T4: 7 of 10

Syntax

The syntax of a language is its grammar, the study of its properly formed expressions. So, in this tutorial we will be looking briefly but very carefully at the structure of SL (sentence logic without names) and PL (0th order logic, the sentences with predicates and names but without "quantifiers"). In tutorial 6, quantifiers are added to the langauge to produce something called "1st order logic".

Our most important task is to say just what objects count as sentences of our languages. We do this by showing how sentences are constructed. Let's begin with just SL.

Syntax for SL

We will call the simplest sentences of this language the atomic sentences of SL. These will include 'A','B','C','D'..., 'Z' (except to avoid confusion with the wedge 'v', we won't use the 'V'). Also, you may add a numeral subscript to any of the upper case letters: so 'A ₁ ' or 'L ₁₆ ' also count as atomic sentences of SL. That way, you'll never run out of atomic sentences. (Though you won't be able to type the subscripted atomic sentences in on the computer.)

The atomic sentences, as their name implies, are used as the basic building blocks for SL sentences. You use them to construct longer sentences, molecular sentences, like the following four sentences:

a. (A&B)
b. ~L 
c. (~L>C)
d. ((A&B)=(~L>C))

Sentence a simply takes two atomic sentences and, aided by the parentheses, connects them together with the '&'. For now, we will use parentheses when connecting pairs of sentences with a binary connective. Later we will drop the "outside" parentheses like those in "(A&B)" as this will cause no confusion.

But we will never use parentheses when adding a tilde. Notice that sentence b has none. We will NOT write "(~L)".

Sentence c. is formed by taking the negation "~L", the atomic sentence "C", and combining them with the horseshoe and the parentheses that go around the entire construct.

Sentence d. is a bit of a mess but its construction is similar to the others. It just takes our sentences a and c, connects them with the triple bar and surrounds the whole with parentheses.

The idea behind the definition of an SL sentence is that we can construct a molecular sentence of SL either by taking any one already constructed sentence and adding a tilde on its left or by taking any pair of already constructed sentences, writing a binary connective between them, and surrounding the result with parentheses.

That's it! Now, which of the following can be so constructed? i.e., which of the following count as Official SL Sentences? Click on all the following which can be constructed by the means just described.

1. (~Z)
2. (~A>B)
3. ~A>B
4. (Z&B&K)
5. ((Z=B)&K)
6. ~(A&B)
7. ~(~A>B)
8. (Z&B)&K
9. (((Z&B)&K)>~L)
10. (z&b)>~l