Unit 3 Lesson 2

Truth Values

In this lesson, we formalise the concept of truth value, establish the foundational principles that govern it, and introduce the truth value assignment as a function.

Learning Objectives

    • Define truth value as a formal object

    • State and apply the Principle of Bivalence

    • State and apply the Law of Non-Contradiction

    • Understand the truth value assignment as a function

    • Distinguish truth from knowledge, belief, and provability

    • Connect truth values to Boolean algebra and computer science

Recap

In the previous lesson, we defined a proposition as a declarative sentence with a well-defined truth value. We noted that every proposition is either true or false. In this lesson, we examine the concept of truth value itself with full rigour.

Truth Value

The truth value of a proposition \(๐‘ƒ\) is the element of the set \({๐‘‡,๐น}\) that is assigned to \(๐‘ƒ\).

We write:

\[๐‘ฃโก(๐‘ƒ)∈{๐‘‡,๐น}\]

where \(๐‘ฃ\) is the valuation function (also called a truth assignment).

  • \(T\) (true) signifies that \(P\) accurately describes reality or follows from accepted axioms
  • \(F\) (false) signifies that \(P\) does not accurately describe reality or contradicts accepted axioms

Notation

Several equivalent notations are used in the literature:

  • \(T\) and \(F\) — standard in mathematical logic (Mendelson, Enderton)
  • \(1\) and \(0\) — standard in Boolean algebra and computer science (Boole, Shannon)
  • \(\top\) and \(\bot\) — common in proof theory and type theory

Throughout this course, we use \(T\) and \(F\) in logical contexts and \(1\) and \(0\) when the connection to computation is emphasised. The correspondence is:

\[ T \leftrightarrow 1 \leftrightarrow \top \]

\[ F \leftrightarrow 0 \leftrightarrow \bot \]

Valuation Function

A valuation (or truth assignment) is a function

\[๐‘ฃ:P→{๐‘‡,๐น}\]

where \(P\) is the set of all propositions under consideration. The function \(v\) assigns to each proposition exactly one truth value.

This is a total function: every proposition in \(P\) receives a value. It is also well-defined: no proposition receives two distinct values simultaneously.

Valuation in Practice

  • Let \(๐‘ƒ \) = "\(2 +3 =5\)". Then \(๐‘ฃโก(๐‘ƒ)  = ๐‘‡\).
  • Let \(๐‘„\) = "7 is even". Then \(๐‘ฃโก(๐‘„)  = ๐น\).
  • Let \(๐‘…\) = "Every prime number greater than 2 is odd". Then \(๐‘ฃโก(๐‘…) = ๐‘‡\).

In each case, the valuation function returns exactly one element of \({๐‘‡,๐น}\).

Principle of Bivalence

Every proposition has exactly one of two truth values: true or false.

This principle consists of two parts:

  • Exhaustiveness: every proposition is either true or false (no gaps).
  • Exclusivity: no proposition is both true and false (no overlaps).

The Principle of Bivalence is a foundational assumption of classical logic. It is distinct from the Law of Excluded Middle introduced in the previous lesson, although they are closely related. The Law of Excluded Middle is a logical law (a tautology within the system); the Principle of Bivalence is a semantic principle about how truth values are assigned.

Unpacking Bivalence

The Principle of Bivalence combines two distinct logical laws:

  • The Law of Excluded Middle (there is no third option)
  • The Law of Non-Contradiction (both options cannot hold simultaneously)

Law of Excluded Middle (Tertium Non Datur)

We recall from the previous lesson that either \(P\) is true, or its negation  is true.  There is no third possibility. This law guarantees that the valuation function \(v\) is total: it is defined for every proposition.

 

Consider the proposition \(P\): "There are infinitely many prime numbers."

By the Law of Excluded Middle, exactly one of the following holds:

  • \(P\) is true (there are infinitely many primes), or
  • \(not \, P\) is true (there are finitely many primes)

In fact, \(v(P) = T\), as demonstrated by Euclid’s proof (c. 300 BCE). But the law holds independently of whether we know the proof: even before Euclid, one of the two alternatives was the case.

Law of Non-Contradiction (Principium Contradictionis)

A proposition and its negation cannot both be true simultaneously. This law guarantees that the valuation function \(v\) is well-defined: each proposition maps to exactly one truth value, never to both.

Let \(P\): "The number 4 is even."

By the Law of Non-Contradiction, it is impossible that:

  • "4 is even" is true, and
  • "4 is not even" is also true

Since \(4 = 2 \times 2\), we have \(v(P) = T\) and \(v(\neg P) = F\). The two values are mutually exclusive, as the law requires.

Truth vs. Knowledge, Belief, and Provability

A critical distinction must be drawn. In classical logic, truth is an objective property of a proposition. It is not contingent upon any epistemic state of the observer.

  • Truth \(\neq\) Knowledge. A proposition may be true even if no one knows it is true. The Goldbach Conjecture ("every even integer greater than 2 is the sum of two primes") is either true or false — regardless of whether a proof has been found.
  • Truth \(\neq\) Belief. A proposition may be true even if everyone believes it to be false, and vice versa. Before the discovery of non-Euclidean geometry, the parallel postulate was universally believed to be a necessary truth. It is not.
  • Truth \(\neq\) Provability. Gödel’s First Incompleteness Theorem (1931) demonstrates that in any sufficiently powerful formal system, there exist true propositions that cannot be proven within that system. Truth and provability are fundamentally distinct concepts.

A Proposition with Unknown Truth Value

Let \(P\): "There exist infinitely many twin primes."

(Twin primes are pairs of primes differing by 2, such as \((3, 5)\), \((11, 13)\), \((17, 19)\).)

As of today, no proof or disproof of this conjecture is known. Nevertheless, by the Principle of Bivalence:

\[ v(P) = T \quad \text{or} \quad v(P) = F \]

The proposition has a truth value. We simply do not know which one it is. Our ignorance does not affect the ontological status of the proposition.

Boolean Domain

The set \(\mathbb{B} = \{0, 1\}\) is called the Boolean domain. Under the identification \(T \leftrightarrow 1\) and \(F \leftrightarrow 0\), the truth value assignment becomes:

\[ v : \mathcal{P} \to \mathbb{B} \]

This formulation is the one used in Boolean algebra, the algebraic structure introduced by George Boole in 1847 and later applied by Claude Shannon (1937) to the design of electrical switching circuits. Every digital computer operates on this two-element algebra.

 

Connection to Computer Science

In programming, the Boolean data type directly implements the truth value set \(\{T, F\}\). Every conditional statement, every loop guard, every assertion reduces to a truth value evaluation.

Consider the following correspondence:

  • A Boolean variable in a program is a propositional variable that holds a truth value
  • An if-statement evaluates a proposition and branches on its truth value
  • A bit in hardware is the physical realisation of an element of \(\mathbb{B}\)

The entire edifice of computation rests upon the binary nature of truth values. This is not a coincidence; it is a direct consequence of the logical foundations we are establishing here.

Remark on Non-Classical Logics

The Principle of Bivalence is not universally accepted. Alternative logical systems reject or modify it:

  • Intuitionistic logic (Brouwer, Heyting) rejects the Law of Excluded Middle. A proposition is true only if a constructive proof exists; it is false only if a refutation exists. If neither exists, no truth value is assigned.
  • Many-valued logics (ลukasiewicz) introduce additional truth values. In three-valued logic, the set of truth values is \(\{T, U, F\}\), where \(U\) represents "undetermined."
  • Fuzzy logic (Zadeh) replaces \(\{T, F\}\) with the continuous interval \([0, 1]\), assigning degrees of truth.

In this course, we work exclusively within classical (bivalent) logic unless explicitly stated otherwise. The reasons are both historical (classical logic is the foundation of standard mathematics) and practical (digital circuits operate on classical Boolean values).

Exercise 1

Determine whether each statement is True or False.

a)

\(10\) is even.

b)

\(3^2 = 6\)

c)

There exists a real number \(x\) such that \(x^2 = 9\).

Exercise 2

Determine whether each statement is true or false.

a)

A valuation assigns exactly one truth value to each propositional variable.

b)

In classical logic, a proposition can have no truth value.

c)

The Boolean type in programming languages is a direct implementation of classical bivalence.

Exercise 3

Answer each question.

a)

The Goldbach Conjecture is currently unproven. What can we say about its truth value in classical logic?

Select the correct statement:

b)

Which logical law guarantees that a proposition cannot be simultaneously true and false?

Select the correct law:

c)

In which logical system can a proposition have a truth value other than \(T\) or \(F\)?

Select:

Exercise 4

Classify each statement as true or false.

a)

A proposition whose truth value is unknown has no truth value.

b)

The Law of Excluded Middle guarantees that the valuation function \(v\) is total.

c)

The Law of Non-Contradiction guarantees that the valuation function \(v\) is well-defined.

d)

In classical logic, a proposition can be neither true nor false.

e)

The Boolean domain \(\{0, 1\}\) is isomorphic to the truth value set \(\{T, F\}\).

Question

The Goldbach Conjecture is unproven. An intuitionistic logician says:

"The conjecture has no truth value, because no constructive proof or refutation exists."

A classical logician says:

"The conjecture is either true or false. We simply do not know which."

Who is correct?

Answer

Both are correct — within their respective logical systems.

The classical logician applies the Principle of Bivalence: every proposition has exactly one truth value, regardless of our epistemic state. The valuation function \(v\) is total and well-defined. The conjecture belongs to \(\mathcal{P}\), therefore \(v(P) \in \{T, F\}\). Our inability to determine which value it holds does not alter this fact.

The intuitionistic logician rejects the Law of Excluded Middle. In intuitionistic logic, a proposition is true only when a constructive proof exists, and false only when a constructive refutation exists. In the absence of either, the proposition receives no truth value.

The disagreement is not about the conjecture itself. It is about the foundational principles governing truth. This is why we stated explicitly that this course operates within classical logic: the choice of logical framework determines what "truth value" means.

  • A truth value is a formal object from the set \(\{T, F\}\) assigned to every proposition
  • The truth value assignment is a total, well-defined function \(v : \mathcal{P} \to \{T, F\}\)
  • The Principle of Bivalence states that every proposition has exactly one of two truth values
  • The Law of Excluded Middle guarantees totality
  • The Law of Non-Contradiction guarantees uniqueness
  • Truth is objective and independent of knowledge, belief, or provability
  • The Boolean domain \(\mathbb{B} = \{0, 1\}\) is the computational counterpart of the truth value set