1. Read the model first
Each lesson opens with a guided explanation so the learner sees what the core move is before any saved response is required.
Propositional Logic
Propositional Logic: Form, Connectives, and Valid Inference
How whole statements combine into logically assessable structures
Students learn to identify atomic and compound statements, master the five connectives of propositional logic, symbolize natural-language arguments, evaluate validity with truth tables, and construct formal proofs using basic inference rules.
Study Flow
How to work through this unit without overwhelm
2. Study an example on purpose
The examples are there to show what strong reasoning looks like and where the structure becomes clearer than ordinary language.
3. Practice with a target in mind
Activities work best when the learner already knows what the answer needs to show, what rule applies, and what mistake would make the response weak.
Lesson Sequence
What you will work through
Lesson 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.
Start with a short reading sequence, study 2 worked examples, then use 15 practice activitys to test whether the distinction is actually clear.
Guided reading2 worked examples15 practice activitys
Lesson 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.
Start with a short reading sequence, study 2 worked examples, then use 15 practice activitys to test whether the distinction is actually clear.
Guided reading2 worked examples15 practice activitys
Lesson 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.
Read for structure first, inspect how the example turns ordinary language into cleaner form, then complete 15 formalization exercises yourself.
Guided reading2 worked examples15 practice activitystranslation support
Lesson 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.
Use the reading and examples to learn what the standards demand, then practice applying those standards explicitly in 15 activitys.
Guided reading2 worked examples15 practice activitysstandards focus
Lesson 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.
This lesson is set up like coached reps: read the sequence, compare yourself with the model, and then work through 15 supported activitys.
Guided reading2 worked examples15 practice activityscoached reps
Lesson 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.
Each lesson now opens with guided reading, then moves through examples and 2 practice activitys so you are not dropped into the task cold.
Guided reading1 worked example2 practice activitys
Rules And Standards
What counts as good reasoning here
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
How arguments get translated into structure
Argument Symbolization Schema
Input form
natural_language_argument
Output form
propositional_argument_form
Steps
- 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.
Common errors
- Using one sentence letter for two different claims that merely share a topic.
- Failing to identify the main connective and symbolizing from the wrong structural layer.
- Applying negation to the wrong part of a compound statement.
Truth-Table Validity Test
Input form
propositional_argument_form
Output form
validity_judgment
Steps
- 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.
Common errors
- Omitting rows and failing to cover all assignments.
- Mislabeling a row as a counterexample without checking that every premise is true in it.
- Confusing the shape of the truth table with the shape of the argument.
Concept Map
Key ideas in the unit
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.
Assessment
How to judge your own work
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.
Mastery requirements
- Classify Atomic And Compound
- Symbolize Propositional Arguments
- Evaluate Truth Functional Validity
- Recognize Logical Equivalences
- Construct Propositional Proofs
History Links
How earlier logicians shaped modern tools
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.
George Boole
Developed an algebraic treatment of logical relations, representing propositions with symbols and reasoning about them with equations.
Gottlob Frege
Gave propositional and predicate logic their first rigorous formulation, distinguishing sense and reference and making inference a matter of explicit symbolic rules.