# What is the difference between inference and deduction?

This question contains my question, but tries to accomplish too much at once; I would like a clear answer to the distinction between inference and deduction.

What is the difference between coming to a conclusion via inference and coming to a conclusion via deduction?

The way I understand it, we deduce conclusions by using inferences. Inferences are statements in the form "if X then Y" and when it turns out previous statements which we assume or have otherwise proven to be true match the "X" part, we call it deduction. Since we used that inference, we say that we obtained our conclusion via inference. (This seems to suggest that while they are different terms, whenever you obtain a conclusion via deduction, you also obtain that conclusion via inference, and vice versa.)

However, other sources claim that deduction must come from originally observed or assumed facts, and that after you deduce one conclusion, you can no longer use that conclusion to "deduce" more; it then becomes "inference".

Is there any widely agreed upon difference between "deduction" and "inference"? If so, what is it? If not, in what ways might the terms differ?

• See Inference : "Inferences are steps in reasoning, moving from premises to logical consequences. Charles Sanders Peirce divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation." Nov 30, 2018 at 8:35
• Deduction : "Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion." Nov 30, 2018 at 8:35
• Deduction is a form of inference.
– user20253
Nov 30, 2018 at 12:45

The term 'deduction' is often used rather loosely in ordinary English. Conan Doyle infamously used it to describe Sherlock Holmes' reasoning, whereas today we would say that what Holmes did was abductive reasoning, which is generally taken to mean reasoning to the best explanation. In logic, we only use 'deduction' to refer to reasoning where there is no possibility of the conclusion being false if the premises are true. It is frequently used, even more narrowly, only in cases where the reasoning relies on formal rules of implication, rather than semantic or model theoretic considerations.

'Inference' is a more general term and refers to any reasoning by which a conclusion is reached from premises. As such, it encompasses both deductive and non-deductive kinds of reasoning. If I see a friend who has been absent for two weeks and notice he has a suntan, I might well infer that he has just returned from holiday. This is not a certain inference, since there are other possible explanations, but it is the most likely. This would be an example of abductive reasoning. If I notice that every morning the sun rises, I might infer that it is likely to do so again tomorrow. Again, this is not certain, but it might be characterised as a plausible inductive inference.

To make matters slightly more confusing, 'inference' is sometimes used for the individual steps within an argument, and logicians traditionally use the term 'rules of inference' for the formal rules, such as modus ponens, that characterise deductive logic. Gilbert Harman, among others, has long argued that this usage is misleading and we should be careful to distinguish between logic and reasoning. He advocates using the term 'rules of implication' for these formal rules.

In any case, deduction and inference have nothing to do with whether your premises are direct observations, assumptions, reported facts, or were themselves inferred from other things. It does not matter where your premises come from.

• To add to the confusion, there is the Deduction Theorem, which is a particular inference involving the discharge of an assumption. Nov 30, 2018 at 8:07

Inference is more abstract: The law says that dog over 20 lbs cannot board a plane. All adult German Shephards weigh over 20 lbs. Therefore, adult German Shephards are not allowed on planes.

A deduction would be more specific to a particular instance: Therefore my dog Mimsy is not allowed on my flight to Florida.

The difference between limbs and roots. Deduction is applying the implication.

Deduction and non-psychological logical relationships

I take the brief answer to be that deduction holds between propositions or statements :

If p then q p

q

This relationship of deductive validity, where the conclusion cannot be false if the premises are true, is non-psychological and holds regardless of anyone thinking it. Inference in contrast is a psychological process of reasoning and is totally dependent on thinking.

Inference and the psychological process of reasoning

'To infer is to change to or take up a position which seems to the thinker to account for or explain the presented data' (Alan White, 'Inference', The Philosophical Quarterly (1950-), Vol. 21, No. 85 (Oct., 1971), pp. 289-302: 292.)

Thus from the fact that my silver has been stolen and only the butler, who has a long history of criminal convictions for theft, and my angelic five-year-old niece, could have stolen it, I infer (I take up the position) that the butler stole the silver. This is my inference to the best explanation. It is (a) psychological and (b) open to error. By contrast deductively validity is non-psychological - a matter of purely logical relationships between propositions or statements - and my inference can be wrong given the presented data whereas in a deductively valid argument the conclusion cannot be false - wrong - given the premises.

Inference can be deductive reasoning (I might infer : 'If p then q; q; therefore p') but can be inductive, abductive or as in the example inference to the best explanation. It is not limited to deductive reasoning. Equally deductive validity is a logical relationship between propositions and statements which holds good whether anyone has reasoned it out or not; it is psychology-free.

This is only a partial answer. The most it attempts is to illustrate how to approach this question: I would like a clear answer to the distinction between inference and deduction.

Answers may differ depending on the logicians one is quoting. One can expect all of these answers to be clear, that is, internally consistent from any particular logician, but not that all logicians will agree on any one definition.

Here is how the authors of forallx use inference: (page 8)

So: we are interested in whether or not a conclusion follows from some premises. Don’t, though, say that the premises infer the conclusion. Entailment is a relation between premises and conclusions; inference is something we do. (So if you want to mention inference when the conclusion follows from the premises, you could say that one may infer the conclusion from the premises.)

For these authors there are subtle differences between entailment and inference.

They use deduction to describe "proof-theoretic" systems, such as "natural deduction", in contrast with semantic arguments using truth tables or interpretations: (page vi)

But entailment is not the only important notion. We will also consider the relationship of being consistent, i.e., of not being mutually contradictory. These notions can be defined semantically, using precise definitions of entailment based on interpretations of the language—or proof-theoretically, using formal systems of deduction.

One thing to note from this is reaching a usable definition of a term may require multiple concepts to keep track of, such as, entailment, consistency, mutually contradictory, semantic, interpretations, and formal system. A full understanding of inference and deduction may require understanding other terms as well.

To see how things might be done differently, Quine uses the two words in the following note: (page 88)

Frege was perhaps the first to distinguish clearly between axioms and the rules of inference whereby theorems are generated from the axioms. Once this distinction is drawn, a recursive characterization of the class of theorems is virtually at hand. But the highly explicit way of presenting formal deductive systems which is customary nowadays dates back only to Hilbert (1922) or Post (1921).

The important thing to observe besides any differences with the previous use of the words is that these terms not only have a definition but they also have a history. One way to acknowledge that history is to associate any definition with whomever is the source of that definition.

So, for a clear answer to the distinction between inference and deduction one needs to further specify which logician's definition of these terms one is interested in.

Because of the differences between logicians I don't have an answer to the final question: Is there any widely agreed upon difference between "deduction" and "inference"? If so, what is it? If not, in what ways might the terms differ?

I suggest, however, given the above, that one doubt any answer one might receive to such questions. Any answer to the differences between these terms should also be associated with the logician providing the description of those differences, because the chief way the terms differ is due to their different sources, that is, the different logicians providing those definitions.

Reference

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

W. V. O. Quine, (1981) Mathematical Logic, Harvard