Why Study Logic?

Although logic is known as the study of correct reasoning, it turns out there are many different logics, each with their own set of reasoning principles.

Let me be clear—this is not a claim that logic should depend on your culture, your race, or your gender! Rather, at a very practical level, different methods of reasoning turn out to be useful in different application domains.

For example, the logic you are most familiar with is called Classical Logic. This is the kind of reasoning used in nearly all math classes, as well as most other situations where you’re expected to make correct and careful arguments.

Why We Might Want Something Different

Suppose we load all of PubMed.gov (a big database of bio-medical research) into a computer programmed to reason using classical logic, and ask the computer,

Based on all these facts, is it true that knowledge of logic increases your lifespan?

The computer will say yes!

That’s encouraging news for anyone taking CS 81, but why? Well, the computer could reason as follows:

1. People who eat fish have significantly lower levels of heart disease. [Study #20332801].
2. People who eat fish do not show a significant reduction in heart disease. [Study #19789394]
3. Contradiction.
4. Therefore, knowledge of logic increases your lifespan.

where the Classical Logic rule that “any conclusion follows logically from a contradiction” gets from step 3 to step 4.

This seems unsatisfactory. But we’d get the same conclusion if we loaded Wikipedia into the computer instead, because there are apparent contradictions there too. (For example, at one point there were pages saying that (1) birds migrate by flying, (2) penguins are birds, and (3) penguins migrate but don’t fly.)

This sort of situation (and the fact that humans can sometimes hold two incompatible beliefs and yet somehow manage not to believe everything) has led philosophers of logic to consider the question, “How can we reason effectively in the presence of contradictions?”

This has led to proposals for various Paraconsistent Logics. Generally speaking, all of these proposals drop the “everything follows from a contradiction” rule, and then vary in what further restrictions they impose.

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Now this was just a simple example of an instance where it can be pragmatically useful to reconsider what “rules of logic” we want to use; we won’t study Paraconsistent Logics any further in CS 81. However, if time permits, we will discuss two other logics with close connections to computation, namely:

• Constructive Logic, which unlike Classical Logic does not build in the assumption that every statement is true or false; and
• Temporal Logic, which extends Classical Logic with a built-in notion of time, allowing us to distinguish whether a statement is “true right now”, or “will always be true”, or “possibly true in the future”, or “will inevitably be true in the future”, or “will always be (possibly true in the future)”, or …, and to reason appropriately.

Justifying Classical Logic

But for now let’s return our focus to Classical Logic, our standard, everyday logic. Classical Logic includes many longstanding reasoning principles that can be traced back to Ancient Greek and Medieval philosophers. Some seem uncontroversial:

• R1: If we know $\cal A$ is true, we can trivially conclude ”$\cal A$ or $\cal B$” is true.
• R2: If we know ”$\cal A$ or $\cal B$” is true, and we know $\cal A$ is false, then we can conclude $\cal B$ is true.

but others are not obvious to everyone, including:

• R3: If we have a contradiction, we can then conclude any statement $\cal C$.

Rule R3 was known as Ex Falso Quodlibet in the Middle Ages. More recent (and less pretty) names include the Principle of Explosion and Bottom Elimination.

Logic is subtle, and relying on intuition for what is acceptable can lead one astray. For example, it can be shown that non-obvious rule R3 is a direct consequence of obvious rules R1 and R2. (As a consequence, in addition to dropping R3, any Paraconsistent Logic has to make the painful decision of whether to give up rule R1, rule R2, or both. Different answers to this question yield different Paraconsistent Logics.)

So the question is, if intuition might be untrustworthy, what objective criteria can we use to explain why rules like R1 and R2 and R3 should be accepted, and to decide what other principles we can allow in everyday reasoning?