# What is the difference between deductive and inductive reasoning?

I have been scouring over the internet in pursuit of a valid elaboration as to the difference between deductive and inductive reasoning, especially when explained using examples. The content that has people have put on the internet is, to some extend, contradictory.

For example
This website gives an example of induction as this:

Jennifer leaves for school at 7:00 a.m. Jennifer is always on time. Jennifer assumes, then, that she will always be on time if she leaves at 7:00 a.m.

The author perceives jennifer's leaving for school at 7 and her arrival on time as occurences to be treated as a particular instances of the premise they mutually establish, that is, her assumption that she will always be on time if she leaves at 7.

However, the same example can also be perceived like this:
The fact that jennifer leaves for school at 7 and that she is always on time, are premises established from her history of departures and arrivals at school and therefore can be treated as generalizations which mutually are responsible for the conclusion or particular instance of these premises, that is her assumption that she will always be on time if she leaves at 7.

Therefore, it can be argued that the example can be a use of deductive reasoning or inductive reasoning.

I think i'm missing something, can someone help clarify said problem.

## 7 Answers

The formulation is a little unclear. What is meant is that Jennifer has always left for school at 7am until now, and has always been on time until now. It is no premise that Jennifer is always on time, nor that she always leaves at 7am.

Formally we write arguments down as a set of premises P1, P2, ..., Pn and a conclusion C. In this example, the implicit argument could be formalised as:

P1: Jennifer left school at 7am on 2015-08-01 and was on time.
P2: Jennifer left school at 7am on 2015-08-02 and was on time.
P3: Jennifer left school at 7am on 2015-08-03 and was on time.
...
Pn: Jennifer left school at 7am on ... and was on time.

∴ C: Jennifer is on time on any day if she leaves at 7am.

This is the classic form of inductive reasoning. An example of deductive reasoning would be:

P1: Jennifer needs at most 15 minutes to go to school.
P2: School starts at 7:15am.

∴ C: Jennifer is on time on any day if she leaves at 7am.

I would like to illustrate the main difference between deductive and inductive reasoning by the following two standard examples:

1) All humans are mortal. Socrates is a human. Hence Socrates is mortal.

The premiss covers more than the conclusion. If the premiss is true, then also the conclusion is true. That's the typical deductive reasoning from logic.

2) Up to now all observed swans are white. Hence all swans are white.

That's the typical inductive reasoning. It attempts to derive from a set of cases a general conclusion. Inductive reasoning is not validated by any type of logic. The present example shows why: Some time after making this reasoning, black swans have been detected in Australia. Hence the conclusion is false.

Deductive reasoning is successfully applied in all sciences where proofs are possible, i.e. in mathematics and in logic itself. Inductive reasoning is applied in all natural sciences. It serves to form general hypotheses, but it cannot prove them.

Added. The inductive reasoning in the Jennifer-example is:

Up to know Jennifer was always on time at school when leaving at 7. Hence also in the future, she will be on time when leaving at 7.

The conclusion does not hold due to several possible reasons, e.g. due to a change of the bus schedule. Nevertheless, the inductive reasoning gives a plausible hypothesis.

The problem with this question is that induction, aside from the mathematical variety, is not reasoning, it is observation. The associated form of reasoning is abduction, which is tracing of possible causes and the estimation of probabilities by combining observations through the plausibility of potential explanations and informal Bayesian intuition.

For the same reason science cannot proceed on a purely inductive basis, but requires the underlying imposition of mechanism and the restraining notion of falsifiability, there is no such thing as inductive reasoning. Without the embracing notion of probability and risk, mere observation, however complete, does not establish any basis for action or belief.

Inductive reasoning makes the conclusion more or less likely, deductive reasoning makes the conclusion certain or necessarily true, given the assumptions. Inductive reasoning may allow for exceptions, unknows or uncertainties, deductive reasoning admits none of these.

Deductive reasoning always starts from a general premise (something about an entire class of things), and arrives at a specific premise (something about one member of that class). To use the standard examples as mentioned by Jo Wehler in his answer, "All humans are mortal. Socrates is a human. Hence Socrates is mortal." is deductive reasoning. The premise is stated over the group (all humans). Socrates is identified as a member of that group. Deductive reasoning is then applied to declare "Socrates is mortal," because he is a member of a group all of whom is mortal.

Inductive reasoning always starts from a specific premise (something about an individual) or a collection of specific premises, and arrives at a general premise (something about an entire group). Again, using the standard examples, "Every swan I have seen is white. Therefore, all swans are white." is an example of inductive reasoning. The premise is about a collection of specific entities (the set of swans I have seen have all been white). Inductive reasoning leads us to a general premise that all swans are white.

As for your specific example, of Jennifer being on time, your deductive example is cheating of sorts. You state (emphasis mine):

The fact that jennifer leaves for school at 7 and that she is always on time, are premises established from her history of departures and arrivals at school and therefore can be treated as generalizations which mutually are responsible for the conclusion or particular instance of these premises, that is her assumption that she will always be on time if she leaves at 7.

In the bolded statement, you have engaged in inductive reasoning. There is nothing in the original axioms which indicate the truthhood of this statement. You have simply made the intuitive leap to a conclusion.

This points out a key to inductive reasoning: it is very often trivial to take an inductive reasoning claim and break it apart into an inductive part followed by a deductive part. Often, when doing so, the inductive assumptions you get to use are smaller and easier to manage.

As an example in math, consider my favorite theorems: Godel's Incompleteness Theorems. They make some frighteningly deep claims about the abilities of mathematical proofs. They claim that many proofs we desperately wish could exist are, in fact, impossible. This would be a gigantic inductive reasoning leap, given that most people actually believed the contrary. His proof breaks that claim up into two parts. The first is an inductive step: he assumed the Peano axioms of arithmetic. For those who are not familiar, these are pretty benign axioms. Handwaving the exactness, they basically describe the ability to count to a number in a very formal manner. It is rare to disagree with the Peano axioms of arithmetic because they're just so natural. Godel then proceeded to attach a giant block of deductive reasoning that proved his claims about the limits of proofs.

This bothered many mathematicians. Godel's results were not popular at the time, but he had succeeded as isolating the inductive portions of his theory to only the most basic of concepts no mathematician dared challenge.

(As a result, many mathematicians have challenged the basic assumptions, and in doing so dodged Godel's claims. However, they have to continuously be cautious. If they accidentally accept his assumptions along the way, the deductive part of his proof comes back in full force)

• "Deductive reasoning always starts from a general premise (something about an entire class of things), and arrives at a specific premise" this is false. '2 is prime; therefore 2 is prime' is deductively valid, but doesn't start from a general premise. And in general your characterization of deductive really weird. Typically we characterize deductively valid arguments as those whose premises can't be true without the conclusion also being true. – possibleWorld Aug 8 '15 at 22:41
• @possibleWorld I would argue that your example is the corner case. "A is A" is a very unusual case to work with. As for your definition of what is "typically characterized," I think your version may be clearer, this is just the way I was taught, by multiple teachers. – Cort Ammon Aug 8 '15 at 23:52
• I'm not sure I know what you mean by corner case. I guess my concern is that casting the deductive/inductive distinction in terms of generality vs specificity of premises doesn't get to the heart of the matter. Part of what we're interested in when we do logic is truth-preservation, and given that it makes sense to talk about arguments which necessitate the truth of their conclusions, given their premises (the deductively valid arguments), vs arguments which don't (the inductive ones). – possibleWorld Aug 9 '15 at 0:04

I think the most concise and informative description I have encountered of the difference between inductive and deductive reasoning is from D Q McInerny's Being Logical, section titled "17. Inductive Argument". He distinguishes the two as follows: Deductive reasoning is productive of necessary conclusions, while inductive argument has the capacity to produce probable conclusions only.

In deductive reasoning, we have a single starting point (major premise), which is assumed to be an established fact - it is always a general statement, e.g. "every natural number except zero is either odd, or even." Given then a minor premise, e.g. "3 is not divisible by two", we can then deduce with absolute certainty that 3 is an odd number. Deductive reasoning can be done with absolute certainty in certain, simple, formal systems - I won't say all formal systems however. :)

In contrast, the premises of an inductive argument are all the particular facts that serve as a body of evidence. "Every day since the creation of the Earth, the sun has rose in the morning". From this body of evidence, we make a reliable generalisation about the data e.g. "The sun will probably rise in the morning". The entire scientific enterprise rests on inductive reasoning. Note, we can never be certain about our inductive conclusions, just assign them various degrees of probability based on the evidence available to us.

In your example above. There has never been an instance where Jennifer has been late if she leaves for school at 7am. Thus, from this body of evidence, you can derive, inductively, a conclusion with a probability of 1. That is, it is certain that Jennifer will be on time if she leaves at 7am. This inductive conclusion now has the status of an established fact and can serve as the basis of the major premise of a deductive argument. This process of going from inductive conclusion, to major premise of a deductive argument is similar to how Newton's Law of Gravitation developed from careful empirical astronomical observation.

Of course, in the real world, we could never assign a probability of 1 to Jennifer being on time, for obvious reasons. So, this example is a bit contrived and misleading. Even "Natural Laws", like Newton's Law of Gravitation, do not allow us to predict the outcome of all events with absolute certainty, e.g special relativity is needed for speeds close to the speed of light etc.

Deductive reasoning would have to find a logical reason why she must be on time if she leaves at 7am. For example, if the school opens at 8am, and the walk to school takes only 30 minutes, that would be a logical reason. (Although the school might decide to open at 7:25am in the summer when it's really hot, for example, so the logic wouldn't be necessarily correct).

The inductive reasoning says: Jennifer was always on time when she left at 7am and we bothered checking, so we conclude that she will always be on time if she leaves at 7am.

I'd want to know more. If Jennifer just started at the school and has gone to school four times so far, then without further information I wouldn't assume she will always be on time. If she went to school for years, and there were cases where she had less than two seconds spare time before the school opened, then I would also not assume that she will always be on time.