Propositions and Truth Values: The Building Blocks of Logic

Logic doesn't care whether your premises are true. It cares whether your reasoning preserves truth.

Start with true premises, apply valid reasoning, end with a true conclusion. That's the guarantee. But to make this work, we need a currency — something to track and preserve.

That currency is truth values: True (T) and False (F).

Every proposition gets exactly one. "Paris is the capital of France" — T. "2 + 2 = 5" — F. No maybes. No sort-ofs. Just T or F.

This binary simplification is what makes logic checkable. We can mechanically verify whether an argument preserves truth by tracking T and F through the reasoning. If true premises can produce a false conclusion, the argument is invalid. Period.

Propositions are statements that can be true or false — the atoms that carry truth values through logical operations.

Not every sentence qualifies. "What time is it?" — no truth value. "Close the door" — no truth value. "Ouch!" — no truth value. Only declarative statements that can be T or F count as propositions.

The Bivalence Principle

Classical logic assumes bivalence: every proposition has exactly one of two truth values — true (T) or false (F).

No proposition is:

• Both true and false
• Neither true nor false
• Partially true
• Sort of false

This is a strong assumption. Real language is vaguer. "John is tall" — true or false? Depends on the context and what counts as tall. Classical logic handles this by assuming such statements have been made precise.

We use variables P, Q, R... to stand for arbitrary propositions.

P might mean "It is raining." Q might mean "The ground is wet."

The variables let us reason about the structure without specifying the content.

Atomic vs. Compound

Atomic propositions are indivisible: they contain no logical connectives.

• "It is raining" (atomic)
• "The ground is wet" (atomic)
• "Socrates is mortal" (atomic)

Compound propositions are built from atomic ones using connectives:

• "It is raining AND the ground is wet" (compound)
• "IF it is raining, THEN the ground is wet" (compound)
• "It is NOT raining" (compound)

The connectives — and, or, not, if-then — are how we build complexity from simplicity.

Why Truth Values?

Logic needs a way to evaluate arguments. Truth values provide it.

An argument is valid if: whenever all premises are true, the conclusion must be true.

To check this, we need to know when propositions are true or false. For atomic propositions, we just assign truth values. For compound propositions, the truth value depends on the components and the connectives.

Truth Value Assignments

A truth value assignment gives every atomic proposition a truth value.

If P = "It is raining" and Q = "The ground is wet," then one possible assignment is:

• P: True
• Q: True

Another possible assignment:

• P: False
• Q: True

Each assignment represents a possible state of the world.

With n atomic propositions, there are 2^n possible truth value assignments. Two propositions: 4 assignments. Three propositions: 8 assignments. The combinations grow exponentially.

The Law of Excluded Middle

Every proposition is either true or false — there's no third option.

P ∨ ¬P is always true.

"It is raining or it is not raining" — this must be true regardless of the weather. One of those has to be the case.

This law is controversial in some alternative logics. Intuitionistic logic, for example, doesn't accept excluded middle. But classical logic treats it as fundamental.

The Law of Non-Contradiction

No proposition is both true and false simultaneously.

¬(P ∧ ¬P) is always true.

"It is raining and it is not raining" — this is always false. A proposition can't contradict itself.

Non-contradiction is less controversial than excluded middle. Most logics keep it. But there are paraconsistent logics that allow contradictions without everything collapsing.

Notation Conventions

Standard symbols for truth values:

• T, 1, ⊤ for true
• F, 0, ⊥ for false

Standard variables for propositions:

• P, Q, R, S... (uppercase)
• p, q, r, s... (lowercase)

We'll use T/F for truth values and uppercase P, Q, R for propositions.

The Foundation

Propositions and truth values are the foundation. Everything else in propositional logic — connectives, truth tables, valid arguments, proofs — builds on this base.

The key insight: logical reasoning is truth-preserving. Valid inferences never take you from true premises to a false conclusion. The truth values of premises constrain the possible truth values of conclusions.

Understanding this foundation lets us analyze any argument's structure and determine whether it's valid or a subtle trick.

Part 2 of the Logic series.

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