An Introductory Example
Let's consider the following statement, or proposition:
John has two children and Mary lives in Scotland
This proposition is a compound proposition, because it is constructed from two subordinate propositions: "John has two children" and "Mary lives in Scotland", which are connected through the conjunction "and".
The constituent propositions "John has two children" and "Mary lives in Scotland" are instead atomic propositions, because they cannot be further decomposed into subordinate propositions.
The conjunction "and" is a logical connective.
Some other logical connectives that can be used to construct propositions from other propositions are:
- Identity - e.g. « it is true that » John has two children
- Negation - e.g. « it is not true that » Mary lives in Scotland
- Disjunction - e.g. John has two children « or » Mary lives in Scotland
- Conditional - e.g. « if » John has two children « then » Mary lives in Scotland
- Biconditional - e.g. John has two children « if and only if » Mary lives in Scotland
Each proposition, be it atomic or compound, can be considered to be correct or incorrect.
For instance, we may consider the proposition "John has two children" to be correct and the proposition "Mary lives in Scotland" to be incorrect.
We can determine the correctness of the compound proposition "John has two children and Mary lives in Scotland" based on the correctness of its composing propositions and on the meaning of the connective "and".
The meaning of the connective "and" can be expressed as follows: given two propositions p1 and p2, the compound proposition "p1 and p2" is correct if and only if both propositions p1 and p2 are correct.
Hence, since "Mary lives in Scotland" is incorrect, "John has two children and Mary lives in Scotland" is incorrect too.
Attributes such as "correct" and "incorrect" that can be associated to propositions are called truth values.
The most commonly used terms for these two truth values are true and false, or alternatively 1 and 0.
The Scope of Classical Propositional Logic
Classical propositional logic is sometimes referred to as propositional logic, but it is actually a subset of propositional logic.
An Informal Definition of Propositional Logic
Propositional logic is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives.
Atomic propositions are handled as indivisible elements, whose sole feature is that of having a truth value, out of a domain of possible truth values which applies to all propositions.
The truth value of compound propositions depends the truth value of the constituent propositions, how the constituent propositions are composed by logical connectives and the meaning of the logical connectives.
What is "Classical" in Classical Propositional Logic
Truth-Functional
Classical propositional logic is a truth-functional logic.
Truth-functional logics are propositional logics where all connectives are truth-functional.
A connective is truth-functional if the truth value of the compound proposition it produces depends only on the truth values of the constituent propositions.
The identity, negation, conjunction, disjunction, conditional and biconditional connectives are all truth-functional.
But let's consider this proposition:
John thinks that Mary lives in Scotland
We may treat "John thinks that" as a connective which is applied to the atomic proposition "Mary lives in Scotland" to produce the compound proposition "John thinks that Mary lives in Scotland".
In this case we cannot derive the truth value of the compound proposition solely from its constituent proposition.
For instance if we consider the proposition "Mary lives in Scotland" to be false, we cannot automatically deduce that the proposition "John thinks that Mary lives in Scotland" is false (nor that it is true).
Hence the "John thinks that" connective is not truth-functional.
Two-Valued
Different types of truth functional logics are characterised by a different number of possible truth values that propositions can take.
Classical propositional logic is bivalent, or two valued: the only truth values that can be associated with a proposition are "true" and "false".
An example of a logic that is not two-valued is a logic that deals with the truth values of "true", "false" and "unknown".
