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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
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.
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
Examines each of the five standard connectives (negation, conjunction, disjunction, conditional, biconditional), how they translate natural-language forms, and the ambiguities students must resolve. The practice in this lesson depends on understanding Logical Connective, 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 formalization practice, classification practice, proof construction, evaluation practice, 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
Correctly symbolize 10 statements covering all five connectives, including at least one exclusive disjunction, one biconditional, and two 'only if' or 'necessary/sufficient' forms.
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.
Foundations
Negation and conjunction: the easy cases
Negation takes a single statement and flips its truth value. If P is true, ¬P is false; if P is false, ¬P is true. In English, negation can appear as 'not,' 'it is not the case that,' 'fails to,' 'never,' and many other forms, but the logic is the same: the compound is true exactly when the component is false.
Conjunction joins two statements with 'and,' 'but,' 'however,' 'although,' or 'while' and is true exactly when both conjuncts are true. Students sometimes think 'but' carries extra meaning, and in rhetoric it does, but in propositional logic 'but' is simply another way to say 'and.' The logic ignores the contrast and keeps only the truth-functional core.
What to look for
- Use negation whenever the English form flips the truth of a claim, regardless of the exact word used.
- Treat 'but,' 'however,' and 'although' as conjunctions for symbolization purposes.
- Watch the scope of any negation you introduce.
Negation flips truth values; conjunction demands both conjuncts be true. The logic strips away rhetorical nuance that English leaves in.
Core ambiguity
Disjunction and the inclusive-versus-exclusive question
The standard disjunction 'P ∨ Q' is inclusive: it is true if P is true, if Q is true, or if both are true. It is false only when both disjuncts are false. This is the default reading in propositional logic, and unless the context clearly forces an exclusive reading, you should use it.
Natural language sometimes uses 'or' exclusively, meaning one or the other but not both. Phrases like 'either...or, but not both' or contexts like a multiple-choice exam ('pick exactly one answer') make the exclusive reading obvious. In those cases, you symbolize the exclusive disjunction as (P ∨ Q) ∧ ¬(P ∧ Q), which captures 'at least one and not both.' When in doubt, go with the inclusive reading and flag the ambiguity explicitly.
What to look for
- Default to inclusive disjunction unless the context clearly demands otherwise.
- When exclusive reading is intended, symbolize it as (P ∨ Q) ∧ ¬(P ∧ Q) rather than inventing a new operator.
- Note the ambiguity in your analysis instead of silently choosing one reading.
Disjunction is inclusive by default. When natural language means exclusive disjunction, spell it out in standard connectives rather than assuming.
Most tricky connective
The conditional and its many English disguises
The conditional P → Q is false only in one case: when P is true and Q is false. In every other case it is true, including the cases where P is false. That treatment of 'false antecedent, anything goes' strikes many students as strange, but it is the only truth-functional reading that makes standard inference rules like modus ponens work reliably.
English expresses conditionals in many forms: 'if P then Q,' 'Q if P,' 'P only if Q,' 'P is sufficient for Q,' 'Q is necessary for P,' and 'whenever P, Q.' They all symbolize as P → Q, but the direction matters. 'P only if Q' is P → Q (Q is necessary for P), while 'P if Q' is Q → P (Q is sufficient for P). Mixing these up is the single most common beginner error in symbolization.
What to look for
- Remember that a conditional is false only when its antecedent is true and its consequent is false.
- Translate 'only if' and 'if' carefully; they point in opposite directions.
- For 'necessary' and 'sufficient' language, ask which event cannot occur without the other.
The conditional has many English faces, and getting its direction right is the single most important symbolization skill in this unit.
Symmetry and precision
The biconditional as mutual implication
The biconditional P ↔ Q is true exactly when P and Q have the same truth value. It expresses mutual implication: P → Q and Q → P at the same time. English phrases like 'if and only if,' 'just in case,' 'exactly when,' and 'is equivalent to' all signal the biconditional, though the first is the most explicit and the others sometimes need context.
A common beginner error is reading 'if and only if' as 'if' alone. That skips the reverse direction and turns a biconditional into a one-way conditional. Another common error is treating mathematical definitions as one-way conditionals when they are meant biconditionally. 'A triangle is equilateral if and only if it has three equal sides' is a biconditional because the definition works in both directions.
What to look for
- Treat 'if and only if' as two conditionals chained together.
- Use P ↔ Q when the context clearly specifies equivalence, especially in definitions.
- Watch for 'just in case,' which is often biconditional in logical writing but can be ambiguous in everyday speech.
The biconditional is mutual implication. Do not strip away either direction when symbolizing.
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.
Logical Connective
An operator such as negation, conjunction, disjunction, conditional, or biconditional that forms a compound proposition from simpler ones.
Why it matters: Connectives are the joints of propositional logic; the truth value of a compound depends entirely on them.
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
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
Only If Versus If
'Only if' points in the opposite direction of 'if.' Mixing them up changes the direction of the entire inference.
Content
- Statement 1: You will pass only if you submit the final exam. (P = pass, S = submit)
- Symbolization 1: P → S. Passing is possible only when S holds, so S is necessary for P.
- Statement 2: You will pass if you submit the final exam. (same letters)
- Symbolization 2: S → P. Submitting is enough to guarantee passing, so S is sufficient for P.
Worked Example
Exclusive Disjunction Spelled Out
Exclusive disjunction is not a separate primitive operator in standard propositional logic. It is captured by a combination of the inclusive disjunction and a conjunction with a negation.
Content
- Statement: Either the report is filed electronically or a signed paper copy is received, but not both.
- Letters: E = filed electronically, S = signed paper copy received.
- Symbolization: (E ∨ S) ∧ ¬(E ∧ S).
- Interpretation: At least one is true, and it is not the case that both are true.
Pause and Check
Questions to use before you move into practice
Self-check questions
- 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?
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.
Formalization Practice
DeductiveTranslate Each Connective
Each statement uses a different English form of a connective. Identify the connective and symbolize the statement using the letters given.
Varied English forms
Use the letters provided for each statement. Identify the connective, note any ambiguity, and give a symbolization that preserves scope.
Statement A
Access is granted only if both an ID and a PIN are provided. (A = access is granted, I = an ID is provided, N = a PIN is provided)
Handle 'only if' carefully; remember which direction it points.
Statement B
Either the report is filed electronically, or a signed paper copy is received, but not both. (E = the report is filed electronically, S = a signed paper copy is received)
Translate the exclusive disjunction explicitly.
Statement C
The alarm is active if and only if the system is armed and the door sensor is closed. (A = the alarm is active, R = the system is armed, D = the door sensor is closed)
Use a biconditional and preserve the scope of the conjunction.
Statement D
It is not the case that if the train is late, the meeting will be canceled. (T = the train is late, M = the meeting will be canceled)
The negation applies to the whole conditional; be careful about its placement.
Choose one of the arguments above, assign sentence letters, and translate the premises and conclusion into symbolic form.
Classification Practice
DeductiveNecessary, Sufficient, or Both?
For each statement, identify whether the relationship it asserts is necessary, sufficient, or both (biconditional), and then symbolize it.
Necessity and sufficiency
For each statement, say which condition is necessary, which is sufficient, or whether the relationship is biconditional. Then symbolize using the letters provided.
Statement A
Being over 18 is necessary for voting. (O = being over 18, V = voting)
If O is necessary for V, then V cannot occur without O.
Statement B
A score above 85 is sufficient for the scholarship. (H = scoring above 85, S = receiving the scholarship)
If H is sufficient for S, then H guarantees S.
Statement C
An animal is a triangle-shaped rectangle if and only if it does not exist. (T = is a triangle-shaped rectangle, E = exists)
Biconditional relationship — both directions hold.
Proof Draft
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Formalization Practice
DeductiveFormalization Drill: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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: The Five Connectives in Depth
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.
Proof Draft
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Analysis Practice
DeductiveSynthesis Review: The Five Connectives in Depth
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 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.
Where students usually go wrong
Treating 'if' and 'only if' as interchangeable.
Reading 'or' as exclusive by default when the context does not force that reading.
Stripping one direction off a biconditional when symbolizing.
Applying negation to only part of a compound when the full compound was meant to be negated.
Historical context for this way of reasoning
George Boole
Boole's algebra of logic gave the five connectives algebraic properties, which is why modern truth tables and equivalences look like arithmetic on truth values.