Strictly speaking: The anatomy of an inference
In logic, to infer is to make a very specific connection between things, based upon your knowledge about each of them. Inference involves rawing a logical conclusion from premises, evidence and sometimes assumptions. A classic example used to introduce this idea is the following logical conclusion: "Socrates is a man, all men are mortal, therefore Socrates is mortal."
Halpern (2003, p.157) described a form of this thought with the following taxonomy:
From this taxonomy of premises, Halpern demonstrated how to determine the validity of an inference while emphasizing that an inference can be logically valid, but not true (if, for example, one of the premises is untrue). To determine whether a conclusion is logically valid, one only needs two additional pieces of vocabulary:
The middle term, which is the category that connects the other two categories. For example, in the inference "All men are mortal, Socrates is a man, therefore Socrates is mortal," the categories are (a) men, (b) mortal, (c) Socrates, and the middle term is "men" because it is the category that connects the other two categories. A simple way to identify the middle term is that the middle term is the category that does not appear in the conclusion.
A category is distributed if a statement applies to everything within that category. In the Socrates example, the category "men are mortal" is distributed.
With these two pieces of vocabulary, we can use the the following algorithm to determine whether a conclusion can be validly drawn from a set of premises (adapted from Halpern, 2003, p.167):
- If one premise is negative, the conclusion must be negative (e.g., "No cars can fly, I use a car to get to work, therefore I do not fly to work.")
- The middle term must be distributed in at least one premise (e.g., "Socrates is a man, all men are mortal, therefore Socrates is mortal.")
- Any term that is distributed in the conclusion must be distributed in at least one premise (e.g., "For me to conclude anything about all students, one of the premises must refer to all students.")
- If both premises are particular, there are no valid conclusions (e.g., "We can conclude nothing knowing only that some students work hard and some Alaskans are students.")
- If one premise is particular, the conclusion must be particular (e.g., "Knowing that some Alaskans are students and all students work hard, we can only conclude that some Alaskans work hard.")
- At least one premise must be affirmative (e.g., we can conclude nothing new from "No Alaskans work hard" and "No Alaskans are students".)
Inferences can also be illustrated using Venn diagrams. The graph below is adapted from Halpern's (2003) visual depiction of logical validity testing, and the interested reader is directed to page 158 for an in-depth discussion of how to represent logical validity tests using diagrams like this.
Not all inferences are created equal
In the real world, we must make inferences in circumstances where definitions are disputed and premises not always crisply distinct. Inferences are therefore frequently based on our own opinions and assumptions and can therefore vary widely in quality and bias. For this reason, it is important that students learn to scrutinize their own inferences and those made by others. Some teachers slow down the inference process to identify which inferences students can make that are supported by specific observations. Others play games in class that require students collect data and draw inferences from those data.
Research findings indicate that inference appears to be a skill that can be improved with training that requires students to organize or cognitively elaborate upon given material. Constable and Klein (2008) found that when individuals were asked to construct a story with a sequential set of information, they were better able to make explanatory and predictive inferences about the given information. Some teachers put a twist on this method, by having students use the data to tell two different stories.
A simple card game can teach students the basic elements of the scientific method, quickly and enjoyably. Students move from wanting to win towards wanting to analyze, hypothesize, and test.
Some have found that the use of diagrams in conjunction with text supports inference skills as well as reduces errors (Butcher, 2006). McGee and Johnson (2003) increased the inferential accuracy of their students using three techniques: helping them identify key words within a text, helping them generate their own questions about a text, and obscuring part of a text and asking students to predict the obscured section contains. This last technique is similar to problematizing part of a text, which other teachers have also found effective.