# “The Conclusion Follows”: How Logical Illiteracy Hides Under the Surface and Interferes with Higher-Order Learning

There is a confusion that I’ve started paying more attention to recently. It’s rarely discussed in logic texts but it shows up frequently in student writing.

The problem is when students use the term “follows” when they should be saying “follows from”.

Example: **“The argument is valid; the conclusion follows the premises.”**

At first I thought this was a minor glitch, but now I think it’s connected to a deeper problem, which is that the logical notion of “following from” is genuinely unfamiliar to many students and takes a while to absorb, and their initial interpretation of the concept of “following” isn’t **logical**, but rather **sequential**.

And this, I think, is symptomatic of a larger problem — that many students manage to get through thirteen or more years of public education and are still, for practical purposes, “logically illiterate”.

Let me explain…

## What “follows from” means in logic

Here’s a schematic argument written in standard form:

**1. premise 1
2. premise 2
:
n. premise n
Therefore, conclusion**

One of the features we look for in a good argument is that the conclusion “follows from” the premises. This is vague as it stands, but we introduce terms like “valid” and “strong” to add precision to the concept of “following from”.

The key point is that “following from” is a logical relation between *statements *(or* sets *of statements). It describes a relation of logical* entailment*. If statement Q follows from statement P, then we’re saying that if P is true then Q is also true, so that given P we can infer or deduce Q.

“Following from” is also consistent with *partial* entailment. If P partially entails Q, then that means that P being true increases the probability that Q is also true. If the probability is high we might still want to say that “Q follows from P”.

So coming back to our schematic argument, we say that the conclusion “follows from” the premises just in case the premises either completely or partially entail the conclusion. We use the terms “valid” and “strong” to distinguish these cases.

[Note: The degree of partial entailment needed to make an argument “strong” is a matter of conventional choice as it turns out, not a matter of logic.]

## How some students interpret “follows from”

Unlike terms like “valid”, “strong” and “weak”, there isn’t a significantly different “everyday” meaning of “to follow from”. The expression “it follows from …” or “it follows that …” always carries the meaning of logical entailment in ordinary English.

This is why it has always puzzled me how common it is for students to drop the “from” and say things like “the conclusion follows the premises”. It’s jarring to my ears to read this out load. Why isn’t it jarring to them?

I think I’ve figured it out. The explanation has two parts.

**(a) The sequential sense of “follows”**

First, take a look again to our schematic argument:

**1. premise 1
2. premise 2
:
n. premise n
Therefore, conclusion**

We write this on the chalkboard to a group of students and say “the conclusion follows from the premises”. But notice that when we write arguments in this form something else is ALSO true: the conclusion FOLLOWS the premises, *sequentially*. We put the conclusion at the bottom of the list of statements, or at the end of the list of statements when written on a single line. Literally, the conclusion *follows* the premises, in the sense that all the premises come before and the conclusion comes at the end.

I swear, it never occurred to me that some students might be hearing me say “the conclusion *follows from* the premises” and interpreting that as “the conclusion *comes after* the premises”. But I’m convinced that this going on in many cases.

What later happens, of course, is that students eventually do internalize the logical usage, but in the meantime there can be a confusion of meanings in their heads, where they’re paying attention to both the ordering of statements when writing arguments in standard form, and the logical properties of those arguments, and are not always clear how to distinguish them.

**(b) A contributing problem: “logical illiteracy”**

Confusion with the sequential sense of “follows” is part of the issue when teaching introductory logic and critical thinking, but I think it’s symptomatic of a broader problem.

We don’t fault students or the general public for not understanding technical vocabulary within an academic field or profession. There’s no shame in not understanding what a “pivot table” is (unless you’re a spreadsheet expert), or what a “synecdoche” is (unless you’re a student of rhetoric), or what a “golgi body” is (unless you’re a biologist), or the difference between “warp” and “weft” (unless you’re a weaver).

But we do expect that students with thirteen-plus years of formal education should be familiar with a certain range of concepts and vocabulary that one would normally encounter over those thirteen-plus years. Public discourse presupposes some shared set of concepts, vocabulary and background knowledge.

The problem is that very few students are ever taught the foundations of logic and argumentation as part of their schooling, and many don’t have opportunity or encouragement to explore these ideas or demonstrate skill in their application.

The result is that for many students, basic logical concepts like “to follow from” simply are not part of their working vocabulary. They experience them as similar to other technical concepts that they acquire only through specialized training.

What I’ve discovered over the years is that many teachers (and I include myself) don’t have a full appreciation of just how “illiterate” some of their students are in this regard. They spend time in class defining technical concepts in their subject matter but presume a working knowledge of critical thinking concepts and communication skills. When students express confusion or struggle with expressing themselves we think the problem is with their mastery of the new technical concepts, but the source of the problem may be deeper. They may be struggling to master more basic thinking skills (logical inference, conceptual analysis, etc.) that they’ve never had to deploy before, and it’s interfering with their ability to learn the classroom material.

Of course this just highlights the need for better critical thinking instruction at earlier stages in the public education system. It’s shameful that in the 21st century, most students going through the public school system (and most college graduates) are never given any formal instruction in basic concepts of argumentation and logical analysis.