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Formal logic: choose a text and test a quantifier claim

The useful output of a first logic course is a defensible judgment: a translation with a key, a proof with legal rule applications, or a countermodel that actually satisfies the premises. This route keeps those three products separate. It uses one proof system consistently and asks you to explain why a plausible quantifier swap fails.

Course topic: Logic

Study route and exercise: University Reading List editorial project. How we curate · Sources checked:

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Scope and responsibility

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 · Edition: 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.

Official free reading ↗

Book record and access options →

Introduction to Logic

Irving M. Copi; Carl Cohen; Victor Rodych · Edition: 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.

Book record and access options →

Language, Proof and Logic

Dave Barker-Plummer; Jon Barwise; John Etchemendy · Edition: 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.

Book record and access options →

Check your preparation

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.

Six steps with a concrete output

These are your own study marks, not an assessment. They are not sent to this site.

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

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.

  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.

Answer and interpretation

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?

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.

  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.

Answer and interpretation

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.

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.

Sources and edition checks

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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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