# Can all inductive arguments be written as deductive arguments?

Whenever I see inductive arguments being used, it seems as though they can be redone by simply making certain assumptions and rephrasing the argument as a deduction from those assumptions.

For example, in this Khan Academy video, Sal says that if you're predicting the population of a town in the future based on the past, that's inductive reasoning. However, I believe that you could also frame this as deductive reasoning by adding in the assumption that "If the population of a town followed a particular pattern up until today, then it will continue to follow that pattern past today".

In fact, it seems that any inductive reasoning can be done with deductive reasoning by adding in some assumption that a particular pattern continues to hold.

Is this the case? If so, is there a formal way of determining what assumptions need to be added? If not, can you give an example of a situation with inductive reasoning where deductive reasoning with certain assumptions cannot model the same reasoning and conclusions?

• The trick is to figure out what "particular pattern" to assume, that is non-deductive and non-trivial, that is the induction (now more commonly called abduction). The rest is a triviality, and can indeed be rephrased deductively (if it works). Commented Dec 6, 2018 at 22:31
• If this were a legitimate meaning of 'deduction', just adding whatever premise you need in order to get the result you want, then you could just take the conclusion as a premise, and all arguments would be deductions. They would also all be valid. Deduction is done after the premises are stated. Popper suggests that science is done by hypothesizing what most powerful premise would best render your induction into a deduction, and seeing if its other consequences bear out. But science is still not deduction.
– user9166
Commented Dec 7, 2018 at 0:39
• @Conifold What if I just assumed that every pattern will continue? Then all I have to deduce is that there is a pattern, and then by that assumption, I can deduce that that pattern will continue. Is there any reason why assuming every pattern will continue is a bad way of rephrasing induction as deduction? Commented Dec 7, 2018 at 1:01
• There is no such thing as "every pattern". How would 1,2,4,... continue? There are infinitely many incompatible patterns, they can't all continue. Commented Dec 7, 2018 at 1:51
• And adding assumptions is induction. Period. It does not matter whether they are easy or hard, they are still not deductions.
– user9166
Commented Dec 11, 2018 at 15:30

...it seems that any inductive reasoning can be done with deductive reasoning by adding in some assumption that a particular pattern continues to hold.

You got it exactly right. The assumption is the Uniformity Principle. Some philosophers have accepted the principle as supported by common observation; others have dropped wet blankets all over it.

According to John Stuart Mill, the “ultimate major premise of all inductions” is the “uniformity of the course of nature” (Mill, p. 224). If a given occurrence caused a certain result, then future similar occurrences will cause similar results (Hume, ¶29).

But Hume concluded that the Uniformity Principle assumes what it sets out to prove (¶30) and ascribed it to “general habit, by which we always transfer the known to the unknown…” (¶84, fn. 19). Mill was satisfied that when “we consult the actual course of nature, we find that the assumption is warranted” (p. 223). Carl Gustav Hempel believed that the Principle is “a rule psychologically guided and stimulated by antecedent knowledge of specific facts, [but] its results are not logically determined by them” (p. 4).

So there you have it. You can be the first kid on your block to save western philosophy by finding a firm foundation for the Uniformity Principle.

Sources:

1. Hempel, Carl Gustav. 1945. Studies in the logic of confirmation I. Mind 54: 1–26.

2. Hume, David. 1902. An enquiry concerning human understanding, 2nd Ed. L.A. Selby-Bigge, ed. [“Hume”] http://www.gutenberg.org/ebooks/9662

3. Mill, John Stuart. 1882. A system of logic, 8th Ed. New York: Harper & Brothers. http://www.gutenberg.org/files/27942/27942-h/27942-h.html#toc47.
4. Wikipedia, Inductive reasoning

Whenever I see inductive arguments being used, it seems as though they can be redone by simply making certain assumptions and rephrasing the argument as a deduction from those assumptions.

For example, in this Khan Academy video, Sal says that if you're predicting the population of a town in the future based on the past, that's inductive reasoning. However, I believe that you could also frame this as deductive reasoning by adding in the assumption that "If the population of a town followed a particular pattern up until today, then it will continue to follow that pattern past today".

In fact, it seems that any inductive reasoning can be done with deductive reasoning by adding in some assumption that a particular pattern continues to hold.

What philosophers call inductive reasoning is just making a guess and then using the consequences of that guess.

Your description of this kind of reasoning:

"If the population of a town followed a particular pattern up until today, then it will continue to follow that pattern past today".

is wrong. For any particular set of data there is an infinite number of ways it could be extended. For example, if the data are (1,2,3) then that might be extended to (1,2,3,4,5,6) or to (1,2,3,2.7,4.2,-10000000.3576). So the idea that the population of the town followed a pattern up to today is ambiguous.

Another answer mentions the "uniformity principle", which is useless because it doesn't specify in what respect the world is uniform.

In reality, knowledge is created by noticing a problem, guessing solutions and criticising the guesses until only one is left and it has no known criticisms. This solution and many related issues were explained by Karl Popper, see the reading list here

http://fallibleideas.com/books#popper

It is not the case in general that inductive reasoning is deductive reasoning from a hidden premise. In particular, the idea that a series of observations that instantiate some pattern can be projected or extrapolated over unknown cases by appeal to a general principle of the uniformity of nature does not hold water. The principle of the uniformity of nature is either false or vacuously true, depending on how you characterise uniformity. As Nelson Goodman pointed out, the future always resembles the past in some respects and does not resemble it in other respects. The crucial question is to discover which properties or predicates are projectible and which not. There is no a priori way to do this, so it is not a deductive activity. Since we can only work from a finite number of observations, there are any number of potential patterns or rules that might fit them.

A more sophisticated answer is given by Quine in his paper Natural Kinds. What we judge to be a pattern is dependent upon the concept of similarity. We observe that two things, properties, or events are similar and we are willing to project a rule on the basis of such similarity. But again, things are always similar in some respects and dissimilar in others, so we have to determine what kinds of similarity matter and what do not. We start our lives with an innate concept of similarity that derives from the 'quality space' that we get from our perceptions. Red appears to be more similar to pink than it is to green, and a rabbit seems more similar to a hare than it is to an elephant. But these similarities are only a starting point. They are true on the face of things, but they may lack the explanatory value of a good scientific theory. Quine offers as an example that we might naively judge that a marsupial mouse is more similar to an ordinary (rodent) mouse than it is to a kangaroo. But for scientific purposes, biologists consider the kangaroo to be more closely connected, because of its evolutionary history. The process of progressively replacing our naive judgements of similarities and patterns with ones that carry a high degree of predictive and explanatory power involves a great deal of scientific labour and is not simply deductive in nature.