What is Probability?

Introduction

In this lecture I want to revisit the first point we raised, which is about **inductive logic**. I want to lay out some terms here so that it’s clear what we’re talking about, and the role that probability concepts play in inductive reasoning.

We distinguish deductive logic from inductive logic. Deductive logic deals with deductive arguments, inductive logic deals with inductive arguments. So what’s the difference between a deductive argument and an inductive argument?

The difference has to do with the logical relationship between the premises and the conclusion. Here we’ve got a schematic representation of an argument, a set of premises from we infer some conclusion.

**1. Premise
2. Premise
:
n. Premise
∴ Conclusion**

That three-point triangle shape is the mathematicians symbol for “therefore”, so when you see that just read it as “premise 1, premise 2, and so on, THEREFORE, conclusion.

Now, in a **deductive argument**, the intention is for **the conclusion to follow from the premises with CERTAINTY**. And by that we mean that IF the premises are all true, the conclusion could not possibly be false. So the inference isn’t a risky one at all — if we assume the premises are true, we’re guaranteed that the conclusion will also be true.

For those who’ve worked through the course on “Basic Concepts in Logic and Argumentation”, you’ll recognize this as the definition of a logically VALID argument. A deductive argument is one that is intended to be valid. Here’s a simple example, well-worn example.

**1. All humans are mortal.
2. Socrates is human.
∴ Socrates is mortal. **

If we grant both of these premises, it follows with absolute deductive certainty that Socrates must be mortal.

Now, by contrast, with **inductive arguments**, we don’t expect the conclusion to follow with certainty. **With an inductive argument, the conclusion only follows with some probability, some likelihood**. This makes it a “risky” inference in the sense that, even if the premises are all true, and we’re 100 % convinced of their truth, the conclusion that we infer from them could still be false. So there’s always a bit of a gamble involved in accepting the conclusion of an inductive argument.

Here’s a simple example.

**1. 90% of humans are right-handed.
2. John is human.
∴ John is right-handed. **

This conclusion obviously doesn’t follow with certainty. If we assume these two premises are true, the conclusion could still be false, John could be one of the 10% of people who are left-handed. In this case it’s highly likely that John is right-handed, so we’d say that, while the inference isn’t logically valid, it is a logically STRONG inference. On the other hand, an argument like this …

**1. Roughly 50% of humans are female.
2. Julie has a new baby.
∴ Julie’s baby is female.**

... is not STRONG. In this case the odds of this conclusion being correct are only about 50%, no better than a coin toss. Simply knowing that the baby is human doesn’t give us good reasons to infer that the baby is a girl; the logical connection is TOO WEAK to justify this inference.

These two examples show how probability concepts play a role in helping us distinguish between logically strong and logically weak arguments.

Now, I want to draw attention to two different aspects of inductive reasoning.

When you’re given an inductive argument there are two questions that have to be answered before you can properly evaluate the reasoning.

The **first** question is this: **How strong is the inference from premises to conclusion? In other words, what is the probability that the conclusion is true, given the premises? **

This was easy to figure out with the previous examples, because the proportions in the population were made explicit, and we all have at least some experience with reasoning with percentages — if 90% of people are right-handed, and you don’t know anything else about John, we just assume there’s a 90% chance that John is right-handed, and 10% chance that he’s left-handed. We're actually doing a little probability calculation in our head when we draw this inference.

This is where probability theory can play a useful role in inductive reasoning. For more complicated inferences the answers aren’t so obvious. For example, if I shuffle a deck of cards and I ask you what are the odds that the first two cards I draw off the top of the deck will both be ACES, you’ll probably be stumped. But you actually do have enough information to answer this question, assuming you’re familiar with the layout of a normal deck of cards. It’s just a matter of using your background knowledge and applying some simple RULES for reasoning with probabilities.

Now, the **other question** we need to ask about inductive arguments isn’t so easy to answer.

The question is, **how high does the probability have to be before it’s rational to accept the conclusion?**.

This is a very different question. This is a question about** thresholds for rational acceptance**, how high the probability should be before we can say “okay, it’s reasonable for me to accept this conclusion — even though I know there’s still a chance it’s wrong”. In inductive logic, **this is the threshold between STRONG and WEAK arguments** — strong arguments are those where the probability is high enough to warrant accepting the conclusion, weak arguments are those where the probability isn’t high enough.

Now, I’m just going to say this up front. THIS is an unresolved problem in the philosophy of inductive reasoning. Why? Because it gets into what is known as the “problem of induction”.

This is a famous problem in philosophy, and it’s about how you justify inductive reasoning in the first place. The Scottish philosopher David Hume first formulated the problem and there’s no consensus on exactly how it should be answered. And for those who do think there’s an answer and are confident that we are justified in distinguishing between strong and weak inductive arguments, the best that we can say is that it’s at least partly a conventional choice where we set the threshold.

To refine our reasoning on this question we need to get into rational choice theory where we start comparing the costs and benefits of setting the bar too low versus the costs and benefits of setting it high, and to make a long story short, that’s an area that I’m not planning on going into in this course.

**In this course we’re going to stick with the first question**, and look at how probability theory, and different interpretations of the probability concept, can be used to assign probabilities to individual claims AND to logical inferences between claims.

With this under our belt we’ll then be in a good position to understand the material on probabilistic fallacies and probability blindness, which is really, really important from a critical thinking standpoint.