Chapter Two, Tutorial Two
Introduction to Sentence
Logic:
SL with Names
This tutorial introduces our first formal language and logic SL.
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.
...of compound sentences are easy to come by. We've been anyalyzing them informally in much of the last tutorial. Conditionals like
If Paul is from Quebec, then he's from Canada.
are prehaps the most important type of compound sentence for logic.
Q: Paul is from Quebec
Cp
Let's begin with the easiest sorts of compounding.
Example one:
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. Next, we look to complications.
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.
Next...
This simple logic is valuable is because it introduces much of the symbolism we will need in this entire course; that is SL. We begin with some examples translating from English into SL.
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:
A&B
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!)
Next, three terms to add to our logical vocabulary:
We will call '&' a "connective". It connects to two sentences making a complex or "molecular" sentence. Simple sentences with no connectives are called "atomic".
Next, we add a symbol for conditional statements.
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.
(A&B)>L
Make sure you see what the new symbols are then...
"L' symolizes their need for a loan. But what's that funny little hook?
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'll find it here: the logic font. (You can find the download on the Logic Café contents under "Tools".)
No horseshoe on your keyboard? That's normal. But you can display it nonetheless when you type in the greater-than symbol: '>'. You'll see a horeshoe. There will be one more connective of SL that has no normal font equivlalent. And there are more special symbols we'll use. So, you should download and install the special font now. It takes only seconds.
Finally...
We've seen
2. If both Agnes will attend law school and Bob will attend law school, then they will need to get a loan.
is translated
(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....