1. Compounding

2. Compounding
So far we’ve looked at some of the basics of formal sentence logic.
And we’ve learned how to translate simple expressions with just one connective (like ‘Karl likes Fred and Fred likes Karl’).
But what about more complex sentences, with more connectives?

3. Compounding
For example, how would we formalize the following sentence of English: ‘Deb doesn’t like Dan and Dan likes Steve’.
Well here’s one way: ‘~A ∧ B’, where A = Deb likes Dan and B = Dan likes Steve.

4. Compounding
Ok, but now what about the sentence: It is not true that both Deb likes Dan and Dan likes Steve. Should we just do the same? ‘~A ∧ B’?
But then we would have the same translation as we did on the previous slide. But these sentences say different things – the first says Deb doesn’t like Dan and Dan likes Steve, while the second just says that it is false that Deb likes Dan and that is is false that Dan likes Steve.

5. Brackets The way to solve this problem is to introduce brackets.
For mathematicians – you will recognize these! They play a similar role in logic as they do in math – they tell us how to read the expression.
So ‘(~A) ∧ B’ means A is false and B is true. But ‘~(A ∧ B)’ means that the whole of ‘A ∧ B’ is false.

6. Brackets
So our first sentence (‘Deb doesn’t like Dan and Dan likes Steve’) becomes ‘(~A) ∧ B’. While our second sentence (‘It is not true that both Deb likes Dan and Dan likes Steve’) becomes ‘~(A ∧ B)’.
The idea is simple – whichever connective falls inside the brackets is to be applied before the connectives outside the brackets. In the first example above, you negate A first and only then form the conjunction. In the second, you form the conjunction first, and then negate the whole thing.

7. Brackets
Brackets can make a huge difference. Consider: (1) P ∨ (Q ∧ R) and (2) (P ∨ Q) ∧ R
(1) says either P is true or both Q and R are true (or both). (2) says either P is true or Q is true (or both), and R is true.

8. Brackets
This is like the difference between the English sentences: (3) Either Jan was there, or both Steve and Jon were there. (4) Either Jan was there or Steve was there. And Jon was there.

9. Brackets
A way to see that these sentences are different is to point out that they are true in different situations. To see this, we can write out a truth table for them, and show that they can have different truth value even for the same assignment of truth values to the relevant atomic sentences.
Let’s write out this truth-table together on the board now.

10. Bracketing conventions
A little note on bracketing conventions. Strictly speaking, there should always be a bracket around the entire sentence of sentence logic. I.e. you should always write ‘(A ∨ B)’ rather than simply ‘A ∨ B’.
But there is a convention that you’re allowed to lose the outermost pair of brackets. This helps keep things simple. But don’t forget to put those brackets back in if you compound the sentence with further connectives!

11. Bracketing conventions
It’s also standard to assume that the negation symbol applies to the smallest component that it is attached to, and that the extra pair of brackets that would go to indicate that can be dropped too.
So instead of ‘(~A) ∨ B’, we just write ‘~A ∨ B’. But ‘~(A ∨ B)’ must have the brackets in the place.

12. Bracketing conventions
Finally, in the Teller textbook, Teller has a convention to the effect that you must use different kind of brackets when you add extra pairs (square brackets, squiggly brackets etc).
We won’t follow that convention – all our brackets will be normal brackets. Just replace his squares and squigglies with ordinary ones!

13. The main connective
It is useful to have the concept of the main connective when thinking about compound sentences. The main connective is just the last connective that was added in constructing the larger sentence out of its component parts.

14. The main connective
So what’s the main connective of: (A ∨ B) ∧ ~C ~((A ∧ B) ∨ (C ∨ D)) (((P ∨ Q) ∧ R) ∨ P) ∧ Q (X ∧ Y) ∨ ~(~(X ∨ Z) ∧ Y)

15. Rules of formation
We can summarize these ideas with some strict rules.
This set of rules gives us a method for constructing compound sentences, and also for telling whether or not any expression involving As and Bs and connectives is a genuine sentence of sentence logic.

16. Rules of formation
Every capital letter 'A', 'B', 'C' is a sentence of sentence logic. Such a sentence is called an Atomic Sentence or a Sentence Letter.
If X is a sentence of sentence logic, so is (~X), that is, the sentence formed by taking X, writing a ‘~' in front of it, and surrounding the whole by parentheses. Such a sentence is called a Negated Sentence.

17. Rules of formation
If X and Y are sentences of sentence logic, so is (X ∧ Y), that is, the sentence formed by writing X, followed by ‘∧', followed by Y, and surrounding the whole with parentheses. Such a sentence is called a Conjunction, and X and Y are called its Conjuncts.
If X and Y are sentences of sentence logic, so is (X ∨ Y), that is, the sentence formed by writing X, followed by ‘∨', followed by Y, and surrounding the whole with parentheses. Such a sentence is called a Disjunction, and X and Y are called its Disjuncts.

18. (Bracketing conventions)
The exceptions to these rules are the bracketing conventions – we leave out the outermost pair of brackets, and we leave out the brackets around a negated sentence even if they are not the outermost set of brackets, because we agree to understand that ‘~’ applies to the shortest full sentence that follows it.

19. Rules of valuation
We can also put together some precise rules for how to work out the truth values of compound sentences if somebody gives us the truth values of the component atomic sentences.

20. Rules of valuation
The truth value of a negated sentence is t if the component (the sentence which has been negated) is f. The truth value of a negated sentence is f if the truth value of the component is t.
The truth value of a conjunction is t if both conjuncts have truth value t. Otherwise, the truth value of the conjunction is f.
The truth value of a disjunction is f if both disjuncts have truth value f. Otherwise, the truth value of the disjunction is t.

21. Examples: are these sentences of sentence logic?
P ∧ ∧ Q
A ∨ ~(B ∨ C)
((A ∨ B) ∧ ~A
(((P ∧ ~Q) ∨ R) ∨ ~~~S)