INTRODUCTION TO PROPOSITIONAL LOGIC
Based on: https://www.coursera.org/course/intrologic
Propositional logic is concerned with propositions and their interrelationships. A proposition is the content of a
sentence that affirms or denies something and is capable of being true or false. In propositional logic there are
two types of sentences: simple sentences (p, q, r, etc.) and compound sentences. Simple sentences express
simple facts about the world, compound sentences express logical relationships among simpler sentences of
which they are composed. Here is an example:
(p∨q) ∧ (¬q∨r)
Compound sentences are formed from simple sentences and logical operators of various sorts. There are six
types of compound sentences in propositional logic.
A negation consists of a negation operator followed by a simple or a compound sentence, all enclosed in
parentheses. For example, given the sentence p we can form the negation of p. The argument of the negation
is often called the target of the negation.
(¬p)
(¬p is true if and only if p is false.)
A conjunction is a sequence of sentences separated by occurrences of the and operator and enclosed in
parentheses. For example, we can form the conjunction of p and q:
(p∧q)
( p∧q is true if p and q are both true; else it is false.)
The constituent sentences in a conjunction are called conjuncts.
A disjunction is a sequence of sentences separated by the occurrence of the or operator, and enlosed in
paratheses. The constituent sentences in a disjunction are called disjuncts.
We can form a disjunction of p and q as shown below:
(p∨q)
(p∨q is true if p or q (or both) are true; if both are false, the statement is false.)
An implication consists of a pair of sentences separated by an implication operator and enclosed in
parentheses. The sentence to the left of the operator is called the antecedent and the sentence to the right is
called the consequent. The implication of p and q is shown here:
(p⇒q)
(p⇒q is true just in the case that either p is false or q is true, or both are true or false.)
A reduction is a reverse of an implication. It consists of a pair of sentences separated by a reduction operator,
which is turned to the left, and enclosed in paratheses. The left sentence of a reduction is called the consquent
and the sentence to the right is called the antecedent.
(p⇐q)
An equivalence or biconditionals is a combination of an implication and a reduction and we can express it as
shown here:
(p⇔q)
(p⇔q is true only if the truth values of p and q agree)
The semantics of logic, i.e the meaning of propositional sentences is similar to the semantics of algebra.
Algebra is unconcerned with the real world significance of variables like x. What is interesting is the
relationships between those variables expressed in our equations. Algebraic methods are designed to respect
these relationships no matter what meanings or values are assigned to the constituent variables.
In a similar way logic is unconcerned with what sentences say about the world being described. What matters
from a logical point of view is the relationship between the truth values of simple sentences and truth values of
compound sentences within which the simple sentences are contained. Moreover the logical reasoning
methods are designed to work no matter what meanings or values are assigned to the proposition constants
used in sentences.
Although logic does not prescribe inherent values for proposition constants, in analysing logic it is useful to
consider the values that logic users associate with proposition constants. Such associations are often called
truth assignments. Formally a truth assignment (i, as in interpretation) in propositional logic is assigning a truth
value to each of the simple sentences of the language. In what follows we use the symbol 1 as a synonym for
true and the symbol 0 as a synonym for false.
negation
p
¬q
1
0
0
1
conjunction
p
q
p∧q
1
1
1
1
0
0
0
1
0
0
0
0
disjunction
p
q
1
1
1
0
0
1
0
0
p∨q
1
1
1
0
implication
p
q
p⇒q
1
1
1
1
0
0
0
1
1
0
0
1
reduction
p
q
1
1
1
0
0
1
0
0
p⇐q
1
1
0
1
equivalence
p
q
1
1
1
0
0
1
0
0
p⇔q
1
0
0
1
Unlike propositional truth assignments, sentential truth assignments are not arbitrary. Given the truth
assignments of proposition constants of a language, the semantics of logic fixes the truth assignments for all
compound sentences. An example:
Interpretation i:
i
p =1
i
q= 0
i
r= 1
Compound sentence:
(p ∨ q) ∧ (¬q ∨ r)
We replace the constants in the formula with their truth values:
(1 ∨ 0) ∧ (¬0 ∨ 1)
(1 ∨ 0) ∧ (1 ∨ 1)
Then we replace both disjunctions with their truth values to get a value of the entire sentence:
1∧1
1
After reading the text, do the tasks or answer the questions below:
1.
2.
3.
4.
What is the relationship between a simple sentence and a compound sentence in propositional logic?
Without the help of the text try to formulate the definition of a proposition / negation / implication. (If
you have problems, check out how it is done in the text. Why was its formulation difficult?)
Give an “everyday” example of a proposition!
Which of the following are examples of a proposition?
a)
“Crocodiles are smaller than Alligators.”
b) “What time is it?”
5.
6.
7.
c)
“Pass the salt, please.”
d)
“If Elvis Presley is alive, then I’m the Pope.”
e)
“Fresca® is the bee's knees.”
How is a sentence in propositional logic different from a sentence in everyday language?
Are the mathematical symbols found in the text different from what you studied at maths? How?
What are the Slovene equivalents of the following notions:
a simple sentence; a compound sentence; an operator; conjunction; implication; antecedent; consequent;
variables; to assign; true; false;
8.
Take a look at the famous legal question: Have you stopped beating your wife?
No matter what your answer is (yes/no), it implies you beat your wife. Show this in the truth table!
p You have stopped.
q You beat your wife.
p
q
¬p
p∧q
¬p∧q
((p∧q) ∨ (¬p∧q)) ⇒ q