Chapter Three
3.4.2 Conjunctions
If we symbolize “I can clone this pig” with the letter “A”, and its negation as “~A”, then we can represent the truth values for “A” and “~A” in a table. Row one of this table says that if A is true, then ~A is false. Row two of the table says that if A is false, ~A is true. A
~A
T
F
F
T
3.4.2 Conjunctions When a statement affirms or denies more than one thing, that statement is a conjunction. In essence, a conjunction claims that all of the statements of which it is composed are true. The individual statements of a conjunction are its conjuncts. These statements could be negative or positive. But when any one of the statements of which a conjunction is composed is false, the whole conjunction is false. For instance, the conjunction “My house is red, and I like to eat buttons,” is only true if both of the individual statements are true; that is, if my house is red and I like to eat buttons. If I don’t like to eat buttons, then the conjunction, “My house is red, and I like to eat buttons” is false. But conjunctions don’t necessarily use the word “and”, and so it is useful to recognize some other indicator words that tell us we’re dealing with a conjunction. Consider the following examples, all of which could be reduced to the conjunction, “I childproofed the house, and children get in the house”: • I’ve childproofed the house, and they still get in. • I’ve childproofed the house, but they still get in. • I’ve childproofed the house, yet they still get in. • Although I’ve childproofed the house, they still get in. • Even though I’ve childproofed the house, they still get in. • I’ve childproofed the house; however, they still get in.
If we symbolize “I’ve childproofed the house” as “A” and “Children get in the house” as “B”, and the conjunction as “A&B”, then the truth table for the conjunction is as follows:
A
B
A&B
T
T
T
T
F
F
F
T
F
F
F
F
From this we can see that the only case where the conjunction “A&B” is true is when both of the individual statements are true. Conjunctions are used when we need to put two or more statements together and treat them all as if they are one single statement. This can make it easier to analyze an argument as a whole. 3.4.3 Disjunctions Disjunctions, like conjunctions, are composed of two or more statements, which could be positive or negative. It is another way to put two statements together and treat them as if they are one statement; but we do this when we know only one of them is true, but we are not sure which one. The statements disjoined in a disjunction are called its disjuncts. In the case of disjunctions, only one of those statements needs to be true to make the disjunction as a whole true. For instance, the statement, “Either I’ll save this money, or I’ll spend it on candy” is true in either of the cases where I save the money or spend it on candy. The statement would be false, however, if instead I bought a motorcycle with the money. All of the following examples are cases of a disjunction: • The hoarder will clean the house, or be evicted. • Either the hoarder will clean the house, or he’ll be evicted. • The hoarder will either clean or be evicted. • Unless the hoarder cleans the house, he will be evicted.
If we symbolize “The hoarder will clean the house” as “A”, and “The hoarder will be evicted” as “B”, then the disjunction as a whole would be represented as “A∨B”. We can summarize the truth of the disjunction in a table. From this we can see that a disjunction is true
51