4-1. Symbolic Logic & Reasoning

The answer is that B is true, and we know that because it was one of the three sentences that was given to us. And now, according to the first sentence, it says that if E or B is true, then A is true. So now we know that A is true. And the second … Read more

Here’s one more example of reasoning and propositional logic. In a particular model of the world, we know the following three sentences are true. E, or B, implies A. A implies J and M. And B. You know these three sentences to be true and that’s all we know. Now, I want you to tell … Read more

The answers. For P and P implies Q, we know that P is true in these bottom two cases. In P implies Q, we saw on the truth table for P implies Q is true in the first, second, and fourth case. So the only case that’s true for both P and P implies Q … Read more

Here’s a quiz. Use truth tables, or whatever method you want to fill in the values of these tables. For each of the values of P and Q, false false, false true, true false, or true true, look at each of these boxes and click on just the boxes in which the formula for that … Read more

The answers are, first, the sentence, if five is an odd number, then Paris is a capital of France, is true in propositional logic. It may sound odd in ordinary English, but in propositional logic, this is the same as true implies true. And if we look on this line, the final line for P … Read more

Here are the truth tables for all the logical connectives. What a truth table does is list all the possibilities for the propositional symbols. So P and Q can be false and false, false and true, true and false, or true and true. Those are the only four possibilities. And then for each of those … Read more

The first logic we would consider is called propositional logic. Let’s jump right into an example, recasting the alarm problem in propositional logic. We have propositional symbols B, E, A, M, and J corresponding to the events of the burglary occurring, of the earthquake occurring, of the alarm going off, of Mary calling, and of … Read more

First answer is yes. This is a good representation of the sentence, Sam has two jobs. It says there exists an x and y. One of them is a job of Sam. The other one is a job of Sam. And crucially, we have to say that x is not equal to y. Otherwise, this … Read more

Now in the early days of AI, there was a lot of excitement and work on logic. Can you give us a bit of historical perspective on the topic. >> AI started in the 50s through the 70s with the idea that what we needed was better languages and systems for writing down logical statements, … Read more

Now I’m going to give you some sentences or axioms in first-order logic. And I want you to tell me if they correctly or incorrectly represent the English that I’m asking about. So tell me yes or no, are these good representations? And the first, I want to represent the English sentence Sam has two … Read more

The answers are. The first sentence is valid. It’s always true. Why is that? Because every model has to have at least one object and we can have both x and y refer to that same object, and so that object must be equal to itself. Second, the left hand side of this implication has … Read more

Now let’s get some practice in first order logic. I’m going to give you some sentences and for each one, I want you to tell me if it is valid. That is always true, satisfiable but not valid. That is there’s some models for which it is true or unsatisfiable, meaning there are no models … Read more

Now let’s go back to the two location vacuum world and represent it in first order logic. So first of all, we can add locations. We can call the left location A and the right location B. And the vacuum V, and the dirt say, D1 and D2. Then we can have relations, the relation … Read more

Now let’s talk about the syntax of first order logic. And like in propositional logic, we have sentences, which describe facts that are true or false. But unlike propositional logic, we also have terms, which describe objects. Now the atomic sentences are predicates corresponding to relations. So we can say, Vowel (A) is an atomic … Read more

How does first order logic work? What does it do? Well, like propositional logic, we start with a model. In propositional logic, a model was a value for each propositional symbol. So, we might say that the symbol, P, was true and the symbol, Q, was false. And that would be a model that corresponds … Read more

We’re going to talk about first order logic, and its relation to the other logics we’ve seen so far, namely propositional logic and probability theory. And we’re going to talk about them in terms of what they say about the world, which we called the ontological commitment of these logics and what types of beliefs … Read more

Propositional logic is a powerful language for what it does. And there are very efficient inference mechanisms for determining validity and satisfiability, although we haven’t discussed them. But propositional logic has a few limitations. First, it can only handle true and false values. No capability to handle uncertainty like we did in probability theory. And … Read more

P and not P is valid. That is, it’s true when P is true because of this, and it’s true when P is false because of this clause. P and not P is unsatisfiable. The symbol can’t be both true and false at the same time. P or Q, or P is equivalent to Q … Read more

Now, for a little more terminology. We say that a valid sentence is one that is true in every possible model for every combination of values of the propositional symbols, and a satisfiable sentence is one that is true in some model and not necessarily in all the models. So what I want you to … Read more

In this next section, Peter will teach us about logic and planning. Peter, what important concepts should our students look out for? >> A logic is a way of describing the world. When we make logical statements, we capture part of what we want to know about the world, and ignore other parts. Any description … Read more