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Propositions, Truth Values and Truth Tables

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  • 0:09 Propositions
  • 0:35 Negation
  • 1:39 Truth Value
  • 3:11 Truth Table
  • 8:47 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson and learn what truth values are and what a truth table looks like. Learn how to go from a proposition to its negation and how that affects the truth values and the truth tables.

Propositions

Maria has a blue dog. Kevin has a purple cat. Joann has a black rat. These are examples of propositions. Propositions, in logic, are statements that can be labeled as either true or false. All of the examples you just heard and saw are complete statements that you can say are either true or false.

Negation

In logic, we sometimes change our original statement to its negative form. We do this by adding a NOT in the statement. So, the negative of 'Maria has a blue dog' is 'Maria does not have a blue dog.' We've added a few words just to make it grammatically correct, but as you can see, we have added a NOT in the statement. We went from stating that something is happening to something that is not happening.

If our original proposition is in the negative form, then the negative form of that statement will be a positive. For example, if our original statement is 'We are not in the year 1990,' then the negative of that statement becomes 'We are in the year 1990.'

Think of the negative as adding a NOT if there is no NOT and deleting the NOT if there is a NOT. You are essentially turning a positive into a negative or a negative into a positive depending on what kind of statement you begin with.

Truth Value

Negating a proposition changes its truth value, whether the statement is true or false. For example, if the statement 'She loves to chase squirrels' is true, then the negative of the statement, 'She does not love to chase squirrels,' is false.

We can create a simple table to show the truth value of a statement and its negation. We will call our statement p and the negation NOT p. We write these in the top row of our truth value table. In the next row, we put T under the p column. Now, if the statement p is true, then its negation NOT p must be false, so we put F in the same row under the NOT p column. In the next row, we put F under the p column and T under the NOT p column since if our original statement is false, then the negation must be true. We can use this truth value table for any logic proposition we come across.

p NOT p
T F
F T

We can look at any proposition and compare it to this truth value table. If our original proposition is false, then its negation is true. If our original proposition is true, then its negation is false. For example, if we know the proposition '2 + 2 = 5' is false, then by looking at the third row in the chart, we can see that the negation '2 + 2 does not = 5' is true.

Truth Table

We can take our truth value table one step further by adding a second proposition into the mix. By adding a second proposition and including all the possible scenarios of the two propositions together, we create a truth table, a table showing the truth value for logic combinations. We will call our first proposition p and our second proposition q. With just these two propositions, we have four possible scenarios.

We can have both statements true; we can have the first statement true and the second false; we can have the first statement false and the second true; and we can have both statements false. For example, if our first proposition, p, is 'Ed is a horse,' and our second proposition, q, is 'Spot is a dog,' then we can have four possible scenarios by combining these two statements. We can have both statements being true, we can have the first statement being true and the second statement false, we can have the first statement being false and the second true, or we can have both statements being false.

The truth table sets all these scenarios up so you can quickly look up your situation to find its truth value. This is just the beginning of our truth table where we set up our scenarios. Keep watching and you will see how to include the truth values for the logic combinations.

p q
T T
T F
F T
F F

There are four logical combinations we can make with these two statements.

1.p AND q. When you have an AND connecting the two simple statements, it means that both statements must be happening at the same time. For example, if our first proposition is 'The room is blue' and our second proposition is 'The lamp is blue,' then p AND q means that both the room and the lamp are blue. In order for this type of 'and statement' to be true, both statements must be true to begin with. We can show this by adding a column to our truth table for p AND q and labeling the row where both p AND q are true with a T and the rest with an F.

p q p AND q
T T T
T F F
F T F
F F F

2.p OR q. When you combine the two propositions with an OR, it means that either or both is happening. If our first proposition is 'The cat is chasing the mouse' and our second proposition is 'The dog is chasing the cat,' combining the two with an OR means that we can see the cat chasing the mouse or we can see the dog chasing the cat or we can see both the dog chasing the cat and the cat chasing the mouse. For this case, if just one of the statements is true, the OR statement will be true. Let's add this information to our truth table under the column p OR q. The only scenario when this is false is when both statements are false to begin with.

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