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.
Predicate Logic
Quantifier Rules in Proof
Introduces the four quantifier inference rules (universal instantiation, existential instantiation, universal generalization, existential generalization) and explains the restrictions that each rule imposes.
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 the four quantifier inference rules (universal instantiation, existential instantiation, universal generalization, existential generalization) and explains the restrictions that each rule imposes. The practice in this lesson depends on understanding Universal Quantifier, Existential Quantifier, Instantiation, and Generalization and applying tools such as Universal Instantiation and Existential Instantiation 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 proof construction, quiz, formalization practice, 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
Construct 4 formal proofs using all four quantifier rules, with every step correctly justified and every rule restriction respected.
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 instantiation
From quantified formulas to usable instances
Quantified formulas cannot be used directly in most propositional-style inference rules. You cannot apply modus ponens to ∀x (Px → Qx) and Pa, because modus ponens needs a conditional with Pa as its antecedent, not a universally quantified conditional. To get there, you need universal instantiation: from ∀x (Px → Qx), you can derive Pa → Qa. Now you have a propositional conditional with the right antecedent, and modus ponens applies.
Universal instantiation is the workhorse rule of predicate proof. When you have a universally quantified premise, your first move is almost always to instantiate it to the individual you care about. The rule is safe: if the universal holds for everything, it holds for any particular thing, so no restrictions apply. You can instantiate to any constant that already appears in your proof or to a fresh constant you introduce for this step.
What to look for
- When you have a universal premise, instantiate to the name you care about.
- Universal instantiation works for any name; no restrictions apply.
- Instantiating does not consume the universal; you can instantiate the same universal multiple times to different names.
Universal instantiation turns a universal claim into a usable statement about a specific individual. It is the first move in most predicate proofs.
Critical restriction
Existential instantiation requires a fresh name
When you have an existential premise like ∃x Px, you know that something satisfies P, but you do not know what. To use the premise, you give the unknown object a name — call it c — and write Pc. This is existential instantiation, and it works only if c is fresh: it must not appear anywhere earlier in the proof. If you reused an existing name, you would be implicitly claiming that the witness of the existential is the same individual as the one named earlier, which is not something the existential premise licenses.
The fresh-name restriction is the single most commonly violated rule in predicate proofs. Students see an existential and want to use a name they already have, especially if the name seems 'relevant.' Resist. Introduce a new letter, use it for the witness, and remember that c is whatever satisfies the existential — you know it exists, but you do not know anything else about it. At the end of the proof, you will typically discharge c via generalization or it will simply remain as a witness.
What to look for
- When you have an existential premise, introduce a fresh constant for the witness.
- Never reuse an existing constant to instantiate an existential.
- Treat the witness as an unknown specific individual: you know it satisfies the existential predicate, but nothing more.
Existential instantiation needs a fresh name. This is not a bureaucratic rule; it prevents you from identifying witnesses that might be different individuals.
Critical restriction
Universal generalization requires arbitrariness
Universal generalization goes the other way: from φ(a), you infer ∀x φ(x). But this is licensed only if a was 'arbitrary' — that is, if nothing specific is known about a beyond the properties derivable from the premises alone. In practice, this means you can generalize from a name you introduced specifically for a derivation step, as long as you never used any premise that singled out that name. If you generalized from a name that came from an existential instantiation, you would be claiming that every individual has the property of the specific witness, which does not follow.
The safest practice is to use universal generalization only on names that you introduced at the start of a subproof for the purpose of reasoning about 'an arbitrary individual.' If the name came from an existential instantiation or from a specific premise, you cannot generalize. When a proof requires generalizing from a name and you are unsure whether the name is arbitrary, trace its history: if every fact derived about it came from universally quantified premises or from assumptions that applied to any object, generalization is safe; otherwise it is not.
What to look for
- Generalize only from names that were introduced as arbitrary.
- Never generalize from a name that came from existential instantiation.
- If you are not sure whether a name is arbitrary, trace its history through the proof.
Universal generalization is powerful but restricted. The name being generalized must be arbitrary, and students should be conservative about when to apply it.
A simple rule
Existential generalization is the easy one
Existential generalization goes from φ(a) to ∃x φ(x). Unlike universal generalization, it has almost no restrictions: if you have established a property for any specific individual, you can conclude that something has that property. The only subtlety is picking the right variable and making sure it does not already appear bound in φ(a), which would cause a clash. In practice, choose a fresh variable for the existential quantifier and you will never run into trouble.
Existential generalization is used more often than students expect. Whenever the goal is existential and you have derived a relevant atomic fact, existential generalization is the concluding step. It is also how you would formally record the move 'I have a specific witness, therefore such a thing exists' — the kind of inference that feels obvious in English but needs an explicit rule in formal proof.
What to look for
- Use existential generalization whenever the goal is existential and you have derived a witness.
- Choose a fresh variable to avoid clashes with existing bound variables.
- Remember that no arbitrariness restriction applies.
Existential generalization is the easy move: any specific fact licenses an existential claim. Use it freely when the goal matches.
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.
Universal Quantifier
The quantifier ∀x, read 'for all x,' which claims that the formula it binds holds for every object in the domain.
Why it matters: Universal quantification is how predicate logic expresses claims of the form 'every S is P' without the distribution machinery of categorical logic.
Existential Quantifier
The quantifier ∃x, read 'there exists x such that,' which claims that the formula it binds holds for at least one object in the domain.
Why it matters: Existential quantification is how predicate logic expresses 'some S is P' and underlies reasoning about witnesses and examples.
Instantiation
A proof move that goes from a quantified claim to a claim about a specific individual; universal instantiation picks any individual, while existential instantiation introduces a fresh name for a witness.
Why it matters: Instantiation is how quantified premises become usable in step-by-step derivations.
Generalization
A proof move that goes from a claim about an individual to a quantified claim; universal generalization is allowed only when the individual was arbitrary, and existential generalization is always allowed when a specific instance has been found.
Why it matters: Generalization is how derivations conclude quantified statements, and its restrictions are the main source of errors in predicate proofs.
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.
Universal Instantiation
From ∀x φ(x), infer φ(a) for any individual constant or previously introduced name a.
Common failures
- Instantiating to a name that has been reserved for existential reasoning.
- Substituting the same variable for two different bound variables by accident.
Existential Instantiation
From ∃x φ(x), infer φ(c), where c is a fresh name not appearing earlier in the proof.
Common failures
- Reusing an existing name for the witness, which can lead to spurious identifications.
- Treating the witness name as if it had arbitrary properties beyond φ(c).
Universal Generalization
From φ(a), where a was introduced as an arbitrary individual, infer ∀x φ(x).
Common failures
- Generalizing from a name introduced by existential instantiation.
- Generalizing from a name that already has specific properties from earlier premises.
Existential Generalization
From φ(a), infer ∃x φ(x) for any individual constant a.
Common failures
- Generalizing to the wrong variable when φ(a) already contains that variable free.
- Using existential generalization to justify a claim that the original individual was unspecified.
Respect Scope and Binding
No variable occurrence may be treated as bound unless a quantifier with that variable actually governs it, and no free variable may be ignored in semantic evaluation.
Common failures
- Claiming a free variable is bound because another quantifier uses the same letter.
- Misplacing a quantifier so that it binds fewer or more occurrences than intended.
Patterns
Use these when you need to turn a messy passage into a cleaner logical structure before evaluating it.
Quantified Translation Schema
Input form
natural_language_claim
Output form
first_order_formula
Steps
- Identify the domain of discourse.
- Identify the predicates needed and their arities (one-place, two-place, etc.).
- Decide whether each part of the claim is universal, existential, or relational.
- Place quantifiers in the order that the English demands.
- Verify that every variable occurrence is bound and that the scope matches the English meaning.
Watch for
- Swapping the order of nested quantifiers.
- Using a two-place predicate as if it were one-place.
- Letting a variable escape the scope of its quantifier.
Predicate Proof Schema
Input form
first_order_argument
Output form
quantifier_sensitive_derivation
Steps
- Formalize premises and conclusion.
- Instantiate universal premises to names that fit the goal.
- When using an existential premise, introduce a fresh witness name.
- Apply propositional inference rules to the instantiated formulas.
- Generalize only when the name is arbitrary (for universal generalization) or when a specific witness is available (for existential generalization).
Watch for
- Generalizing from a name introduced by existential instantiation.
- Reusing a witness name across different existential claims.
- Skipping the instantiation step and trying to manipulate quantified formulas directly.
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
The Socrates Proof
Universal instantiation bridges quantified premises and propositional inference rules. Once you have Hs → Ms, modus ponens applies exactly as in propositional logic.
Content
- Premises: ∀x (Hx → Mx), Hs.
- Goal: Ms.
- Line 1: ∀x (Hx → Mx) (premise).
- Line 2: Hs (premise).
- Line 3: Hs → Ms (universal instantiation from 1, using s).
- Line 4: Ms (modus ponens from 2, 3).
Worked Example
From Existential Premise to Existential Conclusion
Existential instantiation introduces a fresh name, universal instantiation specializes to that name, modus ponens derives the specific fact, and existential generalization wraps up the conclusion. Every step respects its rule's restrictions.
Content
- Premises: ∃x Px, ∀x (Px → Qx).
- Goal: ∃x Qx.
- Line 1: ∃x Px (premise).
- Line 2: ∀x (Px → Qx) (premise).
- Line 3: Pc (existential instantiation from 1, with fresh constant c).
- Line 4: Pc → Qc (universal instantiation from 2, using c).
- Line 5: Qc (modus ponens from 3, 4).
- Line 6: ∃x Qx (existential generalization from 5).
Pause and Check
Questions to use before you move into practice
Self-check questions
- Does every existential instantiation use a fresh name?
- Does every universal generalization use a name that was arbitrary?
- Did I instantiate before applying propositional rules?
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.
Proof Construction
DeductiveApply the Four Rules
For each argument, construct a formal proof using universal instantiation, existential instantiation, universal generalization, or existential generalization as needed, along with propositional rules.
Quantifier proofs
Each argument requires at least one quantifier rule. Number your lines, cite the rule and its inputs, and make sure every instantiation and generalization is legitimate.
Proof A
Premises: ∀x (Px → Qx), Pa. Goal: Qa.
Universal instantiation then modus ponens.
Proof B
Premises: ∀x (Hx → Mx), Hs. Goal: Ms. (The Socrates argument.)
Universal instantiation to s, then modus ponens.
Proof C
Premises: ∃x Px, ∀x (Px → Qx). Goal: ∃x Qx.
Existential instantiation to fresh c, universal instantiation to c, modus ponens, existential generalization.
Proof D
Premises: ∀x Px, ∀x Qx. Goal: ∀x (Px ∧ Qx).
Instantiate both universals to an arbitrary a, conjoin, then universally generalize.
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Quiz
DeductiveScenario Check: Quantifier Rules in Proof
Each question presents a scenario or challenge. Answer in two to four sentences. Focus on showing that you can use what you learned, not just recall it.
Scenario questions
Work through each scenario. Precise, specific answers are better than long vague ones.
Question 1 — Diagnose
A student makes the following mistake: "Reusing an existing name when existentially instantiating." Explain specifically what is wrong with this reasoning and what the student should have done instead.
Can the student identify the flaw and articulate the correction?
Question 2 — Apply
You encounter a new argument that you have never seen before. Walk through exactly how you would apply universal instantiation, starting from scratch. Be specific about each step and explain why the order matters.
Can the student transfer the skill of apply universal instantiation to a genuinely new case?
Question 3 — Distinguish
Someone confuses instantiation with generalization. Write a short explanation that would help them see the difference, and give one example where getting them confused leads to a concrete mistake.
Does the student understand the boundary between the two concepts?
Question 4 — Transfer
The worked example "The Socrates Proof" showed one way to handle a specific case. Describe a situation where the same method would need to be adjusted, and explain what you would change and why.
Can the student adapt the demonstrated method to a variation?
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Formalization Practice
DeductiveFormalization Drill: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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: Quantifier Rules in Proof
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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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 the fresh-name requirement for existential instantiation; it is what keeps the proof honest.
- Do not generalize from any name whose provenance you cannot trace to arbitrary assumptions.
Where students usually go wrong
Reusing an existing name when existentially instantiating.
Generalizing from a name that came from existential instantiation.
Trying to apply propositional rules directly to quantified formulas without instantiating first.
Forgetting to cite the quantifier rule alongside the propositional ones.
Historical context for this way of reasoning
Gerhard Gentzen
Gentzen's natural deduction system, developed in the 1930s, gave us the modern formulation of the four quantifier rules along with their restrictions. Most predicate logic courses still teach the rules essentially in his form.