1. Truth tables

2. Defining our connectives
So we now know that our connectives ~, ∨ and ∧ are equivalents to the English phrases ‘not’, ‘or’ (inclusive) and ‘and’. (Let’s set aside the conditional for now – we’ll return to it later in the course.)
But in logic, we like to be precise. So it’d be nice to get a more specific definition of these connectives.
The way we do this is with a truth table.

3. Truth tables
Truth tables tells us for any complex sentence of formal logic what the truth value will be, given the truth values of its component atomic sentences.
On the left we have all the possible truth values of the atomic sentences, and on the right we have the resulting truth values of the complex sentence.

4. Negation
Like this, which is the truth table definition of negation: A ~A
t f f t
It simply means: if A is true, ~A is false; and likewise, if A is false, ~A is true.

5. Negation
And this is just what we should expect – given that negation is supposed to be an equivalent of the English phrases ‘not’ and ‘it is not the case that’.
Take an example – ‘the chair is by the desk’. When is it the case that ‘it is not the case that the chair is by the desk’ is true? Well, whenever ‘the chair is by the desk’ is false. (And vice versa.)

6. Conjunction
The truth table for conjunction is slightly more complex. A conjunction has two conjuncts – so we need more sentence letters on the left side of our table. We need to make sure that we include all the different possible combinations of the sentence letters.
The table looks like this…

7. Conjunction
A B A ∧ B t t t t f f f t f f f f This just means: A ∧ B is true if and only if both A is true and B is true; in all other cases it is false

8. Conjunction
Again, there’s a nice intuitive rationale for this. When is a sentence like ‘the chair is in the room and the desk is in the room’ true? Well, whenever ‘the chair is in the room’ is true, and ‘the desk is in the room’ is true. If either of those two sentences (or both) is false, then the longer sentence is false.

9. Disjunction
A B A ∨ B t t t t f t f t t f f f
This just means that A ∨ B is true whenever A is true or B is true (or both); otherwise it’s false.

10. Disjunction
And again, the rationale makes sense (providing that we’re all clear that ∨ is supposed to model the inclusive ‘or’, not the exclusive ‘or’).
When is a sentence like ‘Jim’s mother has brown eyes or Jim’s father has brown eyes’ true? Whenever either Jim’s mother has brown eyes, or his father has brown eyes, or both. It’s only false when both ‘Jim’s mother has brown eyes’ and ‘Jim’s father has brown eyes’ are both false.

11. Completing truth tables
It’s important to remember when constructing truth tables that you need to consider all the possible combinations of the relevant sentence letters. This is simple for cases with only two atomic sentence letters, but becomes harder with increasing complexity.
We’ll come back to this when we get to more complex cases – but for now, just remember that you have to make sure the left side of the truth table is complete.

12. Assignments of truth values
It’s useful to stop here to introduce the idea of an assignment of truth values. This is something that will become increasingly useful to us as we do more logic, so it’s worth getting our heads round it now.
An assignment of truth values to a set of sentence letters is a specification for each of those sentence letters of whether they are true or false. Each sentence letter is assigned a truth value.

13. Assignments of truth values
So A = t, B = f, C = t is an assignment of truth values for the sentence letters A, B and C.
It is worth noting that each line of a truth table gives us an assignment of truth values. A full truth table contains every possible assignment of truth values for the relevant sentence letters.
Sometimes, an assignment of truth values will be called a case.

14. Form and content (again)
When we were discussing formalization in the previous session we made a distinction between form and content, and said that in logic we are primarily concerned with the forms that arguments take, rather than their content.
We can make this point again by noticing that truth tables can tell us the truth values of compound sentences given the truth values of their component atomic sentences without our knowing what propositions the atomic sentences represent.

15. Form and content (again)
In order to know what truth vale ~A has, all I need to know is the truth value of A. I don’t have to know that A stands for ‘the chair is by the desk’ or anything like that. (And likewise for our other connectives.)

16. Functionality
Another way of putting this point is to say that our logical connectives are truth functional.
People with a background in mathematics or computer science will be familiar with the idea of functions – the basic point to grasp is that for any input, the function will give you a unique output. For all inputs there is one and only one correct output. (‘x+2’ is a function – for any number x, ‘x+2’ takes a unique value (4, when x is 2, 6 when x is 4, etc.))

17. Functionality
Our logical connectives are the same. For any truth value of A (t or f), our truth table for negation gives us a unique value of ~A (f or t). And likewise, for any possible combination of the values of A and B, our truth table for conjunction gives us a unique value for A ∧ B.
So our logical connectives ~, ∧ and ∨ are all functions from truth values of atomic sentence letters to truth values of compound sentences. They are truth functional.

18. A bonus question…
(This exercise is from the Teller book.) Given these definitions – can you give an example of a declarative compound sentence in English that is not truth functional?

19. Example answer… The light went on because Jeff pushed the button.
Assume that Jeff pushed the button and the light went on.
Then both the atomic sentences are true (‘the light went on’ is true and ‘Jeff pushed the button’ is true). But it is moot whether or not the compound sentence is true. Maybe the light went on because Jeff pushed the button. Maybe it was just a coincidence! There’s not enough information to say.
So there is no unique output of truth variable here – so no functionality!

20. A bonus bonus question…
'Nor’ is sometimes defined in truth functional terms and used as a connective. The symbol ‘↓’ is often used. It works like this: a sentence of the form ‘P ↓ Q’ is true when and only when neither P nor Q are true.
Write out the truth table that defines logical nor.