Alternative Logical Frameworks

Diagnostic lens 1: Assumptions and meaning

Diagnostic lens 2: What an exact test establishes

Diagnostic lens 3: Constructing a counterexample

Strong Kleene unknown remains unknown under negation. It represents lack of information, not an evenly balanced probability distribution. A responsible solution states a semantic domain, separates definitions from empirical observations, and identifies the particular output that would refute the assertion. The student should carry this distinction forward to the worked cases rather than substitute confidence for a visible check.

Default inferences may be retracted after exceptions arise. That is a chosen policy of defeasible reasoning, not a bug in classical consequence. A responsible solution states a semantic domain, separates definitions from empirical observations, and identifies the particular output that would refute the assertion. The student should carry this distinction forward to the worked cases rather than substitute confidence for a visible check.

Modal necessity depends on a relation among possible worlds. Properties like reflexivity and transitivity should be stated, since they determine which modal axioms are valid. A responsible solution states a semantic domain, separates definitions from empirical observations, and identifies the particular output that would refute the assertion. The student should carry this distinction forward to the worked cases rather than substitute confidence for a visible check.

Begin with the following problem: A diagnostic database has true, false, and unknown statuses. Compare “not established true” with “established false” before using a classical two-valued negation. Write the problem in ordinary language first. Identify which words introduce conjunctions, alternatives, negations, quantifiers, or claims of necessity. Then separate object-level statements from the evaluator's assumptions. State a proposed formalization and one plausible alternative. For each, ask what observation or interpretation would distinguish them. Finally, explain which parts can be checked by logic alone and which require additional factual or policy evidence. A correct formal answer must be precise enough for another student to reproduce without guessing the intended meanings.

1. Central idea and precise vocabulary

Intuitionistic logic emphasizes constructions and proofs. Excluded middle and double-negation elimination are not unrestricted theorems, though ordinary modus ponens remains valid. In this lesson, write down the syntax separately from the semantic interpretation. A notation choice should simplify comparison, not hide an extra premise. When a variable occurs more than once, keep its interpretation fixed within the test. If ordinary-language content is underspecified, state a working definition explicitly and report how a different definition could change the result. This practice matters most when a formula is later embedded in software and readers can no longer see the designer’s intention.

2. Formal object and worked walkthrough

Canonical object: Classical: P∨¬P; intuitionistic: not derivable in general.

Worked example: A constructive existence claim asks for a witness or a procedure to obtain one. A classical reductio can establish existence without presenting a specific witness. Start by naming the starting assumptions and the intended result. If this is an entailment claim, try to build an assignment or structure satisfying every premise while falsifying the conclusion. If this is a transformation, calculate both sides under an informative assignment before claiming equivalence, and give a general reason for all remaining cases. If this is a proof step, cite its legitimate introduction or elimination rule and keep the scope of assumptions visible. If it is an application rule, distinguish necessary from sufficient conditions. The same disciplined workflow should be usable by a reader who rejects the story but accepts the formal assumptions.

3. Analysis of the countercase

Common failure: Do not say that intuitionistic logic rejects every classical result; many inferences coincide. Examine this failure in two deliberately separate ways. Semantically, ask whether an exact interpretation can expose the defect. Syntactically, ask whether the proposed step is a member of the permitted rule set. These answers are not always interchangeable: a specific valuation refutes an entailment, while a missing proof does not generally refute it. Do not count merely repeated numerical trials as a universal demonstration. Write a minimal countercase and then test it against each premise independently. If the attempt fails, explain which premise blocked the counterexample and whether it suggests a valid result or merely a poor choice of model.

Boundary question: Which additional principles turn a constructive proof system into a classical one? The answer should identify the exact condition being changed and the conclusion that may or may not survive. Consider domain restrictions, unavailable evidence, ambiguity of language, and the distinction between a pedagogical finite model and a theorem over all interpretations. Many logic mistakes are failures to track one of these four changes, even when every displayed formula looks plausible.

4. Transfer to a real reasoning workflow

Applied scenario: Compare proof obligations for a verification tool with constructive requirements. Draft a tiny audit trail containing a domain declaration, a glossary of symbols, the input facts, the inference or semantic test, and the final classification. State one observable that the formal tool does not calculate. For a database example, do not silently treat absence of a row as a false statement unless a closed-world convention has been adopted. For a scientific claim, distinguish the fact that a sentence was supplied as a premise from the independent evidence needed to establish it. For a system rule, show the outcome of an edge case and identify who is responsible for resolving the specification if two interpretations conflict.

5. Guided exercises with solution guidance

Exercise A — identification and translation. For the situation “Compare proof obligations for a verification tool with constructive requirements.”, write the assumptions as short declarative propositions or predicates. Include at least two alternative translations if the original wording is ambiguous. Solution guidance: first list the relevant people or system states and any excluded cases; next choose atomic predicates before adding connectives; finally identify what additional evidence would decide between rival translations. There may not be a unique answer until the context is fixed.

Exercise B — exact check. Analyze the displayed object Classical: P∨¬P; intuitionistic: not derivable in general. Show a complete derivation if it is a consequence, an exhaustive valuation table if it has only a few propositional atoms, or a fully specified small structure when a semantic countermodel is more appropriate. Solution guidance: state which kind of claim is being assessed first. Then justify each inference, assignment, or interpretation. A correct table row must satisfy the stipulated meanings of each connective; a correct proof must not cross a discharged-assumption boundary.

Exercise C — controlled perturbation. Consider the error “Do not say that intuitionistic logic rejects every classical result; many inferences coincide.” and the question “Which additional principles turn a constructive proof system into a classical one?”. Change exactly one assumption, connective, quantifier dependency, or rule of inference. Determine whether the original conclusion still holds. Solution guidance: make the changed hypothesis visible; compute the new test rather than assuming the old verdict transfers; explain why the result is sensitive or insensitive to the change. Where the statement becomes untestable from provided information, write undetermined, not true or false by guesswork.

1. Central idea and precise vocabulary

A three-valued interpretation can track T, F and U (unknown). Kleene strong negation maps U to U; conjunction takes the minimum in the F

2. Formal object and worked walkthrough

Canonical object: ¬U=U; T∧U=U; F∧U=F.

Worked example: The database knows a permission check is true but the consent record is unavailable. Under a strong Kleene conjunction, permission and unknown consent evaluate unknown, not true. Start by naming the starting assumptions and the intended result. If this is an entailment claim, try to build an assignment or structure satisfying every premise while falsifying the conclusion. If this is a transformation, calculate both sides under an informative assignment before claiming equivalence, and give a general reason for all remaining cases. If this is a proof step, cite its legitimate introduction or elimination rule and keep the scope of assumptions visible. If it is an application rule, distinguish necessary from sufficient conditions. The same disciplined workflow should be usable by a reader who rejects the story but accepts the formal assumptions.

3. Analysis of the countercase

Common failure: Unknown is not automatically equivalent to a probability of 0.5 or to a known falsehood. Examine this failure in two deliberately separate ways. Semantically, ask whether an exact interpretation can expose the defect. Syntactically, ask whether the proposed step is a member of the permitted rule set. These answers are not always interchangeable: a specific valuation refutes an entailment, while a missing proof does not generally refute it. Do not count merely repeated numerical trials as a universal demonstration. Write a minimal countercase and then test it against each premise independently. If the attempt fails, explain which premise blocked the counterexample and whether it suggests a valid result or merely a poor choice of model.

Boundary question: Would a safety-critical policy deny access on U even though the logical result remains U? The answer should identify the exact condition being changed and the conclusion that may or may not survive. Consider domain restrictions, unavailable evidence, ambiguity of language, and the distinction between a pedagogical finite model and a theorem over all interpretations. Many logic mistakes are failures to track one of these four changes, even when every displayed formula looks plausible.

4. Transfer to a real reasoning workflow

Applied scenario: Mark missing readings explicitly in a rule engine. Draft a tiny audit trail containing a domain declaration, a glossary of symbols, the input facts, the inference or semantic test, and the final classification. State one observable that the formal tool does not calculate. For a database example, do not silently treat absence of a row as a false statement unless a closed-world convention has been adopted. For a scientific claim, distinguish the fact that a sentence was supplied as a premise from the independent evidence needed to establish it. For a system rule, show the outcome of an edge case and identify who is responsible for resolving the specification if two interpretations conflict.

5. Guided exercises with solution guidance

Exercise A — identification and translation. For the situation “Mark missing readings explicitly in a rule engine.”, write the assumptions as short declarative propositions or predicates. Include at least two alternative translations if the original wording is ambiguous. Solution guidance: first list the relevant people or system states and any excluded cases; next choose atomic predicates before adding connectives; finally identify what additional evidence would decide between rival translations. There may not be a unique answer until the context is fixed.

Exercise B — exact check. Analyze the displayed object ¬U=U; T∧U=U; F∧U=F. Show a complete derivation if it is a consequence, an exhaustive valuation table if it has only a few propositional atoms, or a fully specified small structure when a semantic countermodel is more appropriate. Solution guidance: state which kind of claim is being assessed first. Then justify each inference, assignment, or interpretation. A correct table row must satisfy the stipulated meanings of each connective; a correct proof must not cross a discharged-assumption boundary.

Exercise C — controlled perturbation. Consider the error “Unknown is not automatically equivalent to a probability of 0.5 or to a known falsehood.” and the question “Would a safety-critical policy deny access on U even though the logical result remains U?”. Change exactly one assumption, connective, quantifier dependency, or rule of inference. Determine whether the original conclusion still holds. Solution guidance: make the changed hypothesis visible; compute the new test rather than assuming the old verdict transfers; explain why the result is sensitive or insensitive to the change. Where the statement becomes untestable from provided information, write undetermined, not true or false by guesswork.

1. Central idea and precise vocabulary

Nonmonotonic reasoning permits conclusions to be withdrawn when new information appears. Default policies are defeasible, unlike ordinary classical entailment. In this lesson, write down the syntax separately from the semantic interpretation. A notation choice should simplify comparison, not hide an extra premise. When a variable occurs more than once, keep its interpretation fixed within the test. If ordinary-language content is underspecified, state a working definition explicitly and report how a different definition could change the result. This practice matters most when a formula is later embedded in software and readers can no longer see the designer’s intention.

2. Formal object and worked walkthrough

Canonical object: Normally P⇒Q; exception E overrides.

Worked example: Assume registered members normally get forum access. A new suspension record defeats the earlier tentative conclusion. The rule must specify precedence and explainability. Start by naming the starting assumptions and the intended result. If this is an entailment claim, try to build an assignment or structure satisfying every premise while falsifying the conclusion. If this is a transformation, calculate both sides under an informative assignment before claiming equivalence, and give a general reason for all remaining cases. If this is a proof step, cite its legitimate introduction or elimination rule and keep the scope of assumptions visible. If it is an application rule, distinguish necessary from sufficient conditions. The same disciplined workflow should be usable by a reader who rejects the story but accepts the formal assumptions.

3. Analysis of the countercase

Common failure: Do not interpret defeasible arrows as unrestricted material implications. Examine this failure in two deliberately separate ways. Semantically, ask whether an exact interpretation can expose the defect. Syntactically, ask whether the proposed step is a member of the permitted rule set. These answers are not always interchangeable: a specific valuation refutes an entailment, while a missing proof does not generally refute it. Do not count merely repeated numerical trials as a universal demonstration. Write a minimal countercase and then test it against each premise independently. If the attempt fails, explain which premise blocked the counterexample and whether it suggests a valid result or merely a poor choice of model.

Boundary question: Can a new fact reverse an earlier inferred state without making the knowledge base inconsistent? The answer should identify the exact condition being changed and the conclusion that may or may not survive. Consider domain restrictions, unavailable evidence, ambiguity of language, and the distinction between a pedagogical finite model and a theorem over all interpretations. Many logic mistakes are failures to track one of these four changes, even when every displayed formula looks plausible.

4. Transfer to a real reasoning workflow

Applied scenario: Design an exception-aware scheduling policy. Draft a tiny audit trail containing a domain declaration, a glossary of symbols, the input facts, the inference or semantic test, and the final classification. State one observable that the formal tool does not calculate. For a database example, do not silently treat absence of a row as a false statement unless a closed-world convention has been adopted. For a scientific claim, distinguish the fact that a sentence was supplied as a premise from the independent evidence needed to establish it. For a system rule, show the outcome of an edge case and identify who is responsible for resolving the specification if two interpretations conflict.

5. Guided exercises with solution guidance

Exercise A — identification and translation. For the situation “Design an exception-aware scheduling policy.”, write the assumptions as short declarative propositions or predicates. Include at least two alternative translations if the original wording is ambiguous. Solution guidance: first list the relevant people or system states and any excluded cases; next choose atomic predicates before adding connectives; finally identify what additional evidence would decide between rival translations. There may not be a unique answer until the context is fixed.

Exercise B — exact check. Analyze the displayed object Normally P⇒Q; exception E overrides. Show a complete derivation if it is a consequence, an exhaustive valuation table if it has only a few propositional atoms, or a fully specified small structure when a semantic countermodel is more appropriate. Solution guidance: state which kind of claim is being assessed first. Then justify each inference, assignment, or interpretation. A correct table row must satisfy the stipulated meanings of each connective; a correct proof must not cross a discharged-assumption boundary.

Exercise C — controlled perturbation. Consider the error “Do not interpret defeasible arrows as unrestricted material implications.” and the question “Can a new fact reverse an earlier inferred state without making the knowledge base inconsistent?”. Change exactly one assumption, connective, quantifier dependency, or rule of inference. Determine whether the original conclusion still holds. Solution guidance: make the changed hypothesis visible; compute the new test rather than assuming the old verdict transfers; explain why the result is sensitive or insensitive to the change. Where the statement becomes untestable from provided information, write undetermined, not true or false by guesswork.

1. Central idea and precise vocabulary

Modal languages introduce operators such as □P and ◇P, evaluated with respect to accessibility between worlds. Truth at one world need not mean necessity across accessible worlds. In this lesson, write down the syntax separately from the semantic interpretation. A notation choice should simplify comparison, not hide an extra premise. When a variable occurs more than once, keep its interpretation fixed within the test. If ordinary-language content is underspecified, state a working definition explicitly and report how a different definition could change the result. This practice matters most when a formula is later embedded in software and readers can no longer see the designer’s intention.

2. Formal object and worked walkthrough

Canonical object: □P ⇒ P only on reflexive frames (system T).

Worked example: A service may work in the present deployment but fail in an accessible failure state. Hence P at the current world alone does not prove □P. Start by naming the starting assumptions and the intended result. If this is an entailment claim, try to build an assignment or structure satisfying every premise while falsifying the conclusion. If this is a transformation, calculate both sides under an informative assignment before claiming equivalence, and give a general reason for all remaining cases. If this is a proof step, cite its legitimate introduction or elimination rule and keep the scope of assumptions visible. If it is an application rule, distinguish necessary from sufficient conditions. The same disciplined workflow should be usable by a reader who rejects the story but accepts the formal assumptions.

3. Analysis of the countercase

Common failure: Do not infer properties of accessibility relations from informal words such as “possible.” Examine this failure in two deliberately separate ways. Semantically, ask whether an exact interpretation can expose the defect. Syntactically, ask whether the proposed step is a member of the permitted rule set. These answers are not always interchangeable: a specific valuation refutes an entailment, while a missing proof does not generally refute it. Do not count merely repeated numerical trials as a universal demonstration. Write a minimal countercase and then test it against each premise independently. If the attempt fails, explain which premise blocked the counterexample and whether it suggests a valid result or merely a poor choice of model.

Boundary question: Which frame property supports the modal axiom □P→P? The answer should identify the exact condition being changed and the conclusion that may or may not survive. Consider domain restrictions, unavailable evidence, ambiguity of language, and the distinction between a pedagogical finite model and a theorem over all interpretations. Many logic mistakes are failures to track one of these four changes, even when every displayed formula looks plausible.

4. Transfer to a real reasoning workflow

Applied scenario: Build a two-world safety requirement model. Draft a tiny audit trail containing a domain declaration, a glossary of symbols, the input facts, the inference or semantic test, and the final classification. State one observable that the formal tool does not calculate. For a database example, do not silently treat absence of a row as a false statement unless a closed-world convention has been adopted. For a scientific claim, distinguish the fact that a sentence was supplied as a premise from the independent evidence needed to establish it. For a system rule, show the outcome of an edge case and identify who is responsible for resolving the specification if two interpretations conflict.

5. Guided exercises with solution guidance

Exercise A — identification and translation. For the situation “Build a two-world safety requirement model.”, write the assumptions as short declarative propositions or predicates. Include at least two alternative translations if the original wording is ambiguous. Solution guidance: first list the relevant people or system states and any excluded cases; next choose atomic predicates before adding connectives; finally identify what additional evidence would decide between rival translations. There may not be a unique answer until the context is fixed.

Exercise B — exact check. Analyze the displayed object □P ⇒ P only on reflexive frames (system T). Show a complete derivation if it is a consequence, an exhaustive valuation table if it has only a few propositional atoms, or a fully specified small structure when a semantic countermodel is more appropriate. Solution guidance: state which kind of claim is being assessed first. Then justify each inference, assignment, or interpretation. A correct table row must satisfy the stipulated meanings of each connective; a correct proof must not cross a discharged-assumption boundary.

Exercise C — controlled perturbation. Consider the error “Do not infer properties of accessibility relations from informal words such as “possible.”” and the question “Which frame property supports the modal axiom □P→P?”. Change exactly one assumption, connective, quantifier dependency, or rule of inference. Determine whether the original conclusion still holds. Solution guidance: make the changed hypothesis visible; compute the new test rather than assuming the old verdict transfers; explain why the result is sensitive or insensitive to the change. Where the statement becomes untestable from provided information, write undetermined, not true or false by guesswork.

1. Central idea and precise vocabulary

Classical entailment permits weakening and explosion in characteristic contexts; relevant and paraconsistent systems may reject certain classical patterns to track premise use or tolerate contradictions. In this lesson, write down the syntax separately from the semantic interpretation. A notation choice should simplify comparison, not hide an extra premise. When a variable occurs more than once, keep its interpretation fixed within the test. If ordinary-language content is underspecified, state a working definition explicitly and report how a different definition could change the result. This practice matters most when a formula is later embedded in software and readers can no longer see the designer’s intention.

2. Formal object and worked walkthrough

Canonical object: Classically, P,¬P ⊢ Q (explosion).

Worked example: A contradictory database can make classical entailment trivial if treated as unrestricted premises. A practical reasoner may instead localize contradictory evidence or adopt a nonclassical calculus. Start by naming the starting assumptions and the intended result. If this is an entailment claim, try to build an assignment or structure satisfying every premise while falsifying the conclusion. If this is a transformation, calculate both sides under an informative assignment before claiming equivalence, and give a general reason for all remaining cases. If this is a proof step, cite its legitimate introduction or elimination rule and keep the scope of assumptions visible. If it is an application rule, distinguish necessary from sufficient conditions. The same disciplined workflow should be usable by a reader who rejects the story but accepts the formal assumptions.

3. Analysis of the countercase

Common failure: Choosing a logic does not fix poor data provenance or ambiguous domain concepts. Examine this failure in two deliberately separate ways. Semantically, ask whether an exact interpretation can expose the defect. Syntactically, ask whether the proposed step is a member of the permitted rule set. These answers are not always interchangeable: a specific valuation refutes an entailment, while a missing proof does not generally refute it. Do not count merely repeated numerical trials as a universal demonstration. Write a minimal countercase and then test it against each premise independently. If the attempt fails, explain which premise blocked the counterexample and whether it suggests a valid result or merely a poor choice of model.

Boundary question: State what properties are lost and gained when rejecting explosion. The answer should identify the exact condition being changed and the conclusion that may or may not survive. Consider domain restrictions, unavailable evidence, ambiguity of language, and the distinction between a pedagogical finite model and a theorem over all interpretations. Many logic mistakes are failures to track one of these four changes, even when every displayed formula looks plausible.

4. Transfer to a real reasoning workflow

Applied scenario: Compare formal requirements for a proof assistant and a dispute-resolution system. Draft a tiny audit trail containing a domain declaration, a glossary of symbols, the input facts, the inference or semantic test, and the final classification. State one observable that the formal tool does not calculate. For a database example, do not silently treat absence of a row as a false statement unless a closed-world convention has been adopted. For a scientific claim, distinguish the fact that a sentence was supplied as a premise from the independent evidence needed to establish it. For a system rule, show the outcome of an edge case and identify who is responsible for resolving the specification if two interpretations conflict.

5. Guided exercises with solution guidance

Exercise A — identification and translation. For the situation “Compare formal requirements for a proof assistant and a dispute-resolution system.”, write the assumptions as short declarative propositions or predicates. Include at least two alternative translations if the original wording is ambiguous. Solution guidance: first list the relevant people or system states and any excluded cases; next choose atomic predicates before adding connectives; finally identify what additional evidence would decide between rival translations. There may not be a unique answer until the context is fixed.

Exercise B — exact check. Analyze the displayed object Classically, P,¬P ⊢ Q (explosion). Show a complete derivation if it is a consequence, an exhaustive valuation table if it has only a few propositional atoms, or a fully specified small structure when a semantic countermodel is more appropriate. Solution guidance: state which kind of claim is being assessed first. Then justify each inference, assignment, or interpretation. A correct table row must satisfy the stipulated meanings of each connective; a correct proof must not cross a discharged-assumption boundary.

Exercise C — controlled perturbation. Consider the error “Choosing a logic does not fix poor data provenance or ambiguous domain concepts.” and the question “State what properties are lost and gained when rejecting explosion.”. Change exactly one assumption, connective, quantifier dependency, or rule of inference. Determine whether the original conclusion still holds. Solution guidance: make the changed hypothesis visible; compute the new test rather than assuming the old verdict transfers; explain why the result is sensitive or insensitive to the change. Where the statement becomes untestable from provided information, write undetermined, not true or false by guesswork.

Challenge. A diagnostic database has true, false, and unknown statuses. Compare “not established true” with “established false” before using a classical two-valued negation. Develop a short technical note with an exact formal statement, one chosen method of evaluation, and a counterexample search. The note should make at least one original decision-table row or proof trace available for checking. Explain why the result would be insufficient to justify a real-world institutional or engineering decision without premise validation. Attach a paragraph comparing the method used in this unit with a method from an earlier unit, or with a possible method to be learned later.

Scoring rubric. Allocate 20 points to definitions and domain, 25 to formalization, 25 to correct derivation or model checking, 15 to counterexamples and limitations, and 15 to clarity and reproducibility. A solution that asserts validity without a general argument or a complete finite enumeration receives at most partial credit. For a claim of invalidity, one genuine countermodel with all premise evaluations shown is decisive. For an application claim, assess the formal result and the evidence for inputs separately.

Instructor follow-up. Ask each student to change one premise and predict whether the final judgment flips. Comparison of predictions with actual outputs identifies memorization without semantic understanding. Archive the original translation, the revised translation, and the reason for revision; these form a transparent record of how the claim was sharpened.