Basic Logic: Principles of Classical Logic
Logic Is Everywhere and Logical AI Just Tries To Capture It in Machines
Many people (still) associate logic with hard-to-understand rules and highbrow explanations of natural phenomena like mathematical thinking and reasoning. This opinion often nurtures the view that Logical AI is something difficult to use and has only a very small place in the vast world of the World Wide Web.
To be sure, most logical rules are more complex than the modus ponens rule we wrote about in our previous article, and many Logical-AI notions are far more abstract (not to say abstruse) than inference and knowledge representation, also discussed there. But, in fact, logical reasoning is what accounts to a great extent for our evolutionary success as a species, in the sense that we learned to identify the logical rules that underlie many natural phenomena and thus gained the ability not only to understand them better but also to predict and even control them. In particular, without logical reasoning, our comprehension of mathematics would be mostly intuitive, and one might doubt whether intuition alone would have taken us to the moon or given us the Internet.
In our first article in this series, we gave an example of a Logical-AI system that very much resembles your computer connected to a printer. Well, your computer is one of the best examples of Logical AI: Unless you can brag that you already own a quantum computer (a very unlikely to impossible possession at present) your computer is very likely a classical one, i.e. an electric circuit composed of wires and logic gates, devices that implement logical operations performed on inputs constituted of 0s and 1s, known as binary inputs, and which output a single binary value 0 or 1. These two values are called truth values, because we identify “1” with truth (the value “true”) and “0” with falsity (“false”). This identification of binary values with truth values is very convenient, allowing us to describe what can be very complex operations by means of truth tables. A very simple logic gate is that of logical negation (NOT), which has the following truth table:
Say, you believe that the statement “The moon is made of cheese” is true, or has truth value 1; then, the statement “The moon is not made of cheese” has truth value 0, or is false. Of course, you can believe that the reverse is the case: “The moon is not made of cheese” is true and “The moon is made of cheese” is false. This is precisely what makes logic so powerful, the ability to apply truth values to (arbitrary) statements by seeing these as propositions, or truth carriers. We typically denote an arbitrary proposition by the lower-case letters p, q, r, …, so that the truth table above is more usually given as:
This very simple logical operation is actually foundational of bivalence, the view that there are only two truth values and they are opposites. According to bivalence, a proposition p cannot be true and false at the same time, a reasoning instance known as the principle of non-contradiction, formalizable as NOT(p AND NOT-p), and either p or its negation must be true (principle of excluded middle: p OR NOT-p). Note in these two principles that you are already composing complex propositions with the logical operations AND and OR, which have the following truth tables:
AND and OR very closely (but not entirely) mimic our use of the English connectives “and” and “or”, especially in scientific or technical settings. For instance, you cannot say of a natural number greater than one that it is both prime and not-prime (also: composite), but it must be either prime or not-prime, right? This bivalence applied to the natural numbers allows us to apply prime factorization to cryptographic protocols to make them more secure, so here you are a good example of logical reasoning allowing us to describe and control natural phenomena in Logical AI.
There are central features of Tau that depend on bivalence, and we will tell you about them in an article to be published later on. Just keep in mind the logical principles that are associated with bivalence.