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“There are two kinds of logic, deductive and bad.”
– Bertrand Russell (1872 – 1970)
a mathematical logician
Russell’s joke points out that induction is far more controversial than deduction. Philosophers have debated about the logical and epistemological status of induction for a few hundred years. Moreover, induction isn’t as exacting and definitive as deduction in terms of how to determine validity.
Deduction. . .
Definition A: deduction brings certainty
Definition B: deduction moves from more general claims to less general claims
Induction. . .
Definition A: induction brings probability
Definition B: induction moves from less general claims to more general claims
Note: Induction’s Definition A & Definition B are used in incompatible ways. See below!
Definition A is the “standard” definition you’ll find in most contemporary textbooks.
Definition B is the “traditional” definition you’ll find in older textbooks or some “liberal arts” textbooks.
Deductive arguments bring certainty: the conclusion to a deductive argument is guaranteed to be true if and only if the premises are true and the argument is valid. So, it’s impossible to derive a false conclusion from true premises and a valid argument.
There’s a sense in which the conclusion is “within” the claims made by the premises. That’s what makes deduction certain. A conclusion is no broader than the premises. That is, the conclusion never makes more sweeping or general claims than the premises.
All conclusions in deductive arguments also invariably have equal “strength.” When looking at different types of deductive arguments, there is no one type that is “stronger” than another. They all produce a conclusion that will be 100% true, provided that the premises are true and the argument is valid.
Understand that deductive arguments “preserve” truth. As long as we start with true premises, any validly derived deduction will likewise be true. Thus truth will be preserved throughout the argument. Then again, it’s obviously possible to start with false premises. Those false premises might, by accident, validly lead to a true conclusion; alternatively, they might validly lead to a false conclusion.
Deduction only guarantees this: true premises + valid argument = true conclusion.
Categorical syllogisms, hypothetical syllogisms, propositional logic, and predicate logic are examples within the sphere of deduction.
What about induction? Keep in mind that induction is highly controversial. Inductive arguments come in a variety of forms. Confusingly, textbooks technically define “induction” differently.
Definition A: Sometimes a textbook will only define induction as an argument that necessarily moves from more particular propositions to more general propositions. For example, we might inductively conclude that the entire population P has attribute X after observing a (sufficient) sample or subset of population P.
Definition B: Other textbooks will define induction broadly to be any probabilistic argument that lacks certainty. For example, an “appeal to a qualified authority” counts as an inductive argument by this definition. We can inductively argue that proposition Y concerning quantum mechanics as being probably true because a Ph.D. quantum physicist says proposition Y is true.
By both definitions, generally speaking, inductive arguments bring probability: the conclusion to an inductive argument is probably true, although it might be false, insofar as the premises are true and the argument is valid. So, there is some chance that the conclusion is false, but it being false is less likely than it being true.
Induction is generally “ampliative.” Here the conclusion does make more sweeping or general claims than the premises. That makes induction “risky.” It goes “beyond” the premises. The conclusion is only probable, not absolutely certain. Thus, it’s possible that a valid induction can bring a false conclusion.
Finally, conclusions can have different “strengths.” Since a wide range of degrees of probability are possible in inductive arguments, some conclusions will be more likely than other conclusions. Determining the exact “strength” of an inductive argument isn’t often clear, however. There’s no “settled science.”
Probable arguments come in a variety of forms. There are statistical arguments, generalizations, arguments via analogy, causal inferences, appeals to authority, etc.
Also, there are textbooks that will include types of “induction” that have a probability of 100%. These bring certainty! Yet they can fit into Definition A. For example, in a “complete enumerative induction,” we might have sampled absolutely everything in a population. We have (for example) a bag of marbles. We count each one as being red; ergo, all of those marbles in that bag are red.[1]
And in Aristotelian-Scholastic logic, great emphasis is placed on the process of abstraction. This is related to the so-called “first act of the mind.” We are “rational animals,” yet we learn about things through experience with the world. Imagine a scientist first discovering that gold has an atomic number of 79. When it comes to this particular inference, though, one sample will do to make a general claim about all gold!
Notice, too, that when we’re grounded in Aristotelian-Scholastic logic, epistemology, and metaphysics, there is more optimism about what induction can do. There’s no good reason to claim that it’s “only likely” that the atomic number is 79 for gold; rather, it is that in fact. Induction can lead to certainty or near certainty in many cases.
Finally, there is something called “mathematical induction.” This is a new beast! My brief point here is: we can get certainty out of it. In number theory, e.g., there are “induction proofs” where a property is shown true for a “base case” test, n, and then showed true for case n+1. Though it arguably isn’t really an “induction” (depending on our definition).[2]
But “mathematical induction” is outside of our present concern.
Let’s be careful! Inconsistencies are present in these definitions. Just be aware of them.
***
[1] To be sure, it can be philosophically argued, no genuine inference took place with that example. An inference should get us something new. The “ergo” part re-stated the premise of “each one as being red.”
[2] For instance: we could claim “mathematical induction” is “deductive” insofar as “deduction” deals with an inferred certain conclusion. Indeed, claiming all of mathematics to be “deductive” is totally reasonable. This includes probability theory. (But note that statistical inferences would still be inductive, despite them making use of mathematics.) Alternatively, it’s reasonable to view “mathematical induction” as “inductive” insofar as it is a process that looks at particular cases to then make a general claim, e.g., about all infinite integers.
***
The Oxford Companion to Philosophy (edited by Ted Honderich) has a useful entry on “Induction” by Michael Cohen. It emphasizes the philosophical questions raised by David Hume on the subject.
It starts out with the traditional definition: “Induction has traditionally been defined as the inference from particular to general.” This corresponds to our Definition B.
It doesn’t explicitly give us Definition A.
Dictionary of Scholastic Philosophy by Bernard Wuellner gives us a concise entry on “Induction.”
It fits perfectly with Definition B. There are three short entries. The third states that induction is “an argument that moves from a more particular premise to a more general conclusion.”
Most contemporary logic textbooks stick with Definition A.
For instance, a popular college-level textbook is A Concise Introduction to Logic by Patrick J. Hurley. It sticks with Definition A.
In inductive arguments, Hurley writes, “the conclusion is claimed to follow only probably from the premises.”
Deductive arguments “are those that involve necessary reasoning…”