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Propositional Logic – A Primer

2. Logical Operators and Truth Tables

In propositional logic, propositions are represented using a propositional variable, which is a symbol that stands for a proposition. Typically, logicians use uppercase letters such as 'P', 'Q', 'R', and so on, to represent propositional variables.

Propositional variables are combined using logical connectives to form more complex propositions, known as compound propositions. The main logical connectives in propositional logic are:

Negation (¬):
The negation of a proposition 'P', denoted by '¬P', is the opposite of 'P'. It is true if 'P' is false and false if 'P' is true.
Conjunction (∧):
The conjunction of two propositions 'P' and 'Q', denoted by 'P ∧ Q', is true if both 'P' and 'Q' are true and false otherwise.
Disjunction (∨):
The disjunction of two propositions 'P' and 'Q', denoted by 'P ∨ Q', is true if at least one of 'P' and 'Q' is true and false if both are false.
Implication (→):
The implication of two propositions 'P' and 'Q', denoted by 'P → Q', is true if 'P' implies 'Q'. That is, if 'P' is true, then 'Q' must also be true. It is false if 'P' is true and 'Q' is false, and true otherwise.
Biconditional (↔):
The biconditional of two propositions 'P' and 'Q', denoted by 'P ↔ Q', is true if 'P' and 'Q' have the same truth value. That is, both are true or both are false. It is false if 'P' and 'Q' have opposite truth values.

These logical connectives combine in various ways to form more complex compound propositions. For example, we can form the proposition:

(P ∧ Q) → (R ∨ ¬S)

by combining conjunction, implication, disjunction, and negation.

Note that in other texts on logic, different symbols for the logical connectives may be given. Here are the main variations you may see:

Negation (¬):
alternatively as ∼, –
Conjunction (∧):
alternatively as ∙, &
Implication (→):
alternatively as ⇒, ⊃
Biconditional (↔):
alternatively as ⇔, ≡

Truth tables are used in propositional logic to show the truth values of compound propositions for all possible combinations of truth values of their component propositions. A truth table has one row for each possible combination of truth values of the component propositions and one column for each component proposition and for the resulting compound proposition. The truth values in the last column of the truth table indicate the truth value of the compound proposition for each combination of truth values of the component propositions.

Importantly, the truth value of a proposition is determined by its truth-functional semantics. That is, the truth value of a compound proposition is determined solely by the truth values of its component propositions and the truth-functional connectives that combine them.

How each of the logical connectives function to determine the truth of a compound proposition is specified in a truth table. For example, the truth table for the negation operator (¬) is:

Truth table – Negation

P ¬P
T F
F T

In this truth table, 'P' represents a proposition that can be either true (T) or false (F). The second column shows the truth value of the negation of 'P', which is true only when 'P' is false. Similarly, truth tables can be constructed for each of the other logical connectives, as shown below.

The truth table for the conjunction operator (∧) is:

Truth table – Conjunction

P Q P ∧ Q
T T T
T F F
F T F
F F F
Book cover: Reasons and Persons by Derek Parfit

The first two columns in the truth table represent the possible truth values for the propositions or expressions being evaluated, 'P' and 'Q'. The propositions being conjoined are known as 'conjuncts'. The column labeled 'P ∧ Q' shows the truth value of the combined expression for each possible combination of truth values for its two conjuncts, 'P' and 'Q'. The truth table shows that the conjunction of 'P' and 'Q' is true only if both 'P' and 'Q' are true, and is false otherwise.

The truth table for the disjunction operator (∨) is:

Truth table – Disjunction

P Q P ∨ Q
T T T
T F T
F T T
F F F

This truth table shows that the disjunction of 'P' and 'Q' is true only if either 'P' is true or 'Q' is true, or both are true, and is false if both 'P' and 'Q' are false. For disjunctions, the propositions or expressions being evaluated are known as 'disjuncts'.

The truth table for the implication operator (→) is:

Truth table – Implication

P Q P → Q
T T T
T F F
F T T
F F T

This truth table shows that the implication from 'P' to 'Q' is true whenever 'P' is false or 'Q' is true, and false otherwise. For implications, the proposition preceding the operator is known as the 'antecedent', while the proposition following is known as the 'consequent'.

The truth table for the biconditional operator (↔) is:

Truth table – Biconditional

P Q P ↔ Q
T T T
T F F
F T F
F F T

This truth table shows that the equivalence of 'P' and 'Q' is true only if 'P' and 'Q' are either both true or both false, and false otherwise.

Evaluating the formal validity of an argument requires first converting the argument given in natural language to symbolic form. Identifying the truth-functional components of a sentence and how they relate logically to the other components of the sentence requires practice with more complex sentences.

Here are some basic examples using simple sentences.

  1. 'If it rains today, I will stay at home.'

    In this example, the proposition 'it rains today' can be represented by the variable 'P' and the proposition 'I will stay at home' can be represented by the variable 'Q'. The statement can then be represented in symbolic form as:

    P → Q
  2. 'The cat is black and white.'

    Here, 'the cat is black' can be represented by the variable 'P' and the proposition 'the cat is white' can be represented by the variable 'Q'. The statement can then be represented in symbolic form as:

    P ∧ Q
  3. 'Either it is sunny today or it is not raining today.'

    In this example, the proposition 'it is sunny today' can be represented by the variable 'P' and the proposition 'it is not raining today' can be represented by the variable 'Q'. The statement can then be represented in symbolic form as:

    P ∨ ¬Q

Once you are able to convert faithfully ordinary language arguments into symbolic form, you are then ready to move onto the next step: using truth tables to evaluate the logical validity of arguments and proofs by showing whether the conclusion logically follows from the premises.

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