Propositional Logic & the Architecture of Proof

Discussion prompt

A proposition is a declarative statement that is definitely true or definitely false - never both, never neither.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Discussion prompt

Explain Two worlds: shape versus meaning by analogy to something with no Foundations of Higher Mathematics in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Discussion prompt

Explain Keep the two questions apart to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Put in order

Put the moves of Worked example: is this string a formula? into the order they have to happen.

1. Atoms are formulas
2. Apply the AND rule
3. Apply the NOT rule
4. Apply the conditional rule
5. Verify by re-parsing top down

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. By the base case, each of p, q, and r is well-formed on its own.

Animation

Shows: Each line of the worked example "is this string a formula?", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The outermost connective is the conditional; its left side is the conjunction and its right side is a negation, each already shown legal. So the string is a well-formed formula.

Sort into buckets

Every line below is part of the definition of bivalence or of unique readability — one or the other, never both. Put each where it belongs.

bivalence

The governing assumption of classical propositional logic; every proposition has exactly one of two truth values, true or false.; This is what makes truth tables finite and total.

unique readability

Every formula is either an atom or has exactly one main connective with uniquely determined immediate sub-formulas.; Parsing is unambiguous - there is only one way to take a formula apart.

b1

The governing assumption of classical propositional logic: every proposition has exactly one of two truth values, true or false. This is what makes truth tables finite and total.

b2

Every formula is either an atom or has exactly one main connective with uniquely determined immediate sub-formulas. Parsing is unambiguous - there is only one way to take a formula apart.

Discussion prompt

Worked example: evaluate under a valuation — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Plug in the atoms

Animation

Shows: Each line of the worked example "evaluate under a valuation", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: A conditional is false only for a true antecedent with a false consequent; here the consequent is true, so the value true is correct. The formula is true under this valuation.

Step through it

Step through Truth tables of not, and, or one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: p is T
2. Step 2: p is T
3. Step 3: p is F
4. Step 4: p is F

Step through it

Step through The material conditional one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: p is T
2. Step 2: p is T
3. Step 3: p is F
4. Step 4: p is F

Predict first

A student writes this, and it looks reasonable:

Claim: if 3 is even, then 3 is odd must be false, because its conclusion is false.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The reader judges the conditional by the consequent alone and ignores the antecedent.

A false antecedent makes the conditional vacuously true, whatever the consequent says.

Why: The reader judges the conditional by the consequent alone and ignores the antecedent. But 3 is even is false, so the promise was never triggered.

Annotate

From Trap: reading a vacuous truth as false — read this one piece at a time. What is each part doing?

On: \( (\mathsf{F} \to \mathsf{F}) \)

• The reader judges the conditional by the consequent alone and ignores the antecedent. But 3 is even is false, so the promise was never triggered.
• A conditional is false only in the true-then-false row. A false antecedent lands us in a true row every time.

Step through it

Step through The biconditional bundles both directions one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: p is T
2. Step 2: p is T
3. Step 3: p is F
4. Step 4: p is F

Match the pairs

Match each term to the definition this lesson gave it — not the one you would guess from the word.

• t1. inductive definition
• t2. valuation
• t3. vacuous truth
• t4. tautology
• t5. contradiction

• d1. A definition with base cases plus construction rules, naming the smallest set closed under those rules - the logic analogue of an algebraic data type in a typed language.
• d2. A function assigning a truth value, true or false, to every atomic proposition. Also called a truth assignment - it is one row of the truth table.
• d3. A conditional with a false antecedent is automatically true, because the promise was never triggered. False antecedent means true conditional, no matter the consequent.
• d4. A formula true under every valuation - the column is all true. Example: p or not p, the law of excluded middle.
• d5. A formula false under every valuation - the column is all false. Example: p and not p.

Why: These are the working definitions of inductive definition, valuation, vacuous truth, tautology, contradiction as Propositional Logic & the Architecture of Proof uses them. Pairing them correctly is the test of whether you could state each one with the slide switched off.

Predict first

Classify the formula below as a tautology, a contradiction, or a contingency.

Commit before you compute: what does Worked example: classify by truth table come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify the one suspicious row

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. An OR is false only when both parts are false.

Animation

Shows: Each line of the worked example "classify by truth table", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: An OR is false only when both parts are false. But a conditional and its converse cannot both be false at once - that would demand p true with q false and simultaneously q true with p false. So the column is all true: the formula is a tautology.

Discussion prompt

Worked example: tautology or merely satisfiable? — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Tabulate it

Step through it

Step through Worked example: tautology or merely satisfiable? one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: p is T
2. Step 2: p is T
3. Step 3: p is F
4. Step 4: p is F

Eliminate the wrong options

Which single valuation makes p implies q false?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. p true, q false
• B. p false, q true
• C. p false, q false
• D. p true, q true

Survives elimination: A

Why: The material conditional is false only when the antecedent is true and the consequent is false - that is the single row p true, q false. Every other combination makes the conditional true.

Check your understanding

Which single valuation makes p implies q false?

• A. p true, q false (correct)
• B. p false, q true
• C. p false, q false
• D. p true, q true

Answer: A

Why: The material conditional is false only when the antecedent is true and the consequent is false - that is the single row p true, q false. Every other combination makes the conditional true.

Why B tempts people

A false antecedent makes the conditional vacuously true, not false.

Why C tempts people

Both false still means a false antecedent, so the conditional is again vacuously true.

Why D tempts people

A true antecedent with a true consequent keeps the promise, so the conditional is true.

Predict first

Worked example: simplify the negation of a conditional is about to be worked. State your hypothesis first: which rule or definition decides this one, and what is the first move it forces? Then watch whether the example agrees with you.

Correct: Rewrite the conditional as a disjunction

Why: Implication-as-or replaces the conditional by not-p or q.

A hypothesis you wrote down is falsifiable; a vague sense of how it will go is not. If the example opens somewhere else, that gap is the thing worth chasing.

Animation

Shows: Each line of the worked example "simplify the negation of a conditional", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The negation of the conditional is true exactly when the conditional is false, namely p true and q false. The conjunction p and not-q is also true only when p true and q false. Same column, so the equivalence is correct.

Predict first

A student writes this, and it looks reasonable:

Claim: if it is a dog, then it is a mammal says the same thing as if it is a mammal, then it is a dog.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Swapping antecedent and consequent gives the converse - a different formula.

A conditional and its converse are not equivalent; the row p false, q true tells them apart.

Why: Swapping antecedent and consequent gives the converse - a different formula. A cat is a mammal but not a dog, so the row p false, q true splits them.

Discussion prompt

The rule this trap just fixed:

A conditional equals its contrapositive, not its converse.

Now break it on purpose. Build a case that violates it and follow the consequences until something visibly fails. Where does the failure first show up — and would you have noticed it if you had not been looking?

Hint: The dangerous rules are the ones whose violation still produces an answer. If yours fails loudly, try to find one that fails quietly.

Predict first

Which formula is logically equivalent to if p then q?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: if not q then not p

Why: A conditional is equivalent to its contrapositive: if p then q shares its truth table with if not q then not p, both false only when p is true and q is false.

Check your understanding

Which formula is logically equivalent to if p then q?

• A. if not q then not p (correct)
• B. if q then p
• C. if not p then not q
• D. if p then not q

Answer: A

Why: A conditional is equivalent to its contrapositive: if p then q shares its truth table with if not q then not p, both false only when p is true and q is false.

Why B tempts people

This is the converse, which swaps antecedent and consequent; it is not equivalent to the original.

Why C tempts people

This is the inverse, which negates both parts without swapping - it equals the converse, not the original.

Why D tempts people

This negates only the consequent, giving the pattern of not(p implies q), the opposite of the conditional.

Put in order

Put the moves of Worked example: check an entailment into the order they have to happen.

1. Tabulate premises and conclusion
2. Keep only premise-satisfying rows
3. Verify the conclusion there

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Lay out both premises and the conclusion across every valuation.

Animation

Shows: Each line of the worked example "check an entailment", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: In that single surviving row the conclusion q is true. Every premise-true row makes q true, so the entailment holds. This is disjunctive syllogism.

Predict first

A student writes this, and it looks reasonable:

Tempting argument: from p implies q and q, conclude p.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: This affirms the consequent. The countermodel p false, q true makes both premises true while the conclusion p is false - so the premises do not guarantee the conclusion.

The valid move is modus ponens: from p implies q and p, conclude q.

Why: This affirms the consequent. The countermodel p false, q true makes both premises true while the conclusion p is false - so the premises do not guarantee the conclusion.

Draw it

Translate both ways. First write the expression above as a sentence with no symbols in it at all. Then cover it, and write your sentence back as notation. If the two versions disagree, the disagreement is the thing to fix.

Check your understanding

Which argument form is INVALID?

• A. from p implies q and q, infer p (correct)
• B. from p implies q and p, infer q
• C. from p implies q and not q, infer not p
• D. from p or q and not p, infer q

Answer: A

Why: Form A affirms the consequent: from a conditional plus its consequent it wrongly infers the antecedent. The countermodel p false, q true makes both premises true but the conclusion false.

Why B tempts people

This is modus ponens, a valid form: affirming the antecedent forces the consequent.

Why C tempts people

This is modus tollens, a valid form: denying the consequent forces denial of the antecedent.

Why D tempts people

This is disjunctive syllogism, a valid form: ruling out one disjunct forces the other.

Predict first

A student writes this, and it looks reasonable:

Reading p or q as excluding the both-true case.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The reader marks the row p true, q true as false, secretly substituting exclusive or for inclusive or.

Inclusive or is true in the both-true row as well.

Why: The reader marks the row p true, q true as false, secretly substituting exclusive or for inclusive or.

Annotate

From Trap: reading 'or' as exclusive — read this one piece at a time. What is each part doing?

On: \( (\mathsf{T} \lor \mathsf{T}) = \mathsf{T} \)

• The reader marks the row p true, q true as false, secretly substituting exclusive or for inclusive or.
• Use XOR only when you truly mean one but not both; the bare or always includes the overlap.

Discussion prompt

Worked example: canonical DNF of XOR — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Select the true rows

Animation

Shows: Each line of the worked example "canonical DNF of XOR", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Row (T,F): first minterm true, so the OR is true - matches. Row (F,T): second minterm true - matches. Rows (T,T) and (F,F): both minterms false, so the OR is false - matches. The DNF reproduces the XOR column exactly.

Predict first

Now build the canonical CNF of the same exclusive or.

Commit before you compute: what does Worked example: canonical CNF of XOR come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify against the table

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Row (T,T): first maxterm is false-or-false, so the AND is false - matches.

Animation

Shows: Each line of the worked example "canonical CNF of XOR", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Row (T,T): first maxterm is false-or-false, so the AND is false - matches. Row (F,F): second maxterm false-or-false - matches. Rows (T,F) and (F,T): both maxterms true, so the AND is true - matches. The CNF reproduces the XOR column exactly.

Predict first

Which formula is the canonical DNF of that truth table?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: (p and not q) or (not p and q)

Why: Canonical DNF ORs the minterms of the true rows: p and not-q for row (T,F), not-p and q for row (F,T). That disjunction reproduces the XOR column.

Check your understanding

Which formula is the canonical DNF of that truth table?

• A. (p and not q) or (not p and q) (correct)
• B. (p and q) or (not p and not q)
• C. (not p or not q) and (p or q)
• D. (p or not q) and (not p or q)

Answer: A

Why: Canonical DNF ORs the minterms of the true rows: p and not-q for row (T,F), not-p and q for row (F,T). That disjunction reproduces the XOR column.

Why B tempts people

This used the false rows instead of the true rows, so it builds XNOR, the negation of XOR.

Why C tempts people

This is the canonical CNF of XOR - a conjunction of maxterms, not a disjunction of minterms.

Why D tempts people

This is a conjunction of OR-clauses (a CNF shape), and it actually encodes the biconditional, again the negation of XOR.

Predict first

Show that NAND alone expresses not, and, or - and hence every Boolean function.

Commit before you compute: what does Worked example: NAND builds everything come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify OR on a row

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Take p false, q false: p NAND p is true, q NAND q is true, and true NAND true is false - matching p or q, which is false there.

Animation

Shows: Each line of the worked example "NAND builds everything", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Take p false, q false: p NAND p is true, q NAND q is true, and true NAND true is false - matching p or q, which is false there. The remaining rows check the same way, so all three connectives are reproduced correctly.

Discussion prompt

Worked example: and/or cannot express not — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: State the invariant

Animation

Shows: Each line of the worked example "and/or cannot express not", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: If some and/or formula equaled not-p, it would be both true (by the invariant) and false (equal to not-p) under the all-true valuation - impossible. Hence no such formula exists: and and or cannot express not.

Predict first

A student writes this, and it looks reasonable:

Claim: since and, or, and the constants let me write a huge variety of formulas, the set {and, or} must be functionally complete.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Completeness means reaching EVERY Boolean function, including negation.

You need a non-monotone ingredient. Adding not restores completeness.

Why: Completeness means reaching EVERY Boolean function, including negation. The monotonicity invariant shows not-p is unreachable, so the set is incomplete no matter how many formulas you build.

Sort into buckets

Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.

Holds up

A proposition is a declarative statement that is definitely true or definitely false - never both, never neither.; Two different worlds are in play, and beginners blur them together.; Ask the two questions on purpose, one at a time.

Breaks

Claim: if 3 is even, then 3 is odd must be false, because its conclusion is false.; Claim: if it is a dog, then it is a mammal says the same thing as if it is a mammal, then it is a dog.

sound

These are stated as this lesson states them — each one survives the edge cases Propositional Logic & the Architecture of Proof puts it through.

flawed

Each of these is lifted from a trap in this deck: reasonable-sounding, and wrong in a way that only shows up once you rely on it.

Check your understanding

Why can {and, or} not express not?

• A. every and/or formula is true when all atoms are true, but not-p is false there (correct)
• B. because and and or each take two arguments while not takes one
• C. because and and or are commutative and not is not
• D. because there are infinitely many Boolean functions to express

Answer: A

Why: The monotonicity invariant: every formula built from atoms with only and/or is true under the all-true valuation, whereas not-p is false there, so no such formula can equal not-p.

Why B tempts people

Arity is irrelevant - you could nest binary connectives freely; the obstruction is monotonicity, not the number of inputs.

Why C tempts people

Commutativity has nothing to do with which functions are expressible; the real invariant is monotonicity.

Why D tempts people

For n inputs there are only finitely many Boolean functions, so cardinality is not the obstacle.

Put in order

Put the moves of Worked example: derive the contrapositive into the order they have to happen.

1. Open the outer conditional
2. Open the inner conditional
3. Set up negation-introduction
4. Fire modus ponens
5. Reach a contradiction
6. Verify by discharging and cross-checking

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Assume p implies q; the remaining goal is to derive not-q implies not-p, which we will later discharge.

Animation

Shows: Each line of the worked example "derive the contrapositive", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Discharge not-q to get not-q implies not-p, then discharge p implies q to get the whole formula, which now depends on no assumptions. Cross-check by table: the formula is false only if p implies q is true while not-q implies not-p is false, but those two are equivalent, so that never occurs - it is a tautology, confirming the derivation.

Discussion prompt

Explain Propositional validity is decidable to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Explain Curry-Howard: proofs are programs by analogy to something with no Foundations of Higher Mathematics in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Comparison matrix

From The logic-to-types dictionary: refill the type column from what you know. The rest of the table is as it appeared.

Discussion prompt

Not every classical tautology comes with a program. Intuitionistic logic keeps only the constructively provable ones.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Eliminate the wrong options

In Curry-Howard, the type corresponding to p or q is?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. a sum (tagged union) type
• B. a product (pair) type
• C. a function type
• D. the empty type

Survives elimination: A

Why: Under Curry-Howard, disjunction corresponds to a sum type: a proof of p-or-q is a tagged value that is either a proof of p or a proof of q, carrying the tag of which side holds.

Check your understanding

In Curry-Howard, the type corresponding to p or q is?

• A. a sum (tagged union) type (correct)
• B. a product (pair) type
• C. a function type
• D. the empty type

Answer: A

Why: Under Curry-Howard, disjunction corresponds to a sum type: a proof of p-or-q is a tagged value that is either a proof of p or a proof of q, carrying the tag of which side holds.

Why B tempts people

A product (pair) type corresponds to conjunction, p and q, not disjunction.

Why C tempts people

A function type corresponds to implication, p implies q, not disjunction.

Why D tempts people

The empty type corresponds to falsehood, the uninhabited proposition, not disjunction.

Discussion prompt

Outside this lesson: where does Propositional Logic & the Architecture of Proof actually turn up? Name one concrete situation — a job, a piece of software someone ships, a decision somebody has to make — and say which part of Pattern: the architecture of proof is doing the work in it.

Hint: Vague is the failure mode here. "Engineering" is not a situation; "deciding whether this build is fast enough to ship" is.

Draw it

One page, no notation unless you need it: draw how these connect — Pattern: the truth-table method · Pattern: table to canonical DNF and CNF · Pattern: the architecture of proof · A proposition has one of two truth values · Two worlds: shape versus meaning. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.