Philosophy 2340
Symbolic Logic

Curtis Brown

Chapter 3 introduces connectives. Now, in addition to atomic sentences, we can construct compound sentences (or "molecular" sentences, if we want to continue the physics metaphor).

We begin with the three "Boolean" connectives, and will add others later. One of them is a one-place connective, namely negation. The other two are two-place connectives, namely conjunction and disjunction. We can understand the meaning of sentences containing these connectives by making use of truth tables.

The first symbol is the negation symbol '¬', which we can read as "it is not the case that," or just "not." This is a one-place operator, meaning that if we prefix it to a sentence, we get another sentence. So, if P is a sentence, then so is ¬P. The negation symbol essentially flip-flops the truth value of a sentence; that is, if P is true, then ¬P is false, and if ¬P is false, then P is true. We can express this by means of a truth table:

P	¬P
T	F
F	T

The other two new symbols are both two-place operators: both of them combine two sentences into a new compound sentence. These symbols are the conjunction symbol ⋀, which we can read as "and," and the disjunction symbol ⋁, which we can read as "or." (This look similar -- don't forget which is which! It makes a huge difference.) If P and Q are sentences, then P⋀Q is also a sentence. We call it a "conjunction," and we call P and Q its "conjuncts." Similarly, if P and Q are sentences, then P⋁Q is a sentence. We call it a "disjunction," and we call P and Q its "disjuncts."

We can indicate the semantics of the conjunction symbol by this truth table:

P	Q	P⋀Q
T	T	T
T	F	F
F	T	F
F	F	F

And we can indicate the semantics of the disjunction symbol as follows:

P	Q	P⋁Q
T	T	T
T	F	T
F	T	T
F	F	F

The only line here that might seem fishy is the first line. Sometimes people interpret "or" in English as exclusive, so that 'P or Q' is false if both P and Q are true. (If the waitress says, "you can have a baked potato or French fries," she probably doesn't mean to include the possibility that you can have both.) Our text offers an argument that even in cases like this, "or" is actually inclusive. (If the waitress says "you can have a baked potato or French fries," and then a few seconds later says, "Oh, or if you like you can have both," we don't understand her as contradicting her earlier assertion. This suggests that the suggestion that we couldn't have both was not a logical implication of her original sentence.) In any case, however "or" works in English, the symbol ⋁ is to be understood as inclusive. If we want an exclusive or as well, we can define it in terms of our other symbols. For example, (P⋁Q) ⋀ ¬(P⋀Q) will do the trick.

The exercises on this material just give you the opportunity to become familiar with the meanings of these connectives and how they work by comparing sentences with worlds. Our text describes this as analogous to the immersion method of learning a new language, as opposed to the translation method. Our strategy in general is to begin by directly comparing our formal language with the world (well, with Tarski's worlds), and only later to translate back and forth between symbolic notation and English.

This section concerns "the game." I don't have much to say about this, but it does provide an interesting way to see what you are missing if you don't see why a sentence has the truth value Tarski's World says it has in a particular world. You can "play the game," holding that it has the truth value it seems to you to have, and then go through a series of steps that should show exactly what it is you've missed. 

This section concerns ambiguity, and how to prevent it by using parentheses.

The simplest thing to do would be to simply require that every compound sentence be built up one step at a time from simpler sentences, putting parentheses around the result every time we use a connective. Thus the negation of a sentence P would be (¬P), the conjunction of sentences P and Q would be (P⋀Q), and so on. We would quickly end up with sentences that contain an awful lot of parentheses! A fairly simple example would be ((¬Tet(a)) ⋀ (¬(Cube(b) ⋁ Dodec(c)))).

Our text allows us to improve readability without introducing ambiguity by several means. First, we eliminate the outermost pair of parentheses around any formula. Second, we eliminate the parentheses around negations. This is all right because we introduce the convention that a negation symbol always negates the shortest complete formula that follow it. (Another way to put this: we have an "order of operations" convention that negation takes priority over the other connectives.)

Third, and more interestingly, we allow strings of conjunctions or disjunctions. In a sense this does introduce ambiguity, but it is an ambiguity that does not matter, because every disambiguation is guaranteed to have the same truth value. (We'll look at this by means of truth tables in class.)

Equivalences: DeMorgan's equivalences and double negation.

DeMorgan's: ¬(P⋀Q) is equivalent to ¬P ⋁ ¬Q; ¬(P⋁Q) is equivalent to ¬P ⋀ ¬Q

Double Negation: ¬¬P is equivalent to P

(It follows from the Double Negation equivalence that ¬¬¬¬¬¬P is equivalent to P, etc.)

Note that these logical equivalences are not rules in our formal system. They can't be used in a formal proof without first deriving them from the fundamental rules. (After we've done that, though, I'll let you use them as "derived rules," i.e. things which, although not basic rules, we know that we could prove from the basic rules if we had to.)

Translation. Things to remember: the English words 'and', 'but', 'however', 'yet', 'nevertheless', and 'moreover' all get translated as '⋀'. It is a good idea to stay fairly close to the English when doing translations, even though very different sentences may still be logically equivalent.