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
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.
Read the explanation sections first, then use the activities to test whether you can apply the idea under pressure.
Start Here
What this lesson is helping you do
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. The practice in this lesson depends on understanding Truth Functionality, Truth Table, Tautology, and Contradiction and applying tools such as Respect the Main Connective and Assign Sentence Letters Consistently correctly.
How to approach it
Read the explanation sections first, then use the activities to test whether you can apply the idea under pressure.
What the practice is building
You will put the explanation to work through evaluation practice, formalization practice, proof construction, analysis 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
Build complete truth tables for 5 arguments and correctly diagnose each as valid or invalid with a counterexample where appropriate, and correctly identify 4 equivalence pairs.
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.
Why truth tables work
Truth functionality and the decision procedure
Propositional logic has a remarkable property: once you know the truth value of every atomic statement, you can mechanically compute the truth value of any compound built from them. This property is called truth functionality, and it is what lets truth tables serve as a decision procedure for validity. There is no guessing, no appeal to meaning, no subjective judgment — just a table and a rule for reading it.
A truth table lists every possible combination of truth values for the atomic statements in an argument. With n atomic statements, there are 2^n rows. You compute the truth value of every premise and the conclusion in each row. The argument is valid if and only if there is no row where every premise is true and the conclusion is false. That is the entire validity test.
What to look for
- List every atomic letter appearing in the argument.
- Build 2^n rows covering all truth-value combinations.
- Compute every premise and the conclusion for each row.
- Check whether any row has all premises true and conclusion false.
Truth tables turn validity testing into a mechanical procedure. No matter how complicated the argument, the test always works the same way.
Three kinds of statements
Tautology, contradiction, and contingency
Once you have a truth table for a compound statement, you can classify it. A tautology is true in every row: for example, P ∨ ¬P. A contradiction is false in every row: for example, P ∧ ¬P. A contingent statement is true in some rows and false in others: for example, P → Q. Every propositional statement falls into exactly one of these three categories.
This classification matters for argument evaluation. If the conclusion of an argument is a tautology, the argument is automatically valid regardless of the premises, because the conclusion cannot be false. If a premise is a contradiction, the argument is also technically valid, because no row can make a contradictory premise true — so the 'all premises true, conclusion false' pattern can never appear. These are extreme cases, but noticing them can simplify analysis.
What to look for
- Check the final column of a truth table: all true, all false, or mixed.
- Tautologies are valid conclusions on their own.
- A contradictory premise makes any argument trivially valid (though not useful).
Every propositional statement is a tautology, a contradiction, or contingent. Recognizing which speeds up evaluation and avoids edge-case confusion.
Working tools
The central equivalences you should memorize
Some logical equivalences come up so often that you should memorize them and recognize them on sight. De Morgan's laws say ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q, and ¬(P ∨ Q) is equivalent to ¬P ∧ ¬Q. These let you push negations inward or pull them outward without changing meaning. Contraposition says P → Q is equivalent to ¬Q → ¬P, which is why modus tollens works. Material implication says P → Q is equivalent to ¬P ∨ Q, which is often the fastest way to turn a conditional into a form you can work with algebraically.
Two more are worth memorizing. Double negation says ¬¬P is equivalent to P. The biconditional expansion says P ↔ Q is equivalent to (P → Q) ∧ (Q → P). Together these equivalences let you transform almost any propositional formula into a more useful form without altering its truth conditions. They are the working vocabulary of propositional manipulation.
What to look for
- Memorize De Morgan's laws, contraposition, material implication, double negation, and biconditional expansion.
- Use equivalences to simplify formulas before testing validity.
- Never apply an 'equivalence' you are not sure is actually truth-preserving.
A small set of equivalences does enormous work. Memorizing them pays off in both speed and accuracy.
Beyond the verdict
Reading truth tables for diagnosis, not just verdict
A truth table tells you whether an argument is valid, but it also tells you why. If the argument is invalid, at least one row shows all premises true and the conclusion false. That row is a counterexample: it describes a possible state of affairs in which the argument fails. Pointing to the counterexample is much more informative than just saying 'invalid.'
Conversely, if the argument is valid, the truth table shows that no such row exists, and you can often see which premise did the logical work. In a modus ponens argument, the premise 'P → Q' rules out the row where P is true and Q is false; the premise 'P' rules out the rows where P is false. Together, they leave only rows where Q is true, which is exactly the conclusion. Reading the table this way gives you an intuition for why the rule is valid, not just a verdict that it is.
What to look for
- For invalid arguments, identify the row that serves as a counterexample.
- For valid arguments, trace which premises rule out which rows.
- Use the truth table as a diagnostic tool, not just a yes-or-no test.
Truth tables do not just say valid or invalid; they explain why. Reading them carefully builds logical intuition you can transfer to proof-building.
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.
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.
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.
Why it matters: Truth tables give an exhaustive, decision-procedure test for validity, consistency, and equivalence in propositional logic.
Tautology
A statement that is true under every possible truth-value assignment to its atomic parts.
Why it matters: Tautologies are logical truths of propositional form; they are the backbone of logical laws and equivalences.
Contradiction
A statement that is false under every possible truth-value assignment to its atomic parts.
Why it matters: Contradictions signal logical impossibility and are used in proofs by reductio ad absurdum.
Logical Equivalence
A relation between two statements that are true under exactly the same truth-value assignments.
Why it matters: Equivalent forms can be substituted for one another and let us simplify and transform arguments.
Validity
The property of an argument whose conclusion cannot be false while all its premises are true.
Why it matters: Validity is the central standard of deductive evaluation, and in propositional logic it can be mechanically tested.
Reference
Open these only when you need the extra structure
How the lesson is meant to unfold
Concept Intro
The core idea is defined and separated from nearby confusions.
Rule Or Standard
This step supports the lesson by moving from explanation toward application.
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
Truth Table for Modus Ponens
Building the truth table exhausts the logical possibilities. The absence of a bad row is not guesswork; it is a proof of validity.
Content
- Atomic letters: P, Q.
- Rows: (T,T), (T,F), (F,T), (F,F).
- P → Q: T, F, T, T.
- P (as premise): T, T, F, F.
- Conclusion Q: T, F, T, F.
- The only rows where both premises are true are (T,T) and neither rules out. Row 1: P → Q is T, P is T, Q is T — no problem. No row has both premises true and the conclusion false, so the argument is valid.
Worked Example
Affirming the Consequent is Invalid
A single counterexample row is enough to prove invalidity. The table also shows exactly what the world would have to look like for the premises to be true and the conclusion false.
Content
- Argument: P → Q, Q ⊢ P.
- Row (F, T): P → Q is true (false antecedent), Q is true, P is false.
- That row has all premises true and the conclusion false.
- Verdict: invalid, with the counterexample showing Q can be true while P is false.
Pause and Check
Questions to use before you move into practice
Self-check questions
- 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?
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.
Evaluation Practice
DeductiveTest Validity by Truth Table
Build a full truth table for each argument and determine whether it is valid. For invalid arguments, identify the counterexample row.
Arguments to test
Build the full truth table for each argument. Decide whether every row with all premises true also has a true conclusion. For invalid arguments, circle the counterexample row.
Argument A
P → Q, P ⊢ Q
This is modus ponens. Confirm validity and see which row each premise rules out.
Argument B
P → Q, Q ⊢ P
This is affirming the consequent. Find the counterexample row where P is false and Q is true.
Argument C
P → Q, ¬Q ⊢ ¬P
This is modus tollens. Verify validity and trace why the row with P true is eliminated.
Argument D
P ∨ Q, ¬P ⊢ Q
This is disjunctive syllogism. Confirm validity and note how the negated premise eliminates a whole column of rows.
Proof Draft
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Evaluation Practice
DeductiveIdentify the Equivalence
For each pair of statements, build a truth table and determine whether the two statements are logically equivalent. If they are, name the relevant equivalence law.
Pairs to compare
Build truth tables for both statements and check whether their final columns match row by row. Name the equivalence law if one applies.
Pair A
¬(P ∧ Q) and ¬P ∨ ¬Q
De Morgan's law.
Pair B
P → Q and ¬Q → ¬P
Contraposition.
Pair C
P → Q and ¬P ∨ Q
Material implication.
Pair D
P ↔ Q and (P → Q) ∧ (Q → P)
Biconditional expansion.
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Formalization Practice
DeductiveFormalization Drill: Truth Tables, Equivalence, and Validity
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: Truth Tables, Equivalence, and Validity
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.
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Evaluation Practice
DeductiveValidity Check: Truth Tables, Equivalence, and Validity
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.
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Proof Construction
DeductiveDeep Practice: Truth Tables, Equivalence, and Validity
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.
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Analysis Practice
DeductiveReal-World Transfer: Truth Tables, Equivalence, and Validity
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.
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Rapid Identification
DeductiveTimed Drill: Truth Tables, Equivalence, and Validity
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.
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Evaluation Practice
DeductivePeer Review: Truth Tables, Equivalence, and Validity
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.'
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Proof Construction
DeductiveConstruction Challenge: Truth Tables, Equivalence, and Validity
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.
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Diagnosis Practice
DeductiveCounterexample Challenge: Truth Tables, Equivalence, and Validity
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.
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Analysis Practice
DeductiveIntegration Exercise: Truth Tables, Equivalence, and Validity
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?
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Diagnosis Practice
DeductiveMisconception Clinic: Truth Tables, Equivalence, and Validity
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.'
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Proof Construction
DeductiveScaffolded Proof: Truth Tables, Equivalence, and Validity
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: Truth Tables, Equivalence, and Validity
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.
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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 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.
Where students usually go wrong
Omitting rows or building an incomplete truth table.
Reading the conclusion column without checking which rows have all premises true.
Claiming validity from a single favorable row rather than absence of a counterexample.
Confusing 'true conclusion in some row' with 'valid argument.'
Historical context for this way of reasoning
Ludwig Wittgenstein
The modern truth-table layout used to test propositional arguments was introduced by Wittgenstein in the Tractatus, building on earlier work by Peirce and others. It made propositional validity a mechanical procedure.