Ash
TNPer
Answers to exercises should be PM'ed directly to me. Discussion of the exercises is positively encouraged.
Logic Lesson 2
Formal Sentential Logic
Sentential Logic, also known as propositional logic, is one of the two forms of classical formal logic. The other is quantifier logic, also known as predicate logic.
Letters can stand for statements:
H = Hyanygo is posting in the Judicial Training Centre.
Capital letters are used for constants while lower case is used for statement variables (will use letters a and b).
Truth values, True and False can be shortened down. So:
TRUE shortened to T
FALSE shortened to F
I will assume that you know basic algebra in all of this.
We can look at five logic operators.
Negation
[c]Not[c]~ x[c]&[c]Conjunction[c]And[c]x & y[c]V[c]Disjunction[c]Or[c]x V y[c]?[c]Conditional[c]
If...then
[c]x ? y[c]?[c]Biconditional[c]
If and only if (iff)
[c]x ? y
NEGATION
A variety of symbols can be used to present negation:
-
~
¬
Let's say something is true, that is it has the truth value T. Now saying truth value over and over again is a bit tiresome, so I'll call it the value instead. Kk?
So going back to the example above,
H = Hyanygo is posting in Albion.
If H is TRUE, then H has the value of T.
What happens if we negate H? In other words, what is the statement with this:
~H
?
H = Hyanygo is posting in Albion.
~H = Hyanygo is NOT posting in Albion.
So what is the value of ~H ? If H is TRUE then ~H is FALSE. You could then say, "well, what's the value of ~~H? It is .... *drumroll* .... TRUE.
We can construct a truth table. I'll get to truth tables later on but this one is pretty simple and straightforward.
If you look at the far left hand column you can see H. In the first row, it is initially TRUE. Go along the row and you can see that H's negation makes it FALSE and so on.
In the second row, we start of with H being FALSE. Going along the row we end up in the column, ~H, this tells us to negate H. The negation of FALSE is TRUE. Going along further we get to ~~H, so we negate the negation, so it becomes FALSE. And so on.
EXERCISE 1
Let E = Earthie is a girl
with E having value T
Q1)What is the truth value for the following?
~~E
~E
~~~E
~~~~~~E
Q2)How would one write, in sentential logic, "Earthie is not a girl"? What is the truth value of this statement?
CONJUNCTION
The conjunction operator, & , is also called the and operator. It may be useful to think of it as joining up two statements with the word and.
For an example, look at this statement:
Paris is the capital of France and six plus six equals twelve
To find out whether this statement is true or false, you need to realise that this statement is in fact made up of two smaller statements. The truth value of the whole statement depends on the truth values of each of the smaller statements. Because both parts of the statement are true (Paris is indeed the capital of France, and six plus six does equal twelve ), the whole statement is true.
But what if one of the statements was false? Let's suppose that Rouen is the capital of France, not Paris. The whole statement then is false.
So how do we write this all out in symbols?
First give each small statement a letter, like so:
P = Paris is the capital of France
S = Six plus six equals twelve
Then we can use the and operator and squash them together:
P & S
The truth value for this whole statement depends on the truth value of each of the smaller statements. If both of these smaller statements are true, then the whole statement is true. If either or both are false, then the whole statement is false.
We can construct a truth table :
To read this table notice that we have to cycle through all the permutations of the truth values of a and b. For example a statement could have a smaller statement inside it that is true and another false. This would correspond to the second row of the table.
a = All men are mortal
b = Earthie is a boy
a is TRUE so we put in its column, T, but b is FALSE so we put in its column, F. And then we look at the last column and see that it asks us to use the and-operator on a, b. Taken as a whole, "All men are mortal and Earthie is a boy" is FALSE, so we put the letter F in the row. So the second row is explained and the rest follow.
EXERCISE 2
Let E = Earthie is a girl
with E having value T
Let M = All men are mortal
with M having value T
Let R = The capital of France is Rouen
with R having value F
Q1)What is the truth value for the following?
M & E
R & M
R & E
Q2)How would one write, in sentential logic, "Earthie is not a girl and the captial of France is Rouen"? What is the truth value of this statement?
Q3)What is the truth value for the following?
~(M & E)
~(R & ~M)
~(~R & ~E)
~(R & ~E) & (~M)
~((M & R) & (~E & R))
DISJUNCTION
Like the and-operator, the or-operator is made up of two smaller statements. The symbol for the or-operator / disjunction operator is V.
Let's go through an example like we did before.
P = Paris is the capital of France
S = Six plus six equals twelve
If we wrote : Paris is the capital of France or six plus six equals twelve , we can write that as:
P V S
Now we need to know if this statement is true. Like the and-operator, when both parts of the statement are true the whole statement is true. However, if only one smaller statement is true then the whole statement is true. Take this example:
Paris is the capital of France or two plus two equals five
This whole statement is true.
However, when both parts are false, the whole statement is false:
Rouen is the capital of France or two plus two equals five
Like the and-operator we can construct a truth table.
We have to be very careful with the or-operator because in English, 'or' has two meanings, it can either be inclusive [this choice or that choice or both] or exclusive [this choice or that choice but not both]. And yes. I'm well aware of the weirdness of explaining 'or' while using it.
In logic, by convention, the or-operator is always inclusive. If both parts of an 'or statement' are true then the whole is true.
EXERCISE 3
Let E = Earthie is a girl
with E having value T
Let M = All men are mortal
with M having value T
Let R = The capital of France is Rouen
with R having value F
Q1)What is the truth value for the following?
M V E
R V M
R V E
Q2)How would one write, in sentential logic, "Earthie is not a girl or the capital of France is Rouen"? What is the truth value of this statement?
Q3)What is the truth value for the following?
~(M V E)
~(R V ~M)
~(~R & ~E) V ~E
~(R & ~E) V (~M)
~((M & R) V (~E V R))
~((~E V ~R) & (M & ~R)) V (M V R)
CONDITIONAL
The symbol ? is called the conditional operator. An example would probably suffice at this time:
If Hyanygo is posting in the Halls of Justice then Hyanygo is a Justice.
Breaking this down to:
A = Hyanygo is posting in the Halls of Justice
J = Hyanygo is a Justice
Connecting the two we have:
A ? J
We can construct a truth table.
Let E = You're in England
Let U = You're in the United Kingdom of Great Britain and Northern Ireland
Let's consider
E ? U
If you are in England, then you are in the United Kingdom of Great Britain and Northern Ireland.
This is true because England is completely inside the United Kingdom of Great Britain and Northern Ireland.
The converse of the statement is
U ? E
If you are in the United Kingdom of Great Britain and Northern Ireland, then you are in England.
This is false because you can be in Scotland. When an if-statement is true, it does not follow necessarily that its converse is true as well.
The inverse of the statement is
~E ? ~U
If you are not in England, then you are not in the United Kingdom of Great Britain and Northern Ireland.
Both parts of the if-statement have been negated. An inverse isn't guaranteed to be true if the if-statement is true. After all, you can still be in Scotland!
The contrapositive of the statement is
~U ? ~E
If you are not in the United Kingdom of Great Britain and Northern Ireland, then you are not in England.
The order has been switched and both parts have been negated. The contrapositive of an if-statement has same truth value as the if-statement. It can also be shown that the inverse of a statement is a contrapositive of the converse.
EXERCISE 4
Let E have value T
Let M have value T
Let R have value F
Q1)What is the truth value for the following?
M ? E
R ? M
R ? E
E ? M
M ? R
E ? R
Q2)What is the truth value for the following?
~(M V E) ? ~(R V ~M)
~(R V ~M) ? (~((~E V ~R) & (M & ~R)) V (M V R))
(~(~R & ~E) V ~E) ? (~(R & ~E) V (~M))
((~(~R & ~E) V ~E) ? (~(R & ~E) V (~M))) ? ((M V R) & (~E ? R))
BICONDITIONAL
Consider this if-statement:
If Abbey has posted in the recruitment thread then Abbey is recruiting.
A = Abbey has posted in the recruitment thread
R = Abbey is recruiting
This statement says:
1. If you see a post, then you know that Abbey is recruiting, but
2. If you see Abbey recruiting, then you can't be sure that there is a post.
This if-statement can be represented in as A ? R, with the arrow pointing in the direction of implication: Post implies recruitment.
Now consider this statement:
Abbey has posted in the recruitment thread if and only if Abbey is recruiting.
This statement is similar to the previous one, but it extends things a bit further. In this case, the statement says:
1. If you see a post, then you know that Abbey is recruiting, and
2. If you see Abbey recruiting, then you know that there is a post in the recruitment thread.
This statement can be represented in as A ? R, with the double arrow providing a clue to its meanings: Both post implies recruitment, and recruitment implies post.
Because the if-operator is also called the conditional operator, the if-and only-if-operator is quite logically called the biconditional operator. Another shorter way of referring to it is the iff-operator.
EXERCISE 5
Let E have value T
Let M have value T
Let R have value F
Q1)What is the truth value for the following?
M ? E
R ? M
R ? E
E ? M
M ? R
E ? R
Q2)What is the truth value for the following?
~(M V E) ? ~(R V ~M)
~(R V ~M) ? (~((~E V ~R) & (M & ~R)) V (M V R))
(~(~R & ~E) V ~E) ? (~(R & ~E) ? (~M))
((~(~R & ~E) V ~E) ? (~(R & ~E) V (~M))) ? ((M V R) & (~E ? R)))
SOME FANCY WORDS
A statement in the form of a & b is called a conjunction. Both parts of the statement are called conjuncts.
A statement in the form of a V b is called a disjunction. Both parts of the statement are called disjuncts.
A statement in the form of a ? b is called an implication. The first part is called the antecedent and the second part is called the consequent.
A statement in the form of a ? b is called a double implication.
