Chapter 2: Logic
Logical deduction: from a set of premises a conclusion can be deduced by logic.
Propositional logic: branch of logic, which takes propositions and considers how they can be combined
and manipulated.
Proposition: a statement that has a truth value, it is either true or false.
In English, a proposition is expressed as a sentence with a subject and a predicate. In the sentence “Delft
is a city”, “Delft” is the subject, and “is a city” is the predicate.
2.1 Propositional Logic
2.1.1 Propositions
Propositional variable: a lower-case letter to represent a proposition (an uppercase letter is used to
represent a compound proposition).
Literal: propositional variable or the negation of it.
To literals are complementary if and only if one is the negation of the other.
2.1.2 Logical operators
Logical operator (Logical connectives): operator that can be applied to one or more propositions to
produce a new proposition. Represented by symbols:
• These symbols do not carry any connation beyond their defined logical meaning;
∧ conjunction, ∨ disjunction and ¬ negation.
Definition. Let 𝑝 and 𝑞 be propositions. Then 𝑝 ∨ 𝑞, 𝑝 ∧ 𝑞 and ¬𝑝 are propositions, whose
truth values are given by the rules:
• 𝑝 ∧ 𝑞 is true when both 𝑝 is true and 𝑞 is true, and in no other case.
• 𝑝 ∨ 𝑞 is true when either 𝑝 is true, or 𝑞 is true, or both 𝑝 is true and 𝑞 are true, and in
no other case.
• ¬𝑝 is true when 𝑝 is false, and in no other case.
•
2.1.3 Precedence rules
Compound proposition: a proposition made up of simpler propositions and logical operators.
In absence of parentheses, the order of evaluation is determined by precedence rules.
Order = ¬ → ∧ → ∨ → left to right evaluation.
Associative operation: a logical operator is an associative operation, when it doesn’t matter which logical
operator is evaluated first (𝑝 ∧ 𝑞 ∧ 𝑟 → ∧ is an associative operation).
Main connective: the connective that is evaluated last in a compound proposition.
2.1.4 Logical equivalence
Truth table: a table that shows the value of one or more compound propositions for each combination of
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠
values of the propositional variables that they contain (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑢𝑡ℎ 𝑡𝑎𝑏𝑙𝑒𝑠 = 22 ).
Situation: a combination of values of the propositional variables.
Two compound propositions are logically equivalent if they always have the same value, no matter what
values are assigned to the propositional variables that they contain.
Ways of showing logically equivalence:
1. Prove 𝑃 → 𝑄 and 𝑄 → 𝑃;
2. Show that 𝑃 and 𝑄 have the same truth tables;
3. Reduce 𝑃 and 𝑄 to a normal form (DNF/CNF) and show their normal forms are equivalent;
, 2.1.5 More logical operators
→ conditional operator, ↔ biconditional operator and exclusive or operator ⊕.
Definition. Let 𝑝 and 𝑞 be propositions. We define the propositions 𝑝 → 𝑞, 𝑝 ↔ 𝑞 and 𝑝 ⊕ 𝑞
according to the truth table:
𝑝 𝑞 𝑝 →𝑞 𝑝 ↔ 𝑞 𝑝⊕𝑞
0 0 1 1 0
0 1 1 0 1
1 0 0 0 1
1 1 1 1 0
2.1.6 Implications in English
Implication/Conditional: 𝑝 → 𝑞 (“𝑝 implies 𝑞”)
• 𝑝 is called the hypothesis/antecedent and 𝑞 is called the conclusion/consequent.
• If the implication 𝑝 → 𝑞 holds, then 𝑝 is sufficient for 𝑞 and 𝑞 is necessary for 𝑝.
• If the statements are 𝑝 → 𝑞 and 𝑞 → 𝑟, it means that 𝑝 → 𝑟 also holds.
2.1.7 More forms of implication
An implication is logically equivalent to its contrapositive (𝑝 → 𝑞 and ¬𝑞 → ¬𝑝).
An implication is not logically equivalent to its converse (𝑝 → 𝑞 and 𝑞 → 𝑝).
An implication is not logically equivalent to its inverse (𝑝 → 𝑞 and ¬𝑝 → ¬𝑞).
Biconditional operator 𝑝 ↔ 𝑞 (“𝑝 is and only if 𝑞”).
2.1.8 Exclusive or
𝑝 ∨ 𝑞 stands for “𝑝 or 𝑞, or both”, while 𝑝 ⊕ 𝑞 stands for “𝑝 or 𝑞, but not both”.
2.1.9 Universal operators
A set of logical operators is functionally complete if and only if all formulas in propositional logic can be
rewritten to an equivalent form that uses only operators from the set.
2.1.10 Classifying propositions
A compound propositions is said to be a tautology if and only if it is true for all combinations of truth values
of the propositional variables which it contains.
A compound proposition is said to be a contradiction if and only if it is false for all combinations of truth
values of the propositional variables which it contains.
A compound proposition is said to be a contingency if and only if it is neither a tautology nor a
contradiction.
Two compound propositions, 𝑃 and 𝑄, are said to be logically equivalent if and only if the proposition 𝑃 ↔
𝑄 is a tautology.
Logical deduction: from a set of premises a conclusion can be deduced by logic.
Propositional logic: branch of logic, which takes propositions and considers how they can be combined
and manipulated.
Proposition: a statement that has a truth value, it is either true or false.
In English, a proposition is expressed as a sentence with a subject and a predicate. In the sentence “Delft
is a city”, “Delft” is the subject, and “is a city” is the predicate.
2.1 Propositional Logic
2.1.1 Propositions
Propositional variable: a lower-case letter to represent a proposition (an uppercase letter is used to
represent a compound proposition).
Literal: propositional variable or the negation of it.
To literals are complementary if and only if one is the negation of the other.
2.1.2 Logical operators
Logical operator (Logical connectives): operator that can be applied to one or more propositions to
produce a new proposition. Represented by symbols:
• These symbols do not carry any connation beyond their defined logical meaning;
∧ conjunction, ∨ disjunction and ¬ negation.
Definition. Let 𝑝 and 𝑞 be propositions. Then 𝑝 ∨ 𝑞, 𝑝 ∧ 𝑞 and ¬𝑝 are propositions, whose
truth values are given by the rules:
• 𝑝 ∧ 𝑞 is true when both 𝑝 is true and 𝑞 is true, and in no other case.
• 𝑝 ∨ 𝑞 is true when either 𝑝 is true, or 𝑞 is true, or both 𝑝 is true and 𝑞 are true, and in
no other case.
• ¬𝑝 is true when 𝑝 is false, and in no other case.
•
2.1.3 Precedence rules
Compound proposition: a proposition made up of simpler propositions and logical operators.
In absence of parentheses, the order of evaluation is determined by precedence rules.
Order = ¬ → ∧ → ∨ → left to right evaluation.
Associative operation: a logical operator is an associative operation, when it doesn’t matter which logical
operator is evaluated first (𝑝 ∧ 𝑞 ∧ 𝑟 → ∧ is an associative operation).
Main connective: the connective that is evaluated last in a compound proposition.
2.1.4 Logical equivalence
Truth table: a table that shows the value of one or more compound propositions for each combination of
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠
values of the propositional variables that they contain (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑢𝑡ℎ 𝑡𝑎𝑏𝑙𝑒𝑠 = 22 ).
Situation: a combination of values of the propositional variables.
Two compound propositions are logically equivalent if they always have the same value, no matter what
values are assigned to the propositional variables that they contain.
Ways of showing logically equivalence:
1. Prove 𝑃 → 𝑄 and 𝑄 → 𝑃;
2. Show that 𝑃 and 𝑄 have the same truth tables;
3. Reduce 𝑃 and 𝑄 to a normal form (DNF/CNF) and show their normal forms are equivalent;
, 2.1.5 More logical operators
→ conditional operator, ↔ biconditional operator and exclusive or operator ⊕.
Definition. Let 𝑝 and 𝑞 be propositions. We define the propositions 𝑝 → 𝑞, 𝑝 ↔ 𝑞 and 𝑝 ⊕ 𝑞
according to the truth table:
𝑝 𝑞 𝑝 →𝑞 𝑝 ↔ 𝑞 𝑝⊕𝑞
0 0 1 1 0
0 1 1 0 1
1 0 0 0 1
1 1 1 1 0
2.1.6 Implications in English
Implication/Conditional: 𝑝 → 𝑞 (“𝑝 implies 𝑞”)
• 𝑝 is called the hypothesis/antecedent and 𝑞 is called the conclusion/consequent.
• If the implication 𝑝 → 𝑞 holds, then 𝑝 is sufficient for 𝑞 and 𝑞 is necessary for 𝑝.
• If the statements are 𝑝 → 𝑞 and 𝑞 → 𝑟, it means that 𝑝 → 𝑟 also holds.
2.1.7 More forms of implication
An implication is logically equivalent to its contrapositive (𝑝 → 𝑞 and ¬𝑞 → ¬𝑝).
An implication is not logically equivalent to its converse (𝑝 → 𝑞 and 𝑞 → 𝑝).
An implication is not logically equivalent to its inverse (𝑝 → 𝑞 and ¬𝑝 → ¬𝑞).
Biconditional operator 𝑝 ↔ 𝑞 (“𝑝 is and only if 𝑞”).
2.1.8 Exclusive or
𝑝 ∨ 𝑞 stands for “𝑝 or 𝑞, or both”, while 𝑝 ⊕ 𝑞 stands for “𝑝 or 𝑞, but not both”.
2.1.9 Universal operators
A set of logical operators is functionally complete if and only if all formulas in propositional logic can be
rewritten to an equivalent form that uses only operators from the set.
2.1.10 Classifying propositions
A compound propositions is said to be a tautology if and only if it is true for all combinations of truth values
of the propositional variables which it contains.
A compound proposition is said to be a contradiction if and only if it is false for all combinations of truth
values of the propositional variables which it contains.
A compound proposition is said to be a contingency if and only if it is neither a tautology nor a
contradiction.
Two compound propositions, 𝑃 and 𝑄, are said to be logically equivalent if and only if the proposition 𝑃 ↔
𝑄 is a tautology.
