Chapter 1 — Mathematical reasoning
Lesson map
Lesson 01 published. Map, example, EX001.
flowchart TD
N0["Learning outcomes"]
N1["1.1 The case and why reasoning matters"]
N2["1.2 Definitions and implications"]
N3["1.3 Counterexamples"]
N4["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4
N4 --> EX["Worked example"]
EX --> Q["Exercise EX001"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["1.1 The case and why reasoning matters"]
N2["1.2 Definitions and implications"]
N3["1.3 Counterexamples"]
N4["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX001"]
[Learning outcomes]
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v
[1.1 The case and why reasoning matters]
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v
[1.2 Definitions and implications]
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v
[1.3 Counterexamples]
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v
[Synthesis]
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v
[Worked example] --> [Exercise EX001]
Learning outcomes
You will (1) reason with definitions and implications; (2) check claims with counterexamples; (3) express token rules precisely.
1.1 The case and why reasoning matters
Sofia Lindqvist is a first-year student at the fictional Harrowgate University, supervised by Dr Kwame Asante. Across four years she will help the fictional Lantern Hill Food Cooperative, about 900 members, run a voucher token on a small permissioned Ethereum-compatible test ledger. Members earn vouchers for volunteer shifts and spend them at the cooperative shop. Every person, address and number in this book is synthetic. Mathematical reasoning comes first because security claims about tokens are claims about rules, and a rule stated loosely cannot be checked. Sofia learns to write each rule so that anyone can test whether it holds.
1.2 Definitions and implications
A definition fixes exactly what a word means; an implication says that if one statement holds, another must follow. Sofia writes the cooperative's first rule as an implication: if a voucher is issued, then a recorded volunteer shift exists for that member. She notes the converse is a different claim: a shift does not guarantee a voucher, because a coordinator may not yet have approved it. Confusing a statement with its converse is a common reasoning error in security reviews. Writing both directions separately shows which one the system must enforce and which one is merely usual practice in the shop.
1.3 Counterexamples
To show a general claim is false, one counterexample is enough. A classmate claims that total vouchers in circulation can never decrease. Sofia finds a counterexample at once: when a member spends vouchers at the shop, they are redeemed and removed, so circulation falls. The corrected claim is that circulation equals vouchers issued minus vouchers redeemed. Searching for counterexamples is a habit that security work depends on, because attackers and accidents both find the cases that designers forgot. Sofia keeps a page in her notebook for each rule, listing the counterexamples she tried and whether any of them broke it.
Worked example
Sofia states issuance as an implication from recorded shifts, separates it from its converse, and corrects a classmate's claim with one redemption counterexample.
Synthetic case. Sofia Lindqvist, Dr Kwame Asante, Harrowgate University, Lantern Hill Food Cooperative and all data are synthetic.
Errors and limits
Treating a rule and its converse as the same claim leads to controls that enforce the wrong direction.
Exercises
- EX001 — Cooperative scale. How many members does Lantern Hill have? Rubric: 3 = about 900;; 2 = 900;; 1 = 9,000;; 0 = none.
- EX002 — Converse error. Why is the converse of the issuance rule a different claim? Rubric: 3 = a shift may exist before a coordinator approves a voucher;; 2 = not automatic;; 1 = same claim;; 0 = none.
- EX003 — Corrected circulation claim. What corrected claim replaces "circulation never decreases"? Rubric: 3 = circulation equals issued minus redeemed;; 2 = redemptions reduce it;; 1 = never decreases;; 0 = none.
Synthesis
Precise definitions and counterexamples turn vague token rules into claims that can be checked.