Defining Propositions

In mathematics, we need a language that is absolute. In everyday English, sentences can be vague, emotional, or open to interpretation. In Discrete Mathematics, we filter those out to find the "Atoms" of logic: Propositions.

1. What is a Proposition?

A Proposition is a declarative sentence that is either True (T) or False (F), but never both at the same time.

Sentence	Is it a Proposition?	Why?
"Kolkata is in West Bengal."	Yes	It is a statement of fact that is True.
"$2 + 2 = 5$"	Yes	It is a statement of fact that is False.
"Close the door!"	No	It is a command, not a statement.
"Is it raining?"	No	It is a question.
"$x + 1 = 2$"	No	We don't know $x$. This is an "Open Sentence."

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2. The Law of Excluded Middle

In classical logic, there is no "maybe." Every proposition exists in a Binary State Space.

• State 1: True ($1$)
• State 0: False ($0$)

This is the "Law of Excluded Middle"—a statement is either True or its negation is True. There is no middle ground.

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3. Representing Propositions (Variables)

To perform mathematical operations on these statements, we assign them lowercase letters, usually starting with $p, q, r,$ and $s$.

• $p$: "The sun rises in the east."
• $q$: "The number 7 is even."

We can then discuss the Truth Value of these variables. For example, the truth value of $p$ is T, and the truth value of $q$ is F.

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4. Why This Matters

By turning sentences into variables ($p$ and $q$), we can stop worrying about the meaning of the words and start focusing on the structure of the logic. This is exactly how computers process information—by reducing every complex instruction down to a series of True/False (1/0) propositions.

# In programming, propositions are 'Booleans'
is_kolkata_in_wb = True
is_seven_even = False

if is_kolkata_in_wb:
    print("This statement is a True Proposition.")

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Summary

• Proposition: A declarative statement that is definitely True or definitely False.
• Truth Value: The classification of a proposition as T (1) or F (0).
• Variables: We use letters like $p$ and $q$ to represent these statements.

Next Step: Now that we have our "Atoms" (Propositions), we need the "Glue" to stick them together.