Propositional Logic (Read Ch. 6)
We’ll begin with propositional connectives. This allows us to build new propositions from old ones (by now, you should know what a proposition is). The connectives we’ll explore are conjunction, disjunction, negation, and conditionals.
Conjunction, “and,” “&” (pages 142–145)
Consider the proposition “John is tall.” It is either true or false. Now, what about “John is tall, and Harry is short”? What happens when we combine two propositions? How do we assess the truth value of that compound sentence? The authors provide a “truth table” for this example at the top of page 143. Abstracting from the form of this example, they provide a “general definition of conjunction” using propositional variables in the truth table at the bottom of page 144. Here are the points to consider in getting from an example to the general truth table on page 144:
Any proposition is either true or false. I said this about “John is tall,” and it is equally true for “Harry is short.”
Since there are two propositions being conjoined under one big proposition, we need to consider the various possible true or false combinations for both propositions.Since either one can be true or false and there are two propositions total, then to make sure you cover all ground, you need to consider four different possibilities. That is, we need to consider the case in which “John is tall” is true and “Harry is short” is true, a case in which “John is tall” is true and “Harry is short” is false, and so on …
We aren’t particularly interested in John and Harry; we are interested in conjunction for all propositions. So instead of analyzing particular propositions, we’ll assign variables. The authors assign “p” and “q” for two distinct propositions—you can choose what you like so long as there are two distinct ones for a conjunction of two propositions (three for three, and so on). (See page 143.) So, the question at hand is “what are the various truth values that we can assign for a conjunction of two propositions, p and q?”
Another detail. We’ll substitute the word “and” with the symbol “&.” Call it laziness D. on the part of logicians. The various truth values for conjunction (“&”) for two propositions, p and q, are found on page 144. It is pretty intuitive, unsurprisingly. A conjunction of two propositions is itself true when both propositions are true. That makes sense. If “John is tall” is true and “Harry is short” is true, then the conjunction “John is tall,
and Harry is short” is true. The conjunction would be false in any other case. Easy!
Pay attention to the cautionary note about when “and” really does represent a conjunction of two propositions and when it does not. For instance, when we say “Serena and Venus are playing each other,” we don’t mean the “and” in that sentence to indicate a conjunction. Otherwise we would be interpreting the sentence as “Serena is playing each other, and Venus is playing each other,” which doesn’t make sense. So, this is the real reason we use “&” instead of “and” to symbolize conjunction: “and” in English doesn’t always mean a conjunction!
Disjunction (pages 150 and 160)
Now that you know what we’re up to, describing the other truth-value connectives is rather easy. “Disjunction” is the connective we call “or.” How do we determine the truth value for a disjunction of propositions, e.g., “John will
win or Harry will win”?
The only time a disjunctive proposition is false is when both disjuncts are false.
On page 150 you have the truth table of both inclusive and exclusive or. In this course, we’ll be mostly discussing inclusive or, which is symbolized ∨.
Negation (pages 150–152)
My students understand negation very well. Many have used it in a tease: “Hey, Jack, I hear that you passed the logic test … NOT!” The “not” is negation. It means that whatever the proposition’s value is (true or false), the negation flips it. So, if the proposition is true, a negation (symbolized “~”) makes it false. If the proposition is false, a negation makes it true.
The truth table for negation is on page 151.
Conditionals (pages 162–165)
We’ve seen conditionals before, in the “if … then… ” statements discussed in an earlier lesson. How do we determine the truth value of conditionals (symbolized with a sideways horseshoe, page 164)? Well, there’s controversy here, as the authors can attest. The authors go through a very nice discussion of how logicians have derived the conventional truth table for conditional. But this is too advanced for beginner logicians. So the description is optional. What is important is that you know that a conditional is true except in cases in which the antecedent (remember that word defined in an earlier lesson?) is true and the consequent false. See the top of
Maybe this joke will clear things up:
Daniel was a teenager about to go on his first date. Naturally, he was nervous and decided to consult his older (and wiser) brother, David, about what to do. Daniel was especially worried about how to break the initial silence and ease into the date.
David’s reply was “That’s easy. There are three topics of conversation that are sure to break the ice: food, family, and philosophy. Don’t forget to use them, and you’ll be fine.”
“Okay, I’ll remember that,” said Daniel (repeating to himself to set it in his memory) “Food, family, and philosophy.”
The next day, Daniel and his date were sitting opposite one another in a booth at the local ice cream parlor. The date was slurping an ice cream soda and not at all paying attention to Daniel. Daniel was sweating profusely; this is exactly the situation he feared. Then, Daniel remembered his brother’s advice about breaking the ice. Daniel blurted out with forced confidence, “Hey, do you like pizza?”
“No,” sneered his date, without even looking up from her soda.
Discouraged but not defeated, Daniel wiped the sweat off his brow, reasserted some self-confidence, and belted out, “Hey, do you have a brother?”
To this, his date looked up at him with a face of indifference and asserted “No” before going back to slurp her soda. Daniel was in agony. He searched his memory for his brother’s last “sure-fire ice-breaker.” Finally, it came to him—philosophy:
“Hey, if you had a brother, would he like pizza?”