1. Truth Functions

2. Relationships A function is a relation that links members of sets.
The set of all x which are in relation R with some y is called the domain of R.
The set of all y such that, for some x, x is in relation R with y is called the range of R.

3. Functions A function is a special kind of relation
It’s a relation where every object, a, from the domain, is related to precisely one object in the range, namely, to the value of the function at a.

4. Domain Codomain Range

5. Functions
Domain
Range
A function is a rule for linking members of the domain to members of the range
It links them in such a way that for every member of the domain there is a unique member of the range to which it connects.
If you know the point at which you start, you know where you end up.

6. Functions
Domain
Range
But what about the links that go from more than one member of the domain to a single member of the range?
Is this ok?
Yes! Because even though in each of these cases we start at different places we still know where we end up!

7. Not a Function!
Domain
Range
But what if it were like this, with links from one member of the domain connecting to more than one member of the range?
Is this ok?
No! Because if it’s like this, then knowing where we start doesn’t tell us where we’re going to end up!

9. The Black Box f
Range
Domain
One at a time, puleeze
So we can think of a function as a black box, a machine that takes members of the domain as inputs and pumps out their values under that function

10. What are functions…really?
A function is a set of ordered pairs: <x, y>
A subset of the Cartesian Product of the domain and codomain.

11. What are functions…really?
Cartesian Product: all possible pairs of shape & color
Shape-Color function: {<triangle, red>, <bar, yellow>, <thingy, green>, <rectangle, red>}

12. Functions & Other Relations?
If it’s not a function, it’s like a gumball machine: you put in your money but you don’t know exactly what will come out.
If it’s a function, if you know what the function is, and know what goes in, then you know exactly what will come out.

13. Functionland Everyone Welcome!
What members?
The members of the domain and range of functions can be anything you please:
They can be numbers…
Or they can be truth values!
Functionland Everyone Welcome!

14. f (x) = x + 1 +1 1
1 goes in…

15. f (x) = x + 1 +1 2
2 comes out…

16. f (x) = x + 1 +1 2
2 goes in…

17. f (x) = x + 1 +1 3
3 comes out…

18. f (x) = x + 1 +1
714
714 goes in…

19. f (x) = x + 1 +1
715
Boring.
715 comes out…
Functions are predictable!

20. Truth Functions f T T F F
Truth functions take truth values as inputs and output truth values.
The 5 connectives of propositional logic represent 5 different truth functions

21. A function is as a function does~ ^2 +1 
Some functions have more than one input
What makes a function the function it is (rather than some other function) is the characteristic pattern of input and output. We can display this by showing tables of values for functions.

22. ^2
f (x) = x2 x
f (x) 1 2 4 3 9 16

23. ~
f (p) = ~p p
f (p) T F
Each of the truth functions represented by the five connectives of propositional logic has a characteristic truth table that makes it the function it is.

24. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
Truth tables for the connectives give us the meanings of ‘not’, ‘and’, ‘or’, ‘if-then’, and ‘if and only if’ in the language of propositional logic--which may not be exactly the same as their meanings in ordinary English.

25. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
p and q are variables standing for sentences.
The columns underneath them represent every possible combination of truth values they may have.

26. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
The other columns represent the values of the truth functions at every given combination of truth values

27. Non-Truth Functional Connectives
All the connectives of propositional logic are truth functional.
But some of the connectives of ordinary English are not truth functional.
These include (among others) English connectives that concern certain mental states and the ordinary English if-then. ?

28. Non-Truth Functional Connectivesp
Sam believes that p T ? F
China is more populous than India.
Sam believes that… T
What do I know?!?!! ?
Sam believes that China is more populous than India.

29. Non-Truth Functional Connectivesp
Sam believes that p T ? F
The moon is made out of green cheese.
Sam believes that… F
Duh, mebbe …dunno. ?
Sam believes that the moon is made out of green cheese.

30. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
Negation reverses truth value:
False True True False
Easy. Just like English.

31. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
OK.
Conjunction is true when both conjuncts are true--otherwise false

32. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
It’s the inclusive or.
Disjunction is false when both disjuncts are false--otherwise true

33. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
Biconditional is true when both sides have same truth value--otherwise false.
It expresses logical equivalence and so tests for sameness of truth value
New one on
me, but OK.

34. Truth Tables for the Connectivesp ~p T F p q
p • q
p  q
p  q
p  q T F
Huh??!? You gotta be kidding!
Conditional is false when antecedent is true and consequent false--otherwise true

35. ‘Paradoxes’ of Material Implicationq
p • q
p  q
p  q
p  q T F
This means that whenever the antecedent is false the whole conditional is true!
‘If the moon is green cheese then 2+2 = 4’ – TRUE
‘If the moon is green cheese then 2+2 = 5’ – TRUE
‘If the moon is green cheese then the moon is not green cheese’ - TRUE

36. ‘Paradoxes’ of Material Implicationq
p • q
p  q
p  q
p  q T F
It also means that whenever the consequent is true the whole conditional is true!
‘If the moon is green cheese then 2+2 = 4’ – TRUE
‘If the moon is not green cheese then 2+2 = 4’ – TRUE
‘If 2+2 = 5 then 2+2 = 4’ - TRUE

37. The conditional of propositional logic is defined by its characteristic truth table--and we can define symbols any way we please.
It just isn’t a perfect translation of the ordinary English if-then because…

38. The English if-then is not truth-functional!
IF-THEN
The conditional of logic is a black box but the IF-THEN of ordinary English is a gum machine!

39. Computing Truth Values
Because the connectives of propositional logic (unlike some of the connectives of ordinary English) are truth functional we can compute the truth values of whole sentences in propositional logic if we know the truth values of their parts.
And sometimes we don’t even need to know the truth values of all their parts to do this.
And sometimes we don’t even need to know the truth values of any of their parts!

40. Computing Truth Value p ~p T F p q
p • q
p  q
p  q
p  q T F
When we compute the truth value of a sentence given truth values assigned to its constituent sentence letters we are computing a composite function!
~ (p • q) = NOT (AND (p, q)) p q T F
AND(p,q) T F
NOT(AND(p,q)) F T

41. Computing Truth Value (A • Y)  (Z  B)F p q
p • q
p  q
p  q
p  q T F
We want to compute the truth value of this sentence given truth values assigned to its constituent sentence letters.
The truth value for the whole sentence will go in a circle under its main connective.
(A • Y)  (Z  B)

42. Computing Truth Value (A • Y)  (Z  B) T F F Tq
p • q
p  q
p  q
p  q T F
We start by assigning truth values to the sentence letters.
How do we know what truth values to assign?
IT TELLS YOU IN THE DIRECTIONS FOR THE EXERCISE!!!
(A • Y)  (Z  B) T F F T

43. Computing Truth Value (A • Y)  (Z  B)F p q
p • q
p  q
p  q
p  q T F
Compound sentences have parts, or subformulas, that are themselves sentences.
The truth value for each subformula goes underneath its main connective--or if its just a sentence letter, underneath it.
(A • Y)  (Z  B)
Truth value for ‘A’ goes here
Truth value for ‘Z  B’ goes here

44. Computing Truth Value (A • Y)  (Z  B) T F F F Tq
p • q
p  q
p  q
p  q T F
Then, working from smaller parts to larger parts, we assign truth values to more subformulas referring to the truth tables.
‘A • Y’ is false because TRUE-FALSE for conjunction is FALSE, as the truth table tells us.
(A • Y)  (Z  B) T F F F T

45. Computing Truth Value (A • Y)  (Z  B) T F F F T Tq
p • q
p  q
p  q
p  q T F
‘Z  B is true because FALSE-TRUE for disjunction is TRUE, as the truth table tells us.
We don’t care about the truth values for A and Y anymore because we’ve used them to compute another truth value.
(A • Y)  (Z  B) T F F F T T

46. Computing Truth Value (A • Y)  (Z  B) T F F F T T Tq
p • q
p  q
p  q
p  q T F
Now we have truth values for both antecedent and consequent of the whole conditional so we compute the truth value for the whole sentence.
FALSE-TRUE for conditional is TRUE so the sentence is TRUE.
(A • Y)  (Z  B) T F F F T T T

47. What’s this exercise for?
This is a preliminary exercise for doing truth tables for compound sentences.
When you compute the truth value of a compound sentence given the truth value of its constituent sentence letters you’re doing one row of a truth table.
Each row of a truth table represents a truth value assignment for its constituent sentence letters.
A complete truth table gives truth values for a sentence in every possible truth value assignment.

48. To be continued…