude.” Under that rule, “All As are Bs; all Bs are Cs; so all As are Cs” would be neither deductive nor inductive, because it uses only general statements. Reasoning from the spiders examined so far to the next spider would land in neither category, too (IEP).
Treating “valid” as “true.” A valid argument with a false premise can lead to a false conclusion, as the toaster example shows. An invalid argument can also happen to have a true conclusion (IEP).
Affirming the consequent: “If it rained, the street is wet. The street is wet. So it rained.” This looks deductive but is invalid, since a sprinkler could have wet the street. The IEP notes this formal fallacy is often mistaken for modus ponens, a valid form (IEP).
Hasty generalization. The IEP says its key error is overestimating how strong an argument is when it rests on too small a sample (IEP). Two friendly neighbors don’t tell you about a whole city.
Treating examples as proof. Rips and Asmuth report that college students asked to prove simple claims about numbers often offer “proof by multiple examples,” and that people sometimes rate these example-based arguments more favorably than correct proofs (Rips and Asmuth).
Taking the best explanation as certain. Abduction isn’t necessary inference (SEP). The best explanation you can think of may not be the right one, especially if you only looked for evidence that fits it. Our guide to confirmation bias explains that trap.

Where philosophers disagree

The definitions above are widely used, but the line between deduction and induction is contested. Lander’s guide defines the two by how much support the premises give: conclusive evidence for deduction, some evidence for induction (Lander University). The IEP article reviews several competing ways of drawing the line, including by the arguer’s intentions, by what the argument claims, and by whether extra premises could change its strength. It finds problems with each one and suggests that philosophers may be better off evaluating arguments without the categorical distinction (IEP). Either way, the useful question in daily life is how strongly the reasons support the conclusion, and asking it is a big part of critical thinking.