Chapter Two, Tutorial One
SL is the logic of compound sentences. So, we will be studying the logical principles resulting from the construction of complicated sentences from simpler ones.
Examples of compound sentences are easy to come by. You discussed these ad nauseum back in grade school. For instance, one could say:
Agnes is to attend law school; as well, Bob will attend law school.
Well, of course, we'd usually put this in a more compact way, say, "Agnes and Bob will attend law school". Or, supposing that Agnes is Agnes Buck and Bob is her husband of the same surname, one might say "The Bucks will both attend law school". In any case, we can understand these as ways to say:
Both Agnes will attend law school and Bob will attend law school.
We will prefer this latter, longer way of putting the point in the next few chapters because it shows a long sentence built out of shorter ones. That's what is meant by a compound sentence.
Compound sentences get more complex, and sometimes, more interesting than our first example. We could take the sentence displayed above and add to it:
If both Agnes will attend law school and Bob will attend law school, then they, the Bucks, will need to get a loan.
And it's easy to understand the logic behind such a statement: assuming for the moment that Agnes and Bob will indeed both attend law school, then their need of a loan is the obvious conclusion.
Much of the logic of compound sentences is really that simple. But it's also an important first step in logic. If we can understand this simple fragment of natural language, we'll be well on our way to grasping the logic for much more sophisticated contexts.
One reason that this simple logic is so valuable is that it introduces much of the symbolism we will need in this entire course; that's SL. We begin with some examples. First the simple statement above,
1. Agnes is to attend law school and, as well, Bob will attend law school.
can be translated into SL symbols with:
Of course, 'A' stands for "Agnes is to attend law school" and 'B' stands for "Bob will attend law school". The funny little symbol, '&', the ampersand, is SL's shorthand way of saying "and". (Can you find this symbol on your keyboard? You will need it!)
We will call '&' a "connective". It connects to two sentences making a complex or "molecular" sentence. Simple sentences with no connectives are called "atomic".
And our slightly more complicated example,
2. If both Agnes will attend law school and Bob will attend law school, then they will need to get a loan.
can be translated with the help of new symbols. First notice that 2. is made up of "If" plus sentence 1, plus "then" and a new sentence about the need for a loan. Assuming their need for a loan can be symbolized with 'L', then the whole of 2 can be symbolized as:
Here the new symbol, '>', the horseshoe, takes the place of "then" in 2. displayed just above. We will say that the horseshoe is a symbol for the "if... then..." of English. (Make sure you download and install the Logic Font so that the horseshoe looks like a horseshoe! You can find the download on the Logic Café home page.)
No horseshoe on your keyboard? That's normal. But you can display it nonetheless when you type in the greater-than symbol: '>'. Of course, you'll need the logic font to see any horseshoes.
Back to logic and the symbolization displayed above: '(A&B)>L'. The parentheses serve to group Agnes and Bob's attending of law school. One first way of thinking about it is that her attending law school along with his attending law school together lead to the need for a loan. In English, we don't use the parentheses, but group by using devices like the comma or the word "together". Look back at 2. above to see how the comma is used to group.
Okay; that's a beginning. Before you move on to some details, guess which of the following correctly symbolizes
3. If they get a loan, then Agnes and Bob will both attend law school.
Click on the best symbolization for 3: