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
Why Predicate Logic Extends Propositional Logic
Motivates the move from propositional to predicate logic by showing arguments that propositional logic cannot capture and introduces the basic vocabulary of predicates, constants, and variables.
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
Motivates the move from propositional to predicate logic by showing arguments that propositional logic cannot capture and introduces the basic vocabulary of predicates, constants, and variables. The practice in this lesson depends on understanding Predicate, Individual Constant, Variable, and Domain of Discourse and applying tools such as Universal Instantiation and Existential Instantiation 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, quiz, 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 identify predicates, arities, and constants in 10 English statements, and write the corresponding atomic formulas for those that do not yet require quantifiers.
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.
Motivation
Arguments that propositional logic cannot see
Consider this argument: 'All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.' It is obviously valid, but propositional logic cannot show that. In propositional logic, each premise is a distinct sentence letter. Premise 1 might be H, premise 2 might be S, and the conclusion might be M. The form H, S ⊢ M is not valid by any propositional rule, because propositional logic cannot see inside the sentences to notice the repeated references to humans and mortality.
Predicate logic solves this by looking inside sentences. 'All humans are mortal' becomes ∀x (Hx → Mx): for every x, if x is a human then x is mortal. 'Socrates is a human' becomes Hs, where s is a constant naming Socrates. The conclusion 'Socrates is mortal' becomes Ms. Now the inference is formally visible: instantiate the universal premise to s, apply modus ponens, and you have Ms. The sentence-internal structure is what predicate logic lets us track.
What to look for
- Recognize that propositional logic treats whole statements as unanalyzed units.
- See that many valid arguments depend on internal sentence structure propositional logic cannot represent.
- Accept that predicate logic adds machinery (predicates, variables, quantifiers) to make that structure visible.
Predicate logic is a strict extension of propositional logic. Everything propositional logic does, predicate logic still does; and predicate logic does many things propositional logic cannot.
Core vocabulary
Predicates and their arguments
A predicate is an expression that says something about one or more objects. 'Is mortal,' 'is tall,' and 'is prime' are one-place predicates: they apply to a single object. 'Loves,' 'is taller than,' and 'divides' are two-place predicates: they apply to a pair of objects. 'Is between,' 'gives to,' and 'introduces to' are three-place predicates. The arity of a predicate is the number of objects it relates, and it matters because a predicate with n arguments cannot be used as one with m ≠ n.
In predicate logic, we write predicates with uppercase letters, and their arguments inside parentheses or as subscripts. 'Socrates is mortal' becomes Ms (or M(s)). 'Alice loves Bob' becomes Lab (or L(a,b)). The first position is filled by the first argument, the second by the second, and so on. Order matters for relational predicates: Lab and Lba mean different things, just as 'Alice loves Bob' and 'Bob loves Alice' mean different things in English.
What to look for
- Identify the arity of every predicate you use.
- Respect the order of arguments in relational predicates.
- Use uppercase letters for predicates and lowercase for constants and variables.
Predicates have arities, and relational predicates care about argument order. Getting either wrong changes the meaning of the formula.
Building blocks
Constants, variables, and the domain of discourse
A constant is a symbol that names a specific individual. We typically use lowercase letters near the start of the alphabet, such as a, b, c, or sometimes meaningful abbreviations like s for Socrates. A variable is a placeholder symbol that does not name any specific individual on its own; we typically use letters near the end of the alphabet, such as x, y, z. Variables become meaningful only when a quantifier binds them or when a context assigns a value.
The domain of discourse is the set of objects that variables are allowed to range over. In one context the domain might be 'all humans,' in another 'all natural numbers,' and in another 'all books in this library.' The same formula ∀x Px can be true in one domain and false in another, so specifying the domain is part of responsible formalization. When the domain is left implicit, always be prepared to state what you mean by 'everything' or 'something.'
What to look for
- Use lowercase letters near the start of the alphabet for constants.
- Use lowercase letters near the end of the alphabet for variables.
- State the domain of discourse explicitly whenever it matters for interpretation.
Constants name; variables hold places; the domain tells you what 'everything' means. Clarity about all three prevents most beginner errors.
Syntax grounding
What atomic formulas look like
An atomic formula in predicate logic is a predicate applied to the right number of constants or variables. Ma is atomic (M is a one-place predicate applied to the constant a). Lab is atomic. Pxy is atomic if we treat P as a two-place predicate, even though it contains variables rather than constants. These atomic formulas are the building blocks; everything else in predicate logic is constructed from them using connectives and quantifiers.
Before you can formalize a complex claim, you need to be comfortable with atomic formulas. Practice reading and writing them until the syntax feels natural. If you can glance at ∀x (Hx → Mx) and immediately see 'for all x, if x has the property H then x has the property M,' you have internalized the basic syntax and are ready to move on to quantifier translation.
What to look for
- Read atomic formulas aloud as English sentences about specific individuals or variables.
- Practice writing atomic formulas for simple English claims.
- Do not move on to quantifiers until the atomic syntax feels natural.
Atomic formulas are the building blocks. Get comfortable with them before layering on quantifiers and connectives.
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.
Predicate
An expression that attributes a property to an object or a relation among several objects; a one-place predicate applies to a single object, while a relational predicate applies to two or more.
Why it matters: Predicates let logic represent internal sentence structure instead of treating whole statements as unanalyzed propositions.
Individual Constant
A symbol that names a specific object in the domain of discourse, typically written with lowercase letters like a, b, c.
Why it matters: Constants pick out particular individuals and let you form atomic formulas about them.
Variable
A symbol like x, y, or z that does not name a specific individual but can range over the domain when bound by a quantifier.
Why it matters: Variables are the placeholders that make generality and existence expressible.
Domain of Discourse
The set of objects that the variables of a predicate-logic formula are taken to range over in a given interpretation.
Why it matters: The domain fixes what 'everything' and 'something' mean in a particular context, and many translation errors come from leaving it unclear.
Reference
Open these only when you need the extra structure
How the lesson is meant to unfold
Hook
A motivating question or contrast that frames why this lesson matters.
Concept Intro
The core idea is defined and separated from nearby confusions.
Worked Example
A complete example demonstrates what correct reasoning looks like in context.
Guided Practice
You apply the idea with scaffolding still visible.
Assessment Advice
Use these prompts to judge whether your reasoning meets the standard.
Mastery Check
The final target tells you what successful understanding should enable you to do.
Reasoning tools and formal patterns
Rules and standards
These are the criteria the unit uses to judge whether your reasoning is actually sound.
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 Argument
Predicates and constants are the raw materials. The universal premise will become a quantified formula in the next lesson, but Hs and Ms are already atomic predicate-logic sentences.
Content
- Argument: 'All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.'
- Predicates: H = is a human (one-place), M = is mortal (one-place).
- Constants: s = Socrates.
- Premise 1 (atomic forms preview): a universal claim about human implying mortal.
- Premise 2: Hs.
- Conclusion: Ms.
Worked Example
Relational Predicates
Order matters for relational predicates. Lab and Lba describe different states of affairs, just as the English sentences do.
Content
- Statement: 'Alice loves Bob but Bob does not love Alice.'
- Predicate: L = loves (two-place).
- Constants: a = Alice, b = Bob.
- Formalization: Lab ∧ ¬Lba.
Pause and Check
Questions to use before you move into practice
Self-check questions
- Have I identified the arity of every predicate?
- Have I respected argument order in relational predicates?
- Is my domain of discourse clear?
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
DeductiveIdentify the Pieces
For each statement, identify the predicates (with their arities), the constants, and any free variables. Then write the atomic formulas.
Statements to analyze
For each statement, list predicates with arities, list constants with what they name, and write the atomic formulas using capital letters for predicates and lowercase for constants.
Statement A
Alice is tall.
One-place predicate T applied to the constant a.
Statement B
Bob is taller than Alice.
Two-place predicate T applied to constants b and a, with the order mattering.
Statement C
7 is prime and 7 is greater than 5.
One-place predicate P and two-place predicate G, connected by conjunction.
Statement D
Chicago is between Seattle and New York.
Three-place predicate B applied to three constants, with the order mattering.
Choose one of the arguments above, assign sentence letters, and translate the premises and conclusion into symbolic form.
Quiz
DeductiveScenario Check: Why Predicate Logic Extends Propositional Logic
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: "Treating a one-place predicate as two-place or vice versa." 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 recognize limits of propositional logic, starting from scratch. Be specific about each step and explain why the order matters.
Can the student transfer the skill of recognize limits of propositional logic to a genuinely new case?
Question 3 — Distinguish
Someone confuses predicate with constant. 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 Argument" 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?
Proof Draft
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Formalization Practice
DeductiveFormalization Drill: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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: Why Predicate Logic Extends Propositional Logic
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 mix up predicates and their arguments; the order in Lab is not interchangeable with Lba.
- Do not move on to quantifiers before atomic syntax feels natural.
Where students usually go wrong
Treating a one-place predicate as two-place or vice versa.
Ignoring argument order in relational predicates.
Confusing constants with variables because both are lowercase.
Formalizing without specifying the domain of discourse.
Historical context for this way of reasoning
Gottlob Frege
Frege's Begriffsschrift introduced the analysis of sentences into function and argument, replacing the subject-predicate grammar of traditional logic with a more mathematically tractable structure.