Proposition and Truth Value

Context: FIT1058_MOC · the atoms of logic · two truth values (T/F, or bits 1/0) · combined by Logical Connectives

Quick Revision

• 🎯 Objective: a statement that is either true or false ➔ the truth-valued atom all of logic is built on.
• 📦 Core Components: proposition ➔ truth value $\{$T,F$\}$ ➔ Boolean variable (a name for one).
• ⚡ Key Constraint: bivalence — exactly one truth value; questions, commands, and self-referential paradoxes are not propositions.

📝 Core

1. The Proposition (Truth-Valued Atom)

• Definition ➔ a declarative statement that is either true or false, never both, never neither.
• Truth value ➔ one of $\{$T,F$\}$ a proposition takes ➔ may be unknown yet still definite.
• Boolean variable ➔ a variable ranging over $\{$T,F$\}$ ➔ often a name for a proposition (“let $X$ be $1+1=2$”, so $X=$T”).

2. Truth Values ↔ Bits

• Two-state abstraction ➔ T/F abstracts any two-state phenomenon (off/on, unmagnetised/magnetised).
• Bit map ➔ $0=$F, $1=$T ➔ lets logic be reasoned about independently of hardware.

3. What Fails to Be a Proposition

• Command ➔ “Come and work for us!” ➔ no truth value.
• Question ➔ not an assertion.
• Paradox ➔ “This statement is false” ➔ self-referential, neither T nor F.

⚖️ Core Decision Matrix

Category	Truth value?	Proposition?	Example
declarative, definite	exactly one	yes	$1+1=2$
declarative, currently unknown	definite (unknown)	yes	”it will rain tomorrow”
command / question	none	no	”close the door”
self-referential paradox	none consistent	no	”this statement is false”

When It Flips: bivalence — every proposition has exactly one truth value — is the assumption the whole module rests on. A Boolean variable ranges over $\{$T,F$\}$; a proposition has a fixed (possibly unknown) value.

📊 Exam Execution Trace

Manual Execution Trace

Classifying statements:

Step / State	Statement	Declarative?	Proposition?	Value
0 (Init)	—	—	—	—
1	$1+1=2$	yes	yes	T
2	”Close the door.”	no (command)	no	—
3	”This statement is false.”	no (paradox)	no	—
4	”It will rain tomorrow.”	yes	yes	unknown

⚠️ Common Mistakes

• 💡 “Unknown” ≠ “not a proposition” ➔ “it will rain tomorrow” is a proposition (definite truth value, merely unknown now); only commands, questions, and paradoxes are excluded.

🧠 Active Recall

What makes a statement a proposition, and why are "Come and work for us!" and "This statement is false" excluded?

• Hint: Definite truth value required.

Answer

• Short answer: A proposition is a declarative statement with exactly one truth value; a command has none, a paradox has no consistent one.
• Why: Bivalence ➔ “this statement is false” is T ⟺ F, so no consistent assignment exists.

How do truth values relate to bits, and what is a Boolean variable?

• Hint: T/F ↔ 1/0.

Answer

• Short answer: $\{$T,F$\}$ correspond to bits $\{1,0\}$; a Boolean variable ranges over $\{$T,F$\}$ and often names a proposition.
• Why: Hardware-independent ➔ logic operations mirror operations on $1/0$, which is why digital hardware computes logic.