Propositional Logic·Intermediate·6 lessons·77 practice activities·~320 min

Propositional Logic: Form, Connectives, and Valid Inference

How whole statements combine into logically assessable structures

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What you'll learn

By the end of this unit, you can…

Classify atomic and compound.
Symbolize propositional arguments.
Evaluate truth functional validity.
Recognize logical equivalences.
Construct propositional proofs.

Lessons

Lesson sequence

1
Atomic and Compound Statements
Introduces propositional logic as the study of how whole statements combine, distinguishes atomic from compound statements, and establishes the discipline of seeing structure before symbolizing.
15 activities5 worked examples
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2
The Five Connectives in Depth
Examines each of the five standard connectives (negation, conjunction, disjunction, conditional, biconditional), how they translate natural-language forms, and the ambiguities students must resolve.
15 activities5 worked examples
Student Pro
3
Symbolizing Whole Arguments
Teaches students to move from natural-language arguments to complete symbolic forms, assigning sentence letters consistently and preserving the inferential structure of the whole argument.
15 activities5 worked examples
Student Pro
4
Truth Tables, Equivalence, and Validity
Introduces truth tables as a decision procedure for propositional validity, establishes the central logical equivalences, and teaches students to use truth tables to diagnose why an argument is valid or invalid.
15 activities5 worked examples
Student Pro
5
Building Short Formal Proofs
Introduces the basic inference rules of propositional proof (modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, simplification, conjunction, and addition) and teaches students to build short formal proofs step by step.
15 activities2 worked examples
Student Pro
6
Capstone: Symbolizing, Proving, and Refuting Propositional Arguments
An integrative lesson that asks students to run the full propositional cycle on mixed arguments: symbolize from English, classify validity, either prove the argument or build a truth-assignment counterexample, and explain the result in plain language.
2 activities1 worked example
Student Pro

How to study

Three moves that work for this unit

Read the explanation

Each lesson opens with a guided walkthrough — read it before the activity.

Study the worked example

Look at why each step follows, not just what the answer is.

Practice with the target in mind

Know which rule applies and what would make the response weak before you start.

Reference materials

Optional context for the unit. Each lesson surfaces the concepts and rules it uses — these are here when you want the bigger picture.

Concept map (11 terms)

Atomic Statement

A declarative sentence that is not further analyzed at the propositional level and is represented by a single sentence letter.

Compound Statement

A statement formed from one or more simpler statements by the use of logical connectives.

Logical Connective

An operator such as negation, conjunction, disjunction, conditional, or biconditional that forms a compound proposition from simpler ones.

Main Connective

The connective with the widest scope in a compound statement, which determines the statement's overall logical form.

Truth Functionality

The property that the truth value of a compound statement is completely determined by the truth values of its component parts.

Truth Table

A systematic listing of every possible assignment of truth values to the atomic parts of a compound statement together with the resulting value of the whole.

Tautology

A statement that is true under every possible truth-value assignment to its atomic parts.

Contradiction

A statement that is false under every possible truth-value assignment to its atomic parts.

Logical Equivalence

A relation between two statements that are true under exactly the same truth-value assignments.

Validity

The property of an argument whose conclusion cannot be false while all its premises are true.

Inference Rule

A schematic pattern that licenses the derivation of a conclusion from one or more premises, such as modus ponens or disjunctive syllogism.

Rules and standards (6)

Respect the Main Connective. A symbolization is acceptable only if the main connective of the symbolic form matches the main connective of the natural-language statement. Common failures: A conditional is symbolized as a conjunction because the student saw two claims joined.; A negation is applied to one conjunct when the whole conjunction was intended to be negated.; A biconditional is read as a one-directional conditional..
Assign Sentence Letters Consistently. Use the same sentence letter for every occurrence of the same atomic claim, and use different letters for distinct claims. Common failures: The same claim is given two different letters in the same argument.; Two different claims are given the same letter because they share a topic.; A negated and an unnegated version of the same claim are assigned different letters..
Truth-Functional Validity Standard. A propositional argument is valid if and only if there is no truth-value assignment on which the premises are all true and the conclusion is false. Common failures: The student treats one favorable row as proof of validity.; The student reads off the truth of the conclusion without checking whether the premises are all true in that row.; The student mistakes invalidity for falsehood or vice versa..
Modus Ponens. From 'P → Q' and 'P', one may derive 'Q'. Common failures: The student affirms the consequent by deriving 'P' from 'P → Q' and 'Q'.; The student derives 'Q' from 'Q → P' and 'P' after confusing the direction of the conditional..
Modus Tollens. From 'P → Q' and '¬Q', one may derive '¬P'. Common failures: The student denies the antecedent by deriving '¬Q' from 'P → Q' and '¬P'.; The student ignores the conditional's direction when the negation is placed on the consequent..
Disjunctive Syllogism. From 'P ∨ Q' and '¬P', one may derive 'Q'; similarly from 'P ∨ Q' and '¬Q', one may derive 'P'. Common failures: The student derives the negated disjunct instead of the remaining disjunct.; The student assumes an exclusive disjunction and draws an unlicensed inference about the second disjunct..

Formalization patterns (2)

Argument Symbolization Schema. From natural_language_argument to propositional_argument_form — Identify the distinct atomic claims used in the argument.; Assign a sentence letter to each distinct atomic claim and record the key.; For each premise and the conclusion, find the main connective.; Symbolize each statement using the assigned letters and the main connective.; Verify that the final symbolization preserves both scope and inferential role..
Truth-Table Validity Test. From propositional_argument_form to validity_judgment — List the atomic letters used in the premises and conclusion.; Enumerate every truth-value assignment to those letters (2^n rows).; Compute the truth value of each premise and the conclusion in every row.; Find any row in which all premises are true and the conclusion is false.; Classify the argument as valid if no such row exists and invalid otherwise..

Full mastery and assessment guidance

Mastery requirements

Classify atomic and compound. Correct Classifications · 10_correct_classifications
Symbolize propositional arguments. Successful Symbolizations · 6_successful_symbolizations
Evaluate truth functional validity. Correct Truth Table Judgments · 5_correct_truth_table_judgments
Recognize logical equivalences. Correct Equivalence Judgments · 4_correct_equivalence_judgments
Construct propositional proofs. Successful Proofs · 4_successful_proofs

Assessment advice

Can I state the main connective in ordinary words before I start symbolizing?
Am I using the same sentence letter every time the same claim reappears?
Have I tracked the scope of every negation?
Do not let the grammar of an English sentence dictate its logical structure; check which connective actually governs the whole.
Do not assign a fresh sentence letter to a repeated claim just because its grammatical form changed.
Did I verify the direction of every conditional I introduced?
Did I handle 'or' consistently and note any ambiguity?
Did I track the scope of every negation?
Do not translate 'P only if Q' as Q → P; the correct direction is P → Q.
Do not default to exclusive disjunction because that is what 'or' usually means in casual speech.
Is my key complete and consistent with every premise and the conclusion?
If I read each symbolic form back into English, does it match the original?
Have I preserved the main connective and the scope of every compound?
Do not skip writing the key; it is the cheapest insurance against symbolization errors.
Do not simplify or clean up the structure during translation; do it later if at all.
Did I include every row for every atomic letter?
Have I checked every row with all premises true to confirm the conclusion is also true?
If the argument is invalid, can I point to the specific counterexample row?
Do not skip rows to save time; an incomplete table cannot establish validity.
Do not confuse the conclusion being true somewhere with the argument being valid everywhere.
Does every derived line match the schema of the rule I cited?
Does my final line match the goal exactly, including any negations?
Can I explain why each step was necessary for reaching the goal?
Do not skip the planning step; backwards reasoning from the goal is the fastest way to find the proof.
Do not apply modus ponens to a line that is not actually a conditional of the right shape.
Did I produce all four outputs for each case?
Did I decide whether to prove or refute before I started writing the proof?
Does my plain-English explanation make sense to someone who does not know the notation?
Burning time on a proof attempt for an invalid argument.
Forgetting that the output of propositional evaluation is a communicable result, not just a proof.

Historical context (3)

Chrysippus. Chrysippus and the Stoics developed the first systematic propositional logic, cataloguing argument forms such as modus ponens and modus tollens long before modern symbolic notation existed. The basic inference rules taught in propositional logic courses today trace directly to Stoic argument patterns.
George Boole. Developed an algebraic treatment of logical relations, representing propositions with symbols and reasoning about them with equations. Boolean algebra underlies truth tables, digital circuit design, and the algebraic manipulation of logical equivalences.
Gottlob Frege. Gave propositional and predicate logic their first rigorous formulation, distinguishing sense and reference and making inference a matter of explicit symbolic rules. The style of formal proof and explicit symbolization used in modern logic textbooks descends from Frege's notation and method.