Truth Propagation

Input: $ARGUMENTS

Overview

Arguments don’t exist in isolation. Each conclusion depends on premises, which depend on other premises, forming a dependency graph. When any assumption’s truth value changes, that change must propagate through all dependent conclusions.

This procedure provides mechanisms for:

1. Computing truth scores from dependencies
2. Detecting when conclusions collapse due to failed assumptions
3. Identifying critical assumptions that would cause cascading failures

Steps

Step 1: Build the Dependency Graph

Start with the conclusion and trace backward:

1. What is the main conclusion/claim?
2. What premises support it? (List ALL, not just the strongest)
3. For each premise: what supports THAT? (Recurse)
4. Continue until you reach:
   • Axioms (accepted without proof)
   • Empirical observations (directly verified)
   • Assumptions (accepted but not verified)

Graph notation:

Conclusion C
├── Premise P1 (AND)
│   ├── Evidence E1 [observed, confidence: 0.9]
│   └── Assumption A1 [assumed, confidence: 0.6]
├── Premise P2 (AND)
│   ├── Premise P2a (OR)
│   │   ├── Evidence E2 [observed, confidence: 0.8]
│   │   └── Evidence E3 [observed, confidence: 0.7]
│   └── Assumption A2 [assumed, confidence: 0.5]
└── Premise P3 (AND)
    └── Axiom X1 [accepted, confidence: 1.0]

Dependency types:

• AND: ALL premises must be true for conclusion to hold
• OR: ANY premise being true is sufficient
• WEIGHTED: Premises contribute proportionally

Step 2: Assign Truth Values

For each leaf node (bottom of the graph):

T = True, F = False, U = Unknown/Uncertain

Step 3: Propagate Upward

Calculate truth values for each non-leaf node:

AND nodes: confidence = minimum of children’s confidences

• If ANY child is False → node is False
• If ALL children are True → node is True (confidence = min)
• If any child is Unknown → node is Unknown (bounded by min)

OR nodes: confidence = maximum of children’s confidences

• If ANY child is True → node is True (confidence = max)
• If ALL children are False → node is False
• If mix → confidence = max of True/Unknown children

WEIGHTED nodes: confidence = weighted average of children’s confidences

Propagate from leaves to root. The root node’s truth value is the conclusion’s truth value.

Step 4: Sensitivity Analysis

For each assumption node, ask: “What if this were false?”

Critical assumptions: Those where flipping to False changes the conclusion. Cascade size: Number of intermediate nodes that change when assumption changes.

Step 5: Identify Vulnerabilities

Single points of failure: AND-dependencies on a single assumption

• If the conclusion requires A AND B AND C, any one failing kills it

Hidden correlations: Assumptions that SEEM independent but aren’t

• A1 and A2 might both depend on the same underlying condition

Confidence gaps: Nodes with low confidence that the conclusion depends on

• The weakest link in an AND-chain determines the chain’s strength

Step 6: What-If Scenarios

Test specific scenarios:

Step 7: Report

TRUTH PROPAGATION:
Conclusion: [main claim]
Current truth value: [T/F/U]
Current confidence: [0-1]

Dependency structure:
[graph visualization]

Critical assumptions (conclusion fails if false):
1. [assumption] — current confidence: [value]
2. [assumption] — current confidence: [value]

Vulnerabilities:
- Single points of failure: [list]
- Weakest links: [lowest confidence nodes in critical paths]
- Hidden correlations: [assumptions that may be linked]

Scenario analysis:
| Scenario | Result | Confidence |
|----------|--------|-----------|
| [scenario] | [T/F/U] | [value] |

Recommendation: [how robust is this conclusion]

When to Use

• Evaluating argument strength
• Testing what-if scenarios (what if assumption X is false?)
• Finding critical weak points in reasoning
• Understanding why a conclusion changed
• → INVOKE: /aex (assumption extraction) to find hidden assumptions
• → INVOKE: /ht (hypothesis testing) to verify uncertain assumptions

Verification

• Dependency graph complete (all premises traced to leaves)
• Truth values assigned to all leaf nodes
• Propagation computed correctly (AND/OR/WEIGHTED rules)
• Sensitivity analysis performed on all assumptions
• Critical assumptions identified
• What-if scenarios tested