"John likes Mary or Steven" can be broken into "John likes Mary" (OR) "John likes Steven"
○
"John likes Mary and Bill" can be broken into "John likes Mary" (AND) "John likes Bill"
○
The truth of the compound sentence is a function of its parts (hence the phrase "truthfunctional" logic")
○
Truth functional compound sentence  sentences that are formed out of other, simpler sentences and whose truth values are related to those simpler sentences.
•
M = def. John likes Mary
B = def. John likes Bill
S = def. John likes Steven
In this chapter, we will use capital letters as abbreviations of sentences:
•
M and B
○
M or S
○
Our original sentences can now be written as:
•
Compound Sentences
Truth Functions
It is true that B
1)
Either B or Eric is tired
2)
B, or B and H
3)
Bill hopes that B
4)
EX:
•
When a variable is repeated, , it must be replaced by the same sentence at each occurrence
•
Truth function  a sentence function that has an additional feature: truth functionality. The truth or falsity of the whole sentence is a function of the truth or
falsity of the sentences replacing the variables in it
•
1 says that "It is true that it is raining", which is true if it is raining and false if it is not.
○
2 says "Either it is raining or Eric is tired". This is not a truth function because if it is raining then the sentence is true but if it not raining the truth of the
sentence is not determined because it depends on whether or not Eric is tired
○
1 and 3 are truth functional because they only contain sentence variables and truth functional words like "and", "or", or "if then".
○
However, 4 is not truth functional because it says "Bill hopes that it is raining" and the truth or falsity of the sentence is entirely a function of what Bill
hopes, and not whether or not it is raining.
○
For example, if B = "It is raining" then:
•
Sentence function  a string of English words and one or more sentence variables. If the sentence variables are replaced by English sentences, the whole string
becomes a sentence in English.
The Five Standard Truth Functions
Q
Not Q
negation
(it is false that Bill is sad)
Q ^ W
Q and W
conjunction
(Bill is late and Mary is tired)
Q v W
Q or W
disjunction
(Bill is late or Mary is tired)
Q > W
If Q then W
implication
(if Bill is late then Mary is tired)
Q <> W
Q if and only if W
equivalence
((Bill is late if and only if Mary is tired)
Truth Tables
Negation
1.
Q
Q
Q
T
F
T
F
T
F
Conjunction
2.
Q W
Q ^ W
T T
T
T F
F
F T
F
F F
F
Disjunction
3.
Q W
Q v W
T T
T
T F
T
F T
T
F F
F
Implication
4.
Q W
Q > W
T T
T
T F
F
Chapter 7 Text Notes
March2612
11:50 PM
Ch.7 Text Notes Page 1
T F
F
F T
T
F F
T
Equivalence
5.
Q W
Q <> W
T T
T
T F
F
F T
F
F F
T
Simple sentence  a sentence containing no truthfunctions. EX: "Bill is hot". They are assigned sentence letters and we write a different letter for each simple
sentence
•
It may also contain other compound sentences as constituents. EX: "If it is raining and the river is swollen, then we stay inside" has two truth functions and
three simple sentences, written as "(q ^ p) > r"
○
Compound sentence  a truthfunctional sentence containing both simple sentences and truth functions. EX: "It is raining and the river is swollen" contains two
simple sentences and the truth function "and". It is written as "q ^ p"
•
The compound sentence "p v p" is an example of a tautology. The truth table of a logical truth will contain only T's in the column under the logical truth
○
Logical truth  a sentence that is true under every possible circumstance: it can never be false. Is also called a tautology.
•
The compound sentence "p ^ p" is an example of a contradiction
○
The negation of a contradiction is a logical truth and the negation of a logical truth is a contradiction.
○
Contradiction  a sentence that is false in every possible circumstance. It can never be true. The truth table of a contradiction will accordingly contain only F's in
the column under the contradiction.
•
We can express an argument as a conditional sentence stating that if the conjunction of the premises is true then the conclusion is true.
○
We make this by conjoining all the premises, and then making a conditional sentence with the conjunction of the premises as the antecedent and the
conclusion as the consequent.
○
This is called the corresponding conditional and is special because the argument is valid if and only if the corresponding conditional is a logical truth
○
An argument is valid if in every possible circumstance in which its premises are true then the conclusion is true as well.
•
(((((premise 1) ^ (premise 2)) ^ (premise 3)) ^ (premise 4)) … ) > (conclusion)
The first step is to produce a compound sentence in which the conclusion is made conditional upon a conjunction of the premises:
○
Then we determine whether this sentence is a tautology or logical truth
○
We can test an argument for validity by performing the corresponding conditional of an argument and doing the truth table of it.
•
Arguments Using Truth Tables
EX: Suppose that P and Q are the constituent simple sentences of the argument and the argument is:
Premise 1 = P v Q "either P or Q"
Premise 2 = P > Q "if P is false then Q is false"
(Therefore) Conclusion = Therefore P
((P v Q) ^ (P > Q)) > P
We write the corresponding conditional of the argument as:
We now write the truth table as:
P Q
((P v Q)
^
( P
>
Q ))
> P
T T
T
T
F
T
F
T
T F
T
T
F
T
T
T
F T
T
F
T
F
F
T
F F
F
F
T
T
T
T
1
5
2
4
3
6
Since the column for the whole sentence contains only T's, the sentence is logically true and a theorem.
When a compound sentence has two or more truth functions, we need to find the one with the widest scope. It will be the one whose associated parenthesis
cover the whole sentence
•
Once we have identified the general form of the sentence, we look inside of it to see if it contains sentences containing truth functions. We continue this process
until all the constituent sentences are simple sentences.
•
Doing Truth Tables
EX 1: (P > Q)
Step 1:
P Q
P > Q
T T
T
T F
F
F T
T
Ch.7 Text Notes Page 2
F F
T
Step 2:
P > Q

(P > Q)
T
F
F
T
T
F
T
F
Then we make a single truth table and put both steps on it:
P Q

(P > Q)
T T
F
T
T F
T
F
F T
F
T
F F
F
T
Ex 2: (P ^ Q) <> (P v Q)
P Q

(P ^ Q)
<>
( P
v
Q )
T T
F
T
T
F
F
F
T F
T
F
T
F
T
T
F T
T
F
T
T
T
F
F F
T
F
T
T
T
T
5
4
6
1
3
2
Translation
Identify the truth functions in the sentences to be translated
i.
Identify the simple sentences in the sentences to be translated
ii.
Write a dictionary of simple sentences, assigning a letter to each sentence starting (for example) with "P"
iii.
Replace the English truth functions with their symbolic logic counterparts, using parentheses and starting with the truth function having the widest scope
iv.
Replace the simple sentences with the sentence letters you have assigned in your dictionary
v.
Steps in translating from English to Symbolic Logic:
EX: If Bill stops at the store while he is on his walk I will be mad.
Step 1: There are two truth functions, ">" and "^" (in the present context the word "while" has the truth functional value of "and"; frequently it means "at the same
time as")
Bill stops at the store
Bill is on his walk
I will be mad
Step 2: There are three simple sentences:
P = def. Bill stops at the store
Q = def. Bill is on his walk
R = def. I will be mad
Step 3: We make a dictionary:
(Bill stops at the store while he is on his walk) > (I will be mad) and then
((Bill stops at the store) ^ (He is on his walk)) > (I will be mad)
Step 4: the truth function ">" has a wider scope than "^" so fits we get:
(P ^ Q) > R
Step 5: By substitution we get
Make a dictionary, assigning an English sentence to each letter
i.
Replace the sentence letters with the sentences in your dictionary
ii.
Rewrite the symbolic logic truth functions as English
iii.
Alter the sentences as much as needed to make it sound natural
iv.
Going from symbolic logic to English is roughly the reverse, the steps we follow are:
EX: (P > Q) v P
Dogs eat meat = def. P
Dogs have fleas = def. Q
Step 1: Dictionary:
Step 2: ((dogs eat meat) > (dogs have fleas)) v (dogs eat meat)
Step 3: If dogs eat meat then dogs have fleas, or dogs don’t eat meat
Step 4: Either dogs eat meat, or if they do then they have fleas
Difficulties in Translation
See pg 12194 in text
Truth Tree Technique
A tree can represent a very large number of possibilities economically, so it is much more practical to analyze complex conditionals using truth trees than by
truth tables
•
Truth tree technique  very similar to the truth table analysis except that it graphs the various possible situations of truth and falsity on a tree instead of a table.
Ch.7 Text Notes Page 3