# Basic Logic: Truth and Consequences

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Logical AI Is All About True Consequences

In our previous article, we wrote that Logical AI gives us true power because in fact, it is the power of truth. This is why we at Tau are betting on Logical AI when so many other AI platforms choose to rely on machine learning instead. That is, we want to empower you with truth.

We all know that when someone lies there are consequences and that these are usually negative and are not limited to one’s nose growing disproportionately long, like poor Pinocchio’s. Shortly put, it doesn’t pay to lie, or to say falsities, so much so that there is even a commandment against this pernicious practice. On the contrary, truth is universally lauded, and speaking the truth typically pays off (even if only at the end).

One of the most central properties of logic is that it is all about true consequences. For instance, suppose that someone states “2 is a prime number” and “3 is a prime number”; both these statements are true (if you don’t think so, then check your elementary math). Because these two statements are true, then it is equally true to state “2 is a prime number and 3 is a prime number” and “either 2 is a prime number or 3 is a prime number.” That is, take p and q as abbreviations for the two statements about the primes 2 and 3; then, we can say that p AND q is a true consequence if both p and q are true, and we can express this in a standard logical metalanguage as

p AND q if and only if ⊨ p and ⊨ q

meaning that the statement “p AND q” is true, formalized as ⊨p AND q, if and only if both p and q are true statements, and we have

p OR q if and only if ⊨ p or ⊨ q

meaning that the statement “p or q” is true, formalized as ⊨p OR q, if at least one of p or q is true.

What is going on here is very powerful, so much so that the symbol “⊨” stands for (truth) consequence. When we write “⊨p AND q”, we are actually saying in the metalanguage (a language that is used to speak about some other language) that the statement “p AND q” is a true consequence in the logical system we are working with. But because falsity is never very far away from truth, we can also formalize falsity in the metalanguage as

p AND q

whenever either p or q is not true, and as

p OR q

whenever both p and q are false. The symbol “⊭” means precisely “is not a true consequence.”

Now, think of a large online platform with a lot of traffic of users writing sentences and wishing to compare their opinions or facts with the other users’, which is what Tau is. The big magic is that it can be found out by the simple means above what conjunctions (AND) are true consequences in a conversation or debate, and what disjunctions (OR) are true consequences there. For instance, you will always have

p OR NOT-p

as any statement, or its negation is always true (a tautology). Of course, you can also find out about those conjunctions and disjunctions that are not true consequences. For instance, you always have it that

p AND NOT-p

because no statement can be both true and false (a contradiction). So, beware of contradictions.

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Read the previous article from the Basic Logic series here.