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 personal outlook, identity, or culture! 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.
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 CS81, but why? Well, the computer could reason as follows:
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?”
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, but if you take more advanced courses on logical reasoning you might delve into the topics further.
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:
but others are not obvious to everyone, including:
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?
Next: 1.2 Propositions