Axiom Architecture Slides Logic Lab / Module 01
Axiom Architecture
Building unshakeable arguments from the ground up by defining our starting points.
01 // STARTING POINTS
The Definition Duel
CASE STUDY: THE HOTDOG
Is a hotdog a sandwich?
"We cannot argue about the answer until we agree on what the words mean."
Must a sandwich have two separate slices of bread?
Does the orientation of the bread matter?
Is a taco a sandwich? A sub?
Awaiting Definition
What is an Axiom?
Core Concept
"A statement or proposition which is regarded as being self-evidently true."
Shared Ground
Axioms are the beliefs you and your audience already agree on.
The Foundation
If your foundation (axioms) is shaky, the whole argument collapses.
Geometry vs. ELA
In math, an axiom is \(a = a\). In ELA, an axiom might be "Humans have a right to privacy."
0 = 0
Rigorous Definitions
To build a proof, you must define your terms exclusively and inclusively.
The Test
Does your definition allow for things that shouldn't be there? (Over-inclusive)
"A sandwich is food between bread."
The Fix
Does your definition exclude things that SHOULD be there? (Under-inclusive)
"A sandwich is two distinct slices of bread with a filling between."
"If you wish to converse with me, define your terms."
— Voltaire (attributed)
The Challenge
Pick a controversial topic and identify the 3 axioms your audience must accept before you even speak.
The Goal
Write a definition for your core topic that is neither too broad nor too narrow.
Definition Duel Worksheet Definition Duel
Axiom Architecture // Phase 01
Logic Lab Station
01
Student Name
Date
The "Hotdog" Dilemma
Before we can argue whether a hotdog is a sandwich, we must establish a formal definition. A good definition is inclusive of all sandwiches and exclusive of all non-sandwiches.
Draft a "Sandwich" Definition:
Is it inclusive?
Does this definition cover a Grilled Cheese? A Hoagie? A Wrap?
Is it exclusive?
Does it accidentally include Pizza? A Burger? A Taco?
The Axiom Architect
An Axiom is a starting point—a shared belief that requires no proof because it is accepted as true by both you and your audience.
Sample Argument:
"Schools should start at 10:00 AM to improve student health."
What axioms must be accepted for this argument to even begin?
1
2
3
(Hint: One axiom might be "The primary purpose of school is to benefit the student.")
Project Blueprint: Your Topic
Choose a topic for your logical proof paper. Write your core claim and its foundational axiom.
My Claim (Conclusion)
Foundational Axiom
Axiom Teacher Guide Axiom Architecture
Lesson 01 // Facilitation Guide
v1.0 // LOGIC-ELA
Learning Objectives
Identify foundational axioms (shared beliefs) in an argument.
Construct rigorous, exclusive, and inclusive definitions for core terms.
Analyze audience beliefs to select effective starting points for a proof.
The Hook: The Hotdog Debate (15 mins)
Begin with a "silent debate." Post two posters: "A hotdog is a sandwich" and "A hotdog is NOT a sandwich." Students place a sticky note. Then, announce the twist: **the debate is paused.**
"Before we argue, we must define 'sandwich.' If we cannot agree on the definition, the argument is impossible."
Key Terminology
Axiom
A starting premise so basic it is accepted without further proof.
Inclusive Def.
Covers all possible examples of the term.
Exclusive Def.
Excludes anything that is NOT the term.
Instructional Sequence
01
Direct Instruction: Slides (20 mins)
Walk through the Axiom Architecture Slides. Focus on the concept that arguments in ELA are "proofs" and definitions are the "code" they run on. Use the inclusive/exclusive tests with simple objects (e.g., "What is a chair?").
02
Workshop: Definition Duel (20 mins)
Students complete Section 1 of the Definition Duel Worksheet. Have them trade definitions and try to "break" them by finding an example that fits the definition but isn't a sandwich, or vice versa.
03
Application: Axiom Mining (25 mins)
Transition to their own writing projects. Students must choose a claim and work backward to find the axiom. **Check for understanding:** If a student's axiom is "Smoking is bad," remind them that's an argument, not an axiom. A better axiom might be "The government has a responsibility to protect public health."
Differentiation & Support
Scaffolding (Struggling Learners)
Provide a list of "Axiom Banks"—standard philosophical starting points (Human Rights, Utilitarianism, Safety vs. Liberty) for students to choose from.
Extension (Advanced Learners)
Challenge students to identify "Cultural Axioms"—beliefs that are accepted in the US but might not be axioms in other cultures.
Chain Reaction Slides Logic Lab / Module 02
Chain Reaction
Mastering the sequence of deductive reasoning: ensuring Step C follows inevitably from Steps A and B.
02 // THE SEQUENCE
The Structural Stress Test
THE JENGA PRINCIPLE
In a sound argument, every claim is a block supporting the Conclusion at the top.
The Logic Check:
"If I remove this premise (block), does the conclusion (tower) fall?"
Is your reasoning sequential?
Are there gaps in the chain?
REMOVED PREMISE
AXIOM
The Syllogism Chain
Logic Syntax
PREMISE 1
All humans are mortal.
PREMISE 2
Socrates is human.
CONCLUSION
Socrates is mortal.
The conclusion is "baked into" the premises. It is inevitable.
Blueprint Protocol
1
State the Axiom
Start with the definition or shared belief everyone agrees on.
2
The Connecting Link
Connect a specific case to that general axiom.
3
The Derived Truth
State the conclusion that is now logically unavoidable.
[IF P → Q] + [P] = [Q]
Logic Tower Worksheet Logic Tower
Chain Reaction // Phase 02
Name: __________________________
Date: ________
01. Structural Analysis
Rearrange the following scattered claims into a valid 3-step logical chain (Syllogism).
A. Humans have an inherent right to self-expression.
B. Therefore, internet access must be protected.
C. In the modern era, self-expression requires internet access.
Premise 1:
Premise 2:
Conclusion:
02. Blueprint Construction
Apply your project topic from Lesson 1. Break your primary argument into a 4-step chain. Ensure each step is dependent on the one before it.
1
Axiom (Starting Point)
2
Connecting Link (Specific Case)
3
Development (The "So What?")
4
The Derived Conclusion
Stress Test Protocol
Read your Tower above. If you removed Step 2, could you still prove Step 4? If the answer is "Yes," your chain is broken because Step 2 wasn't actually necessary.
Self-Evaluation:
Logic Tower Answer Key Logic Tower Answer Key
Lesson 02 // Teacher Reference
LOGIC-ELA-REF-02
01. Structural Analysis (The Syllogism Scramble)
Correct Sequence:
Premise 1
A. Humans have an inherent right to self-expression. (The Axiom/General Rule)
Premise 2
C. In the modern era, self-expression requires internet access. (The Connecting Link/Condition)
Conclusion
B. Therefore, internet access must be protected. (The Derived Truth)
Teacher Notes: Blueprint Construction
Common Pitfalls
Circular Logic: Students restate their axiom as the conclusion.Missing Links: Students jump from the axiom to the conclusion without the "connecting link" (Premise 2).Non-Axioms: Students use a controversial opinion as an axiom.
Stress Test Guidance
When students perform the "Stress Test," encourage them to find a counter-example . If Step 2 can be removed and the conclusion still "feels" true, it usually means the student is relying on intuition or emotion rather than derivation .
Modus Ponens Slides Logic Lab / Module 03
The Modus Ponens Engine
Using the "If-Then" structure to force readers into a logical corner.
03 // PERSUASIVE FORMS
Modus Ponens
THE "MODE OF AFFIRMING"
The Formal Structure:
Step 1: If P, then Q.
Step 2: P is true.
Step 3: Therefore, Q must be true.
Why it works in ELA:
If your reader accepts the If-Then rule and you prove the specific condition has been met, they have NO CHOICE but to accept your conclusion.
It creates a "rhetorical trap" for the opposition.
Closing Argument Analysis
"Members of the jury, the law states that if a witness has been caught in a lie, then their entire testimony must be discarded. We have shown you video evidence that Mr. Smith lied about his whereabouts on Tuesday. Therefore, you must discard everything he said on the stand."
The Rule (P → Q)
If witness lies, testimony is out.
The Evidence (P)
We proved the witness lied.
The Conclusion (Q)
The testimony is out.
Logical Syntax Errors
Affirming the Consequent
"If it is raining, the street is wet. The street is wet. Therefore, it is raining."
The Error:
Just because Q is true, doesn't mean P caused it. Maybe a fire hydrant broke!
Correct Modus Ponens
"If it is raining, the street is wet. It is raining. Therefore, the street is wet."
The Truth:
This sequence is 100% valid every time.
If-Then Engine Worksheet If-Then Engine
Modus Ponens // Phase 03
Name: __________________________
Date: ________
01. Constructing the Form
Convert your argument's core reasoning into a strict Modus Ponens structure.
If P (The Rule)
Example: If a community prioritizes safety over privacy, then surveillance must be increased.
And P (The Case)
Example: Our community has explicitly stated that safety is our #1 priority.
Therefore Q
Example: Therefore, our community must increase surveillance.
02. Rhetorical Implementation
Now, take the rigid structure from above and write it as a persuasive paragraph. Use "bridge words" (since, clearly, follows that) to make the logic feel natural but unavoidable.
03. The Defense
How could an opponent try to break your "Engine"? Which part of your Modus Ponens is the most vulnerable?
Attack the Rule (P → Q)
Why might the "If-Then" not always be true?
Attack the Case (P)
How could someone prove P is actually false?
Absurdity Proof Slides Logic Lab / Module 04
The Absurdity Proof
Mastering Reductio ad Absurdum : Proving your point by showing the impossibility of its opposite.
04 // INDIRECT LOGIC
Reductio ad Absurdum
REDUCTION TO THE ABSURD
Sometimes, the best way to prove you are Right is to temporarily pretend you are Wrong.
The Logic Loop:
Assume the Opposite of your claim is true.
Follow that assumption to its Logical Conclusion .
Reveal the Contradiction or Absurdity that results.
Therefore, the opposite must be False .
The Contradiction
If a claim leads to something impossible (e.g., 0=1), the claim itself is impossible.
In-Situ Challenge
Live Demo
"Prove you are currently in a classroom."
ASSUMPTION
I am NOT in a classroom.
DEDUCTION
Then I am outside, in a park, or at home.
THE ABSURDITY
But I am touching a desk, looking at a whiteboard, and surrounded by 30 students. If I were home, these things would be hallucinations. Therefore, the assumption is false.
The "Slippery Slope" Fix
Why use it?
Exposes Extremes: Shows that an opponent's logic, if followed consistently, leads to disaster.Closes the Gap: It removes the "middle ground" by making the alternative look ridiculous.
Example in Debate:
"If we accept the argument that any minor inconvenience justifies breaking a law, then the very concept of 'law' becomes a mere suggestion, and society descends into a state where stoplights are optional and taxes are a choice."
Contradiction Quest Worksheet Contradiction Quest
Reductio ad Absurdum // Phase 04
Name: __________________________
Date: ________
01. Step Zero: The Assumption
To prove your claim by contradiction, you must first articulate its exact opposite.
My Actual Claim
(Example: High school students should have mandatory financial literacy classes.)
The Counter-Assumption
02. Descending to Absurdity
If the Counter-Assumption above were true, what would be the logical results? Follow the chain until you find a result that is impossible, unethical, or ridiculous.
1
Result of Counter-Assumption:
2
Second Order Result:
!
The Contradiction / Absurdity
03. Synthesis
Write a short "Indirect Proof" paragraph using the data above. Start with: "Suppose for a moment that [Counter-Assumption] were true..."
Defense Protocol Slides 01011001 01000101 01010011 00001010 01001110 01001111 00001010 01001100 01001111 01000111 01001001 01000011 01011001 01000101 01010011 00001010 01001110 01001111 00001010 01001100 01001111 01000111 01001001 01000011 01011001 01000101 01010011 00001010 01001110 01001111 00001010 01001100 01001111 01000111 01001001 01000011
Logic Lab / Module 05
The Logical Defense Protocol
Peer-review as "Compiling": Identifying syntax errors, logic breaks, and structural vulnerabilities.
05 // STRESS TEST
The Compiler Mindset
DEBUGGING THE ARGUMENT
In this session, we don't care if we Agree or Disagree with the opinion.
Your Goal:
"Does the argument follow its own internal rules? Can I find a gap in the derivation?"
Syntax Error
Undefined terms or contradictory axioms.
Logic Break
Claim 3 does not follow from Claim 2.
Compiled Successfully
The conclusion is logically unavoidable.
The Stress Test Procedure
1. Probe Axioms
Can I reject the starting point without being irrational?
2. Test Definitions
Can I find a counter-example that fits the definition but shouldn't?
3. Audit Chains
Is there a 'hidden' premise that was never stated?
4. Verify Modus
Did they commit the 'Affirming the Consequent' error?
// PROTOCOL INITIATED // NO EMOTIONAL INTERACTION ALLOWED //
Logic Compiler Checklist Logic Compiler
Error Detection Report // Final Protocol
UNIT-05-STRESS
AUTHOR: ____________________
COMPILER: __________________
"Compiler Instructions: Evaluate the structural integrity of the argument. Do not provide feedback on writing style or personal agreement. Look only for logical breaks."
01. Foundations
Axiom Stability:
Is the axiom actually self-evident or universally accepted by the intended audience?
Definition Boundary:
Is the core term defined clearly? Does it pass the "exclusive/inclusive" tests?
02. Chain of Reasoning
Infallible Sequence:
Does every claim lead directly to the next? Identify any "missing links."
Modus Ponens Protocol:
Is the "If-Then" engine logically valid? (No affirming the consequent).
03. Reductio Analysis
True Contradiction:
Does the assumption of the opposite actually lead to an absurdity, or just a different opinion?
Compiler Debug Log
Identified Logic Breaks:
Synthesis / Final Verdict:
Compile Pass
Compile Fail
Logical Proof Rubric The Logical Proof
Summative Assessment Rubric
Scale
100 PTS
01
Foundational Architecture (20 pts)
Inadequate (0-10) Developing (11-15) Proficient (16-18) Mastery (19-20) Axioms are not identified or are just controversial opinions. Core terms remain undefined. Axioms are identified but shaky. Definitions are present but fail the inclusive/exclusive tests. Axioms are stable and shared. Definitions are rigorous and precise. Axioms are unassailable for the audience. Definitions create a flawless logical field.
02
Structural Derivation (40 pts)
Inadequate (0-20) Developing (21-30) Proficient (31-36) Mastery (37-40) Reasoning is disjointed. Conclusion does not follow from the premises. Multiple "jumps" in logic. A basic chain exists but has gaps. Modus Ponens structure is attempted but contains errors. Arguments follow a clear syllogistic sequence. Conclusion feels unavoidable. Flawless derivation. Every claim is a necessary block for the next. Total structural integrity.
03
Indirect Logic Application (20 pts)
Inadequate (0-10) Developing (11-15) Proficient (16-18) Mastery (19-20) No attempt at Reductio ad Absurdum. Direct claims only. Attempted Reductio, but it leads to a "straw man" rather than a logical contradiction. Effective proof by contradiction showing the invalidity of the counter-position. Sophisticated use of Reductio to expose the fundamental impossibility of the opposition.
04
Rhetorical Integration (20 pts)
Logic is not enough; it must be readable. This criterion evaluates the flow of the proof into a cohesive, persuasive document that respects the conventions of formal ELA writing.
