Read
Build the mental model
Move through the guided explanation first so the central distinction and purpose are clear before you evaluate your own work.
Propositional Logic
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.
Focus on understanding the core distinction first, then use the examples to see how the idea behaves in actual arguments.
Start Here
What this lesson is helping you do
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. The practice in this lesson depends on understanding Atomic Statement, Compound Statement, Main Connective, and Truth Functionality and applying tools such as Respect the Main Connective and Assign Sentence Letters Consistently correctly.
How to approach it
Focus on understanding the core distinction first, then use the examples to see how the idea behaves in actual arguments.
What the practice is building
You will put the explanation to work through classification practice, analysis practice, formalization practice, proof construction, evaluation practice, rapid identification, and diagnosis practice activities, so the goal is not just to recognize the idea but to use it under your own control.
What success should let you do
Correctly classify 10 statements as atomic or compound, name the main connective of every compound, and consistently assign sentence letters to atomic parts across 5 short arguments.
Reading Path
Move through the lesson in this order
The page is designed to teach before it tests. Use this sequence to keep the reading, examples, and practice in the right relationship.
Study
Watch the move in context
Use the worked examples to see how the reasoning behaves when someone else performs it carefully.
Do
Practice with a standard
Only then move into the activities, using the pause-and-check prompts as a final checkpoint before you submit.
Guided Explanation
Read this before you try the activity
These sections give the learner a usable mental model first, so the practice feels like application rather than guesswork.
Orientation
What propositional logic studies
Propositional logic treats complete statements as its basic units. It does not look inside a statement to analyze individual terms or objects; that is the job of predicate logic, which comes later. Instead, propositional logic asks how whole statements combine through connectives like 'and,' 'or,' 'if...then,' and 'not,' and how those combinations determine the truth of more complex claims.
This restricted focus is a deliberate choice. By treating each distinct claim as a single unit, propositional logic makes the structure of an argument much easier to see. Complicated English sentences become short symbolic formulas, and those formulas can be manipulated and tested by rules you can learn and apply without guesswork.
What to look for
- Remember that propositional logic works at the level of whole statements, not individual objects.
- Treat each distinct claim as a unit to be labeled, not a sentence to be analyzed further.
- Expect symbolic clarity to replace English ambiguity as you work.
Propositional logic is the logic of how whole statements combine. The payoff is structural clarity that natural language alone rarely gives you.
Core distinction
Atomic statements and the discipline of labeling
An atomic statement is one you are treating as logically simple at the propositional level. You are not claiming that the sentence is philosophically basic or that it could never be analyzed further. You are making a working choice: for the purposes of this analysis, this claim counts as a single unit and gets a single sentence letter.
The practical rule is strict. Two occurrences of the same claim get the same letter. Two different claims get different letters. If a claim appears once as a negation and once as a positive assertion, it is still the same atomic claim underneath, so it gets the same letter and the negation is expressed by the connective, not by changing the letter.
What to look for
- Use the same sentence letter whenever the same claim reappears, in any grammatical form.
- Use different letters for different claims, even when they share a topic.
- Do not bury a negation inside a letter; let the negation symbol carry it.
Atomic-statement labeling is disciplined bookkeeping. It preserves the identity of claims so that structure can be tracked across a whole argument.
Core skill
Compound statements and the main connective
A compound statement is built from one or more atomic statements and one or more connectives. The main connective of a compound is the operator with the widest scope: the one that, if you removed it, would leave behind two (or, for negation, one) smaller statements that together make up the whole.
Students often identify the wrong main connective because they fixate on the first or most obvious operator rather than the one governing the entire claim. A good habit is to describe the statement in words before symbolizing it. If you find yourself saying 'this is a conditional whose antecedent is a conjunction,' you have already done the structural work; the symbolization will follow almost mechanically.
What to look for
- Name the main connective in ordinary words before symbolizing.
- Ask which connective would have to be removed last if you were peeling the statement apart.
- Use parentheses freely to keep scope visible once you symbolize.
The main connective sets the logical shape of a compound statement. Identifying it first prevents most common symbolization errors.
Big picture
Form matters more than topic
Two arguments about completely different topics can share exactly the same logical form, and propositional logic is built to expose that shared form. When you see 'If it rains the field closes; it is raining; so the field closes' and 'If the server is down the service fails; the server is down; so the service fails,' both arguments have the form P → Q, P, therefore Q. They stand or fall together as far as propositional validity is concerned.
This is why a disciplined student resists the temptation to evaluate an argument by whether its subject matter is familiar. Surface content is a distraction; structure is the thing that logic can actually test. Training your eye to see structure first is one of the core skills you are building in this unit.
What to look for
- When comparing arguments, ignore topic and compare symbolized forms.
- Remind yourself that familiarity with a topic is not a logical virtue.
- Treat unfamiliar topics as an opportunity to practice reading form cleanly.
Propositional logic rewards readers who can see the shared shape of arguments underneath different subject matter.
Core Ideas
The main concepts to keep in view
Use these as anchors while you read the example and draft your response. If the concepts blur together, the practice usually blurs too.
Atomic Statement
A declarative sentence that is not further analyzed at the propositional level and is represented by a single sentence letter.
Why it matters: Atomic statements are the basic building blocks from which compound propositions are constructed.
Compound Statement
A statement formed from one or more simpler statements by the use of logical connectives.
Why it matters: Compound statements are where logical structure lives, and tracking them correctly is the whole point of propositional analysis.
Main Connective
The connective with the widest scope in a compound statement, which determines the statement's overall logical form.
Why it matters: The main connective tells you the basic shape of the proposition and therefore governs how it participates in inference.
Truth Functionality
The property that the truth value of a compound statement is completely determined by the truth values of its component parts.
Why it matters: Truth functionality is what makes propositional logic mechanically analyzable with truth tables.
Reference
Open these only when you need the extra structure
How the lesson is meant to unfold
Hook
A motivating question or contrast that frames why this lesson matters.
Concept Intro
The core idea is defined and separated from nearby confusions.
Worked Example
A complete example demonstrates what correct reasoning looks like in context.
Guided Practice
You apply the idea with scaffolding still visible.
Assessment Advice
Use these prompts to judge whether your reasoning meets the standard.
Mastery Check
The final target tells you what successful understanding should enable you to do.
Reasoning tools and formal patterns
Rules and standards
These are the criteria the unit uses to judge whether your reasoning is actually sound.
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.
Patterns
Use these when you need to turn a messy passage into a cleaner logical structure before evaluating it.
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.
Watch for
- 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.
Watch for
- 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.
Worked Through
Examples that model the standard before you try it
Do not skim these. A worked example earns its place when you can point to the exact move it is modeling and the mistake it is trying to prevent.
Worked Example
Identifying the Main Connective
Even when the antecedent contains its own connective, the main connective is still the operator with the widest scope. Here, the conditional governs the whole statement.
Content
- Statement: If the backup finishes and the checksum matches, the restore can begin.
- Structure: This is a conditional whose antecedent is a conjunction.
- Atomic parts: B = the backup finishes, C = the checksum matches, R = the restore can begin.
- Main connective: the conditional.
Worked Example
Negation Inside Versus Outside
Whether the main connective is a negation or a conjunction depends on scope. The same words can form very different compound statements, and missing that changes the logic.
Content
- Statement 1: Nora is not on the roster, and the match will be rescheduled.
- Structure 1: A conjunction whose first conjunct is a negation. Main connective: the conjunction.
- Statement 2: It is not the case that Nora is on the roster and the match will be rescheduled.
- Structure 2: A negation whose scope is a conjunction. Main connective: the negation.
Pause and Check
Questions to use before you move into practice
Self-check questions
- 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?
Practice
Now apply the idea yourself
Move into practice only after you can name the standard you are using and the structure you are trying to preserve or evaluate.
Classification Practice
DeductiveAtomic or Compound?
Classify each statement as atomic or compound. For compound statements, name the main connective in plain English before symbolizing anything.
Statements to classify
For each statement, say whether it is atomic or compound at the propositional level. If it is compound, name its main connective and list the atomic parts.
Statement A
The library closes at nine tonight.
Ask whether any connective is needed to express this claim.
Statement B
If the power stays on, the exam will run on schedule.
Find the main connective and the atomic parts it joins.
Statement C
Nora is not on the roster, and the match will be rescheduled.
Notice that the negation is inside a conjunction, so the main connective is the conjunction.
Statement D
It is not the case that both the printer is jammed and the toner is empty.
Notice that the negation here applies to the whole conjunction, not just one conjunct.
Statement E
Either the witness is mistaken or the log was tampered with, but not both.
Decide whether this is a plain disjunction, an exclusive disjunction, or something more complex.
Proof Draft
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Analysis Practice
DeductiveDescribe the Structure
For each complex statement below, describe its structure in ordinary words before symbolizing anything. Say which connective is the main connective and what its scope contains.
Structures to describe
Talk through the structure of each statement in one or two sentences. You should be able to say something like 'This is a conditional whose antecedent is a conjunction.'
Statement A
If the backup finishes and the checksum matches, the restore can begin.
Notice that the antecedent of the conditional is itself a conjunction.
Statement B
Either the team wins both games or the season is over.
Notice that the first disjunct is a conjunction.
Statement C
The alarm sounds if and only if the door is open and the system is armed.
Notice the biconditional, and be careful about the scope of the conjunction on one side of it.
Proof Draft
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Formalization Practice
DeductiveFormalization Drill: Atomic and Compound Statements
Translate each natural-language argument into formal notation. Identify the logical form and check whether the argument is valid.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Argument 1
If the server crashes, then the backup activates. If the backup activates, then an alert is sent. The server crashed. What follows?
Argument 2
Either the contract is valid or the parties must renegotiate. The contract is not valid. What follows?
Argument 3
All databases store records. This system does not store records. What can we conclude about this system?
Choose one of the arguments above, assign sentence letters, and translate the premises and conclusion into symbolic form.
Proof Construction
DeductiveProof Practice: Atomic and Compound Statements
Construct a step-by-step proof or derivation for each argument. Justify every step with the rule you are applying.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Prove
From premises: (1) P -> Q, (2) Q -> R, (3) P. Derive R.
Prove
From premises: (1) A v B, (2) A -> C, (3) B -> C. Derive C.
Prove
From premises: (1) ~(P & Q), (2) P. Derive ~Q.
Proof Draft
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Evaluation Practice
DeductiveValidity Check: Atomic and Compound Statements
Determine whether each argument is deductively valid. If invalid, describe a counterexample where the premises are true but the conclusion is false.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Argument A
All philosophers study logic. Socrates is a philosopher. Therefore, Socrates studies logic.
Argument B
If it rains, the ground is wet. The ground is wet. Therefore, it rained.
Argument C
No birds are mammals. Some mammals fly. Therefore, some things that fly are not birds.
Argument D
Either the door is locked or the alarm is on. The door is not locked. Therefore, the alarm is on.
Proof Draft
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Proof Construction
DeductiveDeep Practice: Atomic and Compound Statements
Work through these challenging exercises. Each one requires careful application of formal reasoning. Show your work step by step.
Challenging derivations
Prove each conclusion from the given premises. Label every inference rule you use.
Challenge 1
Premises: (1) (A & B) -> C, (2) D -> A, (3) D -> B, (4) D. Derive: C.
Challenge 2
Premises: (1) P -> (Q & R), (2) R -> S, (3) ~S. Derive: ~P.
Challenge 3
Premises: (1) A v B, (2) A -> (C & D), (3) B -> (C & E). Derive: C.
Challenge 4
Premises: (1) ~(P & Q), (2) P v Q, (3) P -> R, (4) Q -> S. Derive: R v S.
Proof Draft
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Analysis Practice
DeductiveReal-World Transfer: Atomic and Compound Statements
Apply formal logic to real-world contexts. Translate each scenario into formal notation, determine validity, and explain the practical implications.
Logic in the wild
These scenarios come from law, science, and everyday reasoning. Formalize and evaluate each.
Legal reasoning
A contract states: 'If the product is defective AND the buyer reports within 30 days, THEN a full refund will be issued.' The product was defective. The buyer reported on day 35. The company denies the refund. Is the company's position logically valid?
Medical reasoning
A diagnostic protocol states: 'If the patient has fever AND cough, test for flu. If the flu test is negative AND symptoms persist for 7+ days, test for bacterial infection.' A patient has a cough but no fever. What does the protocol require?
Policy reasoning
A policy reads: 'Students may graduate early IF they complete all required courses AND maintain a 3.5 GPA OR receive special faculty approval.' Due to the ambiguity of OR, identify the two possible readings and explain what difference they make.
Proof Draft
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Rapid Identification
DeductiveTimed Drill: Atomic and Compound Statements
Work through these quickly. For each mini-scenario, identify the logical form, name the rule used, and state whether the inference is valid. Aim for accuracy under time pressure.
Quick-fire logic identification
Identify the logical form and validity of each argument in under 60 seconds per item.
Item 1
If the reactor overheats, the failsafe triggers. The failsafe triggered. Therefore, the reactor overheated.
Item 2
All licensed pilots passed the medical exam. Jenkins is a licensed pilot. Therefore, Jenkins passed the medical exam.
Item 3
Either the encryption key expired or someone changed the password. The encryption key did not expire. Therefore, someone changed the password.
Item 4
If taxes increase, consumer spending decreases. Consumer spending has not decreased. Therefore, taxes have not increased.
Item 5
No insured vehicle was towed. This vehicle was towed. Therefore, this vehicle is not insured.
Item 6
If the sample is contaminated, then the results are unreliable. The sample is contaminated. Therefore, the results are unreliable.
Proof Draft
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Evaluation Practice
DeductivePeer Review: Atomic and Compound Statements
Below are sample student responses to a logic exercise. Evaluate each response: Is the formalization correct? Is the proof valid? Identify specific errors and suggest corrections.
Evaluate student proofs
Each student attempted to prove a conclusion from given premises. Find and correct any mistakes.
Student A's work
Premises: P -> Q, Q -> R, P. Student wrote: (1) P [premise], (2) P -> Q [premise], (3) Q [MP 1,2], (4) Q -> R [premise], (5) R [MP 3,4]. Conclusion: R. Student says: 'Valid proof by two applications of Modus Ponens.'
Student B's work
Premises: A v B, A -> C. Student wrote: (1) A v B [premise], (2) A -> C [premise], (3) A [from 1], (4) C [MP 2,3]. Conclusion: C. Student says: 'Since A or B is true, A must be true, so C follows.'
Student C's work
Premises: ~P v Q, P. Student wrote: (1) ~P v Q [premise], (2) P [premise], (3) ~~P [DN 2], (4) Q [DS 1,3]. Conclusion: Q. Student says: 'I used double negation then disjunctive syllogism.'
Student D's work
Premises: (P & Q) -> R, P, Q. Student wrote: (1) P [premise], (2) Q [premise], (3) (P & Q) -> R [premise], (4) R [MP 1,3]. Conclusion: R. Student says: 'Modus Ponens with P and the conditional.'
Proof Draft
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Proof Construction
DeductiveConstruction Challenge: Atomic and Compound Statements
Build complete proofs or arguments from scratch. You are given only a conclusion and some constraints. Construct valid premises and a rigorous derivation.
Build your own proofs
For each task, create a valid argument with explicit premises and step-by-step derivation.
Task 1
Construct a valid argument with exactly three premises that concludes: 'The network is secure.' Use at least one conditional and one disjunction in your premises.
Task 2
Build a valid syllogistic argument that concludes: 'Some scientists are not wealthy.' Your premises must be universal statements (All X are Y or No X are Y).
Task 3
Create a proof using reductio ad absurdum (indirect proof) that derives ~(P & ~P) from no premises. Show every step and justify each with a rule name.
Task 4
Construct a chain of conditional reasoning with at least four steps that connects 'The satellite detects an anomaly' to 'Emergency protocols are activated.' Make each link realistic and name the domain.
Proof Draft
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Diagnosis Practice
DeductiveCounterexample Challenge: Atomic and Compound Statements
For each invalid argument below, construct a clear counterexample -- a scenario where all premises are true but the conclusion is false. Then explain which logical error the argument commits.
Find counterexamples to invalid arguments
Each argument appears plausible but is invalid. Prove invalidity by constructing a specific counterexample.
Argument 1
If a student studies hard, they pass the exam. Maria passed the exam. Therefore, Maria studied hard.
Argument 2
All roses are flowers. Some flowers are red. Therefore, some roses are red.
Argument 3
No fish can fly. No birds are fish. Therefore, all birds can fly.
Argument 4
If the alarm sounds, there is a fire. The alarm did not sound. Therefore, there is no fire.
Argument 5
All effective medicines have been tested. This substance has been tested. Therefore, this substance is an effective medicine.
Proof Draft
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Analysis Practice
DeductiveIntegration Exercise: Atomic and Compound Statements
These exercises combine deductive logic with other topics and reasoning styles. Apply formal logic alongside empirical evaluation, explanation assessment, or problem-solving frameworks.
Cross-topic deductive exercises
Each scenario requires deductive reasoning combined with at least one other skill area.
Scenario 1
A quality control team uses this rule: 'If a batch fails two consecutive tests, it must be discarded.' Batch 47 failed Test A and passed Test B, then failed Test C. Formally determine whether the rule requires discarding Batch 47, and discuss whether the rule itself is well-designed from a problem-solving perspective.
Scenario 2
A researcher argues: 'All peer-reviewed studies in this meta-analysis show that X reduces Y. This study shows X reduces Y. Therefore, this study will be included in the meta-analysis.' Evaluate the deductive form, then inductively assess whether the meta-analysis conclusion would be strong.
Scenario 3
An insurance policy states: 'Coverage applies if and only if the damage was caused by a covered peril AND the policyholder reported it within 72 hours.' A policyholder reported water damage after 80 hours, claiming the damage was not discoverable sooner. Apply the formal logic of the policy, then consider whether the best explanation supports an exception.
Scenario 4
A hiring algorithm uses: 'If GPA >= 3.5 AND experience >= 2 years, then advance to interview.' Candidate X has GPA 3.8 and 18 months experience. Formally determine the outcome. Then evaluate: is the algorithm's rule inductively justified? What evidence would you want?
Proof Draft
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Diagnosis Practice
DeductiveMisconception Clinic: Atomic and Compound Statements
Each item presents a common misconception about deductive logic. Identify the misconception, explain why it is wrong, and provide a correct version of the reasoning.
Common deductive misconceptions
Diagnose and correct each misconception. Explain the error clearly enough for a fellow student to understand.
Misconception 1
A student claims: 'An argument is valid if its conclusion is true. Since the conclusion "Water is H2O" is obviously true, any argument concluding this must be valid.'
Misconception 2
A student says: 'Modus Tollens and denying the antecedent are the same thing. Both involve negation and a conditional, so they must work the same way.'
Misconception 3
A student writes: 'This argument is invalid because the conclusion is false: All cats are reptiles. All reptiles lay eggs. Therefore, all cats lay eggs.'
Misconception 4
A student argues: 'A sound argument can have a false conclusion, because soundness just means the argument uses correct logical rules.'
Misconception 5
A student claims: 'Since P -> Q is equivalent to ~P v Q, we can derive Q from P -> Q alone, without knowing whether P is true.'
Proof Draft
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Proof Construction
DeductiveScaffolded Proof: Atomic and Compound Statements
Build proofs in stages. Each task gives you a partially completed derivation. Fill in the missing steps, justify each one, and then extend the proof to a further conclusion.
Step-by-step proof building
Complete each partial proof, then extend it. Every step must cite a rule.
Scaffold 1
Premises: (1) (A v B) -> C, (2) D -> A, (3) D. Partial proof: (4) A [MP 2,3]. Your tasks: (a) Complete the proof to derive C. (b) If we add premise (5) C -> E, extend the proof to derive E.
Scaffold 2
Premises: (1) P -> (Q -> R), (2) P, (3) Q. Partial proof: (4) Q -> R [MP 1,2]. Your tasks: (a) Complete the proof to derive R. (b) If we add premise (5) R -> ~S, extend to derive ~S. (c) If we also add (6) S v T, what can you derive?
Scaffold 3
Premises: (1) ~(A & B), (2) A. Your task: Prove ~B step by step. Hint: You may need to use an assumption for indirect proof. Show the subproof structure clearly.
Scaffold 4
Premises: (1) P v Q, (2) P -> R, (3) Q -> S, (4) ~R. Your tasks: (a) Derive ~P from (2) and (4). (b) Using (a), derive Q from (1). (c) Using (b), derive S. (d) Name each rule used.
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Analysis Practice
DeductiveSynthesis Review: Atomic and Compound Statements
These exercises require you to combine everything you have learned about deductive reasoning. Each scenario tests multiple skills simultaneously: formalization, rule application, validity checking, and proof construction.
Comprehensive deductive review
Each task combines multiple deductive skills. Show all your work.
Comprehensive 1
A software license agreement states: 'The software may be used commercially if and only if the licensee has purchased an enterprise plan and has fewer than 500 employees, or has received written exemption from the vendor.' Formalize this using propositional logic, determine what follows if a company has an enterprise plan and 600 employees with no exemption, and identify any ambiguity in the original text.
Comprehensive 2
Construct a valid argument with four premises and one conclusion about data privacy. Then create an invalid argument about the same topic that looks similar but commits a formal fallacy. Finally, prove the first is valid and show a counterexample for the second.
Proof Draft
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Truth-Table Builder
Enter a propositional formula using variables (P, Q, R...) and connectives. Separate multiple formulas with commas to compare them.
~ or ¬& or ∧| or ∨-> or →<-> or ↔
| P | Q | P → Q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Contingent2 variables · 4 rows · 3 true
Proof Constructor
Build a formal proof step by step. Add premises, apply inference rules, cite earlier lines, and derive your conclusion.
Add premises and derived steps above, or load a template to get started.
\u2588 Premise\u2588 Derived step
Symbols:\u00AC not\u2227 and\u2228 or\u2192 if-then\u2194 iff\u2234 therefore
Animated Explainers
Step-by-step visual walkthroughs of key concepts. Click to start.
Read the explanation carefully before jumping to activities!
Further Support
Open these only if you need extra help or context
Mistakes to avoid before submitting
- 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.
Where students usually go wrong
Calling a statement atomic when it contains an easily spotted connective.
Calling a statement compound when it is genuinely atomic and the 'connective' is part of the predicate.
Fixating on the first connective rather than the one with the widest scope.
Changing the sentence letter assigned to a claim when the claim reappears in negated form.
Historical context for this way of reasoning
Chrysippus
Chrysippus and the Stoics were the first to study propositional inference systematically, analyzing statements as wholes and cataloguing valid argument forms that remain central to propositional logic today.