# Formal logic: choose a text and test a quantifier claim — Study worksheet

[Study guides](https://unireadinglist.com/guides/formal-logic/)

[Course topic: Logic](https://unireadinglist.com/subjects/philosophy/#course-logic)

Sources checked: 2026-11-03

Study route and exercise: University Reading List editorial project.

For first study of classical truth-functional and first-order logic. The edition map uses forall x: Calgary Fall 2023, Copi/Cohen/Rodych 15e (2019), and Language, Proof and Logic 2e (2011). The free forallx landing page serves changing revisions; the linked fixed F23 source establishes this route’s chapter map. Modal logic, induction and metatheory are optional later branches, not prerequisites for the exercise.

## Choose your starting point

- [ ] I want a free proof-and-model route
  Choose forallx. Download a working copy and record its revision before starting; otherwise exercise numbers and rule names can drift between classmates. The route below names Fall 2023 parts and chapter titles, and you should compare them with your copy.

- [ ] I need everyday arguments as well as symbols
  Choose Copi when your course also covers definitions, fallacies, syllogisms and induction. Do not read all of that as mandatory preparation for a first-order proof course; select the symbolic chapters with the syllabus.

- [ ] I learn by building and checking worlds
  Choose LPL if the model-building and proof software will be used regularly and you can establish the required access. A second-hand printed copy alone may not supply a usable registration key; the free route remains a complete alternative for the tasks here.

## Compare the three books

### forall x: Calgary. An Introduction to Formal Logic

P. D. Magnus; Tim Button; Robert Trueman; Richard Zach; with contributions by J. Robert Loftis and Aaron Thomas-Bolduc · Fall 2023 · 2023
Free, focused formal introduction

Choose it when: You want truth tables, interpretations and Fitch-style deduction in one openly available route.
Use another route when: Your required assessments use another proof calculus and you cannot reconcile rule conventions.
Selected reading: Fall 2023 Parts I–VII are the core; postpone modal logic and metatheory.
Limits and trade-offs: The author site’s current free files and newest print recommendation are distinct from the stored Fall 2023 paperback. Freeze the version rather than treating all revisions as one edition.

#### Try before choosing

Using the selected Fall 2023 FOL interpretation material, translate “each participant has a reviewer” and “one reviewer reviews every participant”. Build a two-participant interpretation before attempting a proof.
Check your work: Write both quantifier orders and list every true assignment pair. A model that separates the sentences matters more than fluent paraphrase; use the edition’s own notation.

Your notes:

________________________________

ISBN: 9798859963584

[Official free reading](https://forallx.openlogicproject.org/)

[Book record and access options](https://unireadinglist.com/subjects/philosophy/forall-x-calgary/)

### Introduction to Logic

Irving M. Copi; Carl Cohen; Victor Rodych · 15 · 2019
Broad reasoning reference

Choose it when: You need informal argument analysis and traditional categorical logic alongside symbolic methods.
Use another route when: You want the shortest direct sequence from truth tables to first-order natural deduction.
Selected reading: 15e chapters 1–2 orient argument analysis; chapters 8–10 cover the symbolic route.
Limits and trade-offs: Breadth brings a different allocation of space. Its methods-of-deduction conventions should not be mixed unlabelled with a Fitch exercise; translate and justify the rule correspondence.

#### Try before choosing

After 15e chapters 8–10, test A → B against B → A and ¬B → ¬A with one complete truth table. Label the invalid inference, not merely the false sentence.
Check your work: All four valuations appear; A=false, B=true refutes the converse inference, while the contrapositive has the same truth values as A → B.

Your notes:

________________________________

ISBN: 9781138500860

[Book record and access options](https://unireadinglist.com/subjects/philosophy/introduction-to-logic-copi/)

### Language, Proof and Logic

Dave Barker-Plummer; Jon Barwise; John Etchemendy · 2 · 2011
Interactive first-order courseware

Choose it when: You benefit from testing worlds and checking proofs and your course supports the package.
Use another route when: You need unrestricted free access or assume every used copy includes transferable grading.
Selected reading: 2e chapters 1–8 build propositional methods; 9–14 introduce quantification and proofs.
Limits and trade-offs: 9781575866321 identifies the paperback package; CSLI lists a separate electronic ISBN. Current web applications and nontransferable keys must be checked with Openproof, rather than inferred from an old CD description.

#### Try before choosing

Use the second-edition quantification/proof route to annotate a short existential-witness subproof: which names are temporary, and which conclusions may leave the subproof? Do this on paper if software access is absent.
Check your work: The exported conclusion contains no temporary witness name, and that name occurs in no outside undischarged assumption. A software green tick cannot replace this explanation.

Your notes:

________________________________

ISBN: 9781575866321

[Book record and access options](https://unireadinglist.com/subjects/philosophy/language-proof-and-logic/)

## Check your preparation

### Must I already know algebra?

Your notes:

________________________________

### Which notation should my notebook use?

Your notes:

________________________________

### What counts as checking an invalidity claim?

Your notes:

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## Six steps with a concrete output

- [ ] 1. Separate argument from truth

Selected reading: forallx Fall 2023 Part I, Arguments and The scope of logic; Copi 15e chapter 1 is an alternative.

Make this: Write two arguments of the same form with different subject matter. Label premises, conclusion and factual assumptions.

Check your work: You distinguish validity from actual premise truth and do not call a false sentence an invalid argument.

Your notes:

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- [ ] 2. Translate without hiding ambiguity

Selected reading: forallx Fall 2023 Part II, especially First steps to symbolization and Ambiguity; LPL 2e chapters 3 and 7 offer comparison.

Make this: Give a symbolization key and two translations of one ambiguous sentence. Describe a case where the readings differ.

Check your work: Every sentence letter has a fixed interpretation and the parentheses expose the intended main connective.

Your notes:

________________________________

- [ ] 3. Search for a falsifying row

Selected reading: forallx Fall 2023 Part III, Complete truth tables and Semantic concepts; Copi 15e chapter 8 provides another method.

Make this: Build a complete table for an argument with three sentence letters. Mark only rows where every premise is true, then inspect the conclusion.

Check your work: You inspect all eight rows or justify a sound shortcut; you do not test only the actual-world assignment.

Your notes:

________________________________

- [ ] 4. Make every proof dependency visible

Selected reading: forallx Fall 2023 Part IV, Basic rules for TFL and Constructing proofs; LPL 2e chapters 5–6 are optional.

Make this: Write a conditional proof, indent its assumption, cite each rule and close the subproof explicitly.

Check your work: No line depends on a discharged assumption unless a rule permits it; a checker’s acceptance is accompanied by an explanation.

Your notes:

________________________________

- [ ] 5. Give quantifiers a domain

Selected reading: forallx Fall 2023 Parts V–VI, Multiple generality and Using interpretations; LPL 2e chapters 9–11 provide model practice.

Make this: Translate the assignment claim below and draw its relation as arrows. Test both quantifier orders on the same finite domain.

Check your work: You can identify a different witness for each participant and explain why that does not supply one shared witness.

Your notes:

________________________________

- [ ] 6. Repair the claim and justify the repair

Selected reading: forallx Fall 2023 Part VII, Proofs with quantifiers and Proofs and semantics; LPL 2e chapters 12–13 give a parallel route.

Make this: Complete the original example, then write the valid weaker conclusion’s derivation in your chosen calculus. Name the restrictions on the temporary witness.

Check your work: Your countermodel and proof answer different claims. The final universal statement contains no witness name that escaped its existential subproof.

Your notes:

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## Worked exercise: Does each participant having a reviewer imply a shared reviewer?

Let P(x) mean participant, R(y) reviewer, T(y) trained, and M(x,y) assigned to. Premises: ∀x(P(x) → ∃y(R(y) ∧ M(x,y))) and ∀y(R(y) → T(y)). Does ∃y(T(y) ∧ ∀x(P(x) → M(x,y))) follow? If not, test the repair ∀x(P(x) → ∃y(T(y) ∧ M(x,y))).

Assumptions: Use classical first-order logic with a nonempty domain and unrestricted quantifiers. P, R and T are predicates, not separate sorts. The scenario is invented. “Assigned to” has only its listed ordered pairs: no uniqueness, symmetry or extra real-world rules are assumed.

Your notes:

________________________________

## Change an assumption: Does uniqueness make the reviewer shared?

Keep the original two premises and nonempty classical domain. First add only “each participant is assigned to at most one reviewer”. Does this force the shared-reviewer conclusion? Then replace that extra assumption by “exactly one reviewer exists in the whole domain”: ∃w(R(w) ∧ ∀y(R(y) → y = w)). Test again, including an empty participant set.

Your notes:

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## Before moving on

- Hand in a keyed translation with its domain, one competing interpretation and a case distinguishing them.
- Produce a complete truth-table test and a rule-cited proof with properly discharged assumptions.
- Submit the countermodel, evaluation of every premise and conclusion, and a valid repaired quantifier argument with the witness restrictions stated.

## Answer and interpretation

### Must I already know algebra?

No advanced mathematics is needed for this route. You must follow a recursive definition, distinguish a symbol from what it names, and keep track of assumptions. Practise reading parentheses before attempting longer proofs.

### Which notation should my notebook use?

Write a one-page key for ¬, ∧, ∨, →, ↔, ∀ and ∃, the domain and every predicate’s arity. Put your chosen proof rules on the same page. A change in glyph is often harmless; a change in rule restrictions is not.

### What counts as checking an invalidity claim?

Specify a nonempty domain, predicate extensions and relations. Evaluate every premise as true and the conclusion as false. One unsuccessful attempt to prove a conclusion establishes nothing; one fully checked countermodel establishes invalidity.

### Does each participant having a reviewer imply a shared reviewer?

1. Take domain \{a,b,c,d\}, P = \{a,b\}, R = T = \{c,d\}, and M = \{(a,c),(b,d)\}. Thus a is assigned to c and b to d; these are the only assignments.
2. The first premise is true: c witnesses the existential for a and d for b. For c and d the antecedent P is false. The second premise is true because every member of R is in T.
3. The proposed conclusion is false. Candidates a and b fail T; b is not assigned to c, meaning M(b,c) is false; a is not assigned to d, meaning M(a,d) is false. No candidate meets the entire conjunction.
4. To derive the repair, take arbitrary u and temporarily assume P(u). The first premise supplies ∃y(R(y) ∧ M(u,y)). Open existential elimination with a fresh witness v and assumption R(v) ∧ M(u,v).
5. Conjunction elimination gives R(v) and M(u,v). Universal elimination on the second premise gives R(v) → T(v); conditional elimination gives T(v). Conjoin T(v) ∧ M(u,v), then existentially introduce ∃y(T(y) ∧ M(u,y)).
6. That last sentence contains no v, so close existential elimination. Discharge P(u) by conditional introduction, then universally generalize u. The fresh v occurs in no undischarged premise outside its subproof, and u is arbitrary rather than a named participant.

The shared-reviewer conclusion is invalid; the four-object interpretation is a countermodel. The repaired conclusion is valid: each participant is assigned to some trained object, possibly a different one for each participant.

Test the result: Recheck all four candidate shared reviewers, not just c. Also test P empty: both premises may be true without any T, so the original existential conclusion cannot be rescued by assuming “everyone” guarantees someone exists. This finite countermodel refutes the original claim; validity of the repair comes from the general derivation, not finite sampling.

### Change an assumption: Does uniqueness make the reviewer shared?

1. Local uniqueness does not repair the argument: the original four-object model already gives each participant exactly one assigned reviewer, but the reviewers differ. Its premises and the added condition are true while the shared conclusion is false.
2. Global existence and uniqueness is stronger. Choose its reviewer w. The original second premise yields T(w). For any participant u, the first premise gives a reviewer v with M(u,v); global uniqueness gives v=w, so substitution yields M(u,w).
3. Thus the same trained w works for every participant. If P is empty, the universal assignment condition is vacuous, but w and T(w) still exist because the added global condition and second premise establish them.

At-most-one reviewer per participant is insufficient. Exactly one reviewer in the whole domain is sufficient with the original premises. Existence matters: merely “at most one reviewer exists” still permits no reviewers and no participants, leaving the existential shared-reviewer conclusion false.

Test the result: Give a one-object interpretation with P=R=T=∅ to refute the existence-free version. State which premise supplies a witness before using it; uniqueness alone cannot create one.

## Common mistakes

### Swapping ∀ and ∃ without a new argument

∀x∃y permits the witness to depend on x. ∃y∀x requires one object to work for every x. Draw the arrows before asserting equivalence.

### Letting a temporary witness escape

A name introduced for existential elimination is local. Derive a conclusion that no longer contains it before closing the subproof.

### Calling a proof checker a translation checker

A legal derivation proves the submitted formal sequent. It does not establish that the sequent faithfully expresses an ambiguous original sentence or that its premises are factual.

## Sources and edition checks

- [Open Logic Project: editions and official free access](https://forallx.openlogicproject.org/)
  Confirms free PDF/HTML access and warns that development files change. Its current print recommendation is Fall 2025; this guide does not substitute that for the stored Fall 2023 edition.

- [Author repository: fixed Fall 2023 source](https://github.com/rzach/forallx-yyc/tree/9372a1824af84d02b910f51de1e7a8ca570ca164)
  Author-controlled F23 tag, resolved to commit 9372a1824af84d02b910f51de1e7a8ca570ca164, supplies the fixed edition label, parts and chapter titles. The current free PDF is a separate revision.

- [Routledge: Introduction to Logic, 15th edition](https://www.routledge.com/Introduction-to-Logic/Copi-Cohen-Rodych/p/book/9781138500860)
  Publisher supplies 15e authors, copyright 2019 and the argument/symbolic/inductive contents. Availability of advertised companion exercises is not promised here.

- [CSLI: Language, Proof and Logic, second edition](https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575866321.shtml)
  Publisher confirms second-edition scope, paperback/electronic ISBN distinction and nontransferable registration after activation.

- [University of Chicago Press: second-edition contents](https://press.uchicago.edu/ucp/books/book/distributed/L/bo12734345.html)
  Official distributor supplies the 19-chapter contents used for the alternate readings; older desktop-platform prose is not treated as current software guidance.

- [Openproof: package access and registration keys](https://www.gradegrinder.net/Store/store.html)
  Current first-party store describes web applications, the paperless package and nontransferable keys. The countermodel and repaired proof are original, independent of the software.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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