The inverse slip: A → B is not NOT A → NOT B

Q: Why do the inverse and the converse have the same truth table?

Because NOT A -> NOT B is the contrapositive of B -> A: swap and negate both sides of one and you get the other. The contrapositive page shows that operation preserves truth values, so the two columns coincide, true on rows 0, 2 and 3.

A -> B against its inverse NOT A -> NOT B. The equivalence check fails with the counterexample A = FALSE, B = TRUE; the inverse turns out to have the same column as the converse.

The inverse negates both sides of an implication without swapping them. “If you pay, you get access” becomes “if you do not pay, you get no access”, which sounds like the same policy but is a separate rule: free trials exist. The engine's reading of the original is “if A is true, then B is true”. On the four-row table the implication is true on rows 0, 1 and 3, Σm(0, 1, 3); the inverse is true on rows 0, 2 and 3, exactly the converse's column.

The equivalence check is violated with the counterexample A = FALSE, B = TRUE: the original is true, since A is false, and the inverse is false, since NOT A holds and NOT B does not. The table marks two differing rows, 1 and 2, A = FALSE, B = TRUE and A = TRUE, B = FALSE. The checks on the implication are as on its own page: not a tautology, counterexample A = TRUE, B = FALSE; satisfiable, witness A = FALSE, B = FALSE; no simplification law applies; both normal forms are NOT A OR B; 2 NAND gates, 5 NOR gates.

The inverse is the logic behind denying the antecedent, the invalid argument on its own page, and behind the common belief that a rule's negation is its mirror image. In access policies it is dangerous in the lenient direction: the original only promises access to payers, and says nothing about non-payers, so a system that denies everyone else is applying the inverse, not the rule. The thing to watch is that the inverse is the contrapositive of the converse, which is why the two slips have identical columns and are refuted by the same two rows.

implication := A -> B · outputinverse := NOT A -> NOT B · output

Exact checks

Always true (tautology)? No Counterexample: A = T, B = F · Exhaustively checked and solver cross-checked · 3 rows

Always false (contradiction)? No Counterexample: A = F, B = F · Exhaustively checked and solver cross-checked · 1 rows

Can be true? Yes Example assignment: A = F, B = F · Exhaustively checked and solver cross-checked · 1 rows

Are implication and inverse equivalent? No Counterexample: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (4 rows)

The inverse slip: A → B is not NOT A → NOT B
#	A	B	implication	inverse
0	FALSE	FALSE	TRUE	TRUE
1	FALSE	TRUE	TRUE	FALSE
2	TRUE	FALSE	FALSE	TRUE
3	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerimplication is true exactly when the following holds: if A is true, then B is true.
In every other case, implication is false.
About the operators:
- An “if … then” condition only constrains the case where its “if” part holds: when the “if” part is false, the condition is satisfied whatever the “then” part is. It does not say the reverse.

Intermediateimplication is true exactly when the following holds: if A is true, then B is true.
In every other case, implication is false.

Advancedimplication is true exactly when the following holds: if A is true, then B is true [material implication →].
In every other case, implication is false.
Formal notation: A → B

Logic diagram

Simplification, step by step

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

A -> B

Karnaugh map

A \ B

1m0

1m1

0m2

1m3

cells m0, m1NOT A
cells m1, m3B

NOT A OR B · Derived NOT A OR B. Exactly equivalent on every assignment.

Minterms and maxterms

implication(A, B) = Σm(0, 1, 3)implication(A, B) = ΠM(2)

Variable order: A, B · 4 rows

Normal forms

Conjunctive normal form (CNF)

NOT A OR B

1 clauses, 2 literals · complete · Implication elimination ×1

Disjunctive normal form (DNF)

NOT A OR B

2 terms, 2 literals · complete · Implication elimination ×1

NAND-only and NOR-only

NAND-only form

NAND(A, NAND(B, B))

2 gates · verified equivalent on every row

NOR-only form

NOR(NOR(NOR(A, A), B), NOR(NOR(A, A), B))

5 gates · verified equivalent on every row

The same rule as code

Python

# -*- coding: utf-8 -*-
# Generated by Tangibool (tangibool.python/1). Do not edit by hand; regenerate from the source revision.
# Model: not stated
# Source revision: reference
# Interpretation status: not stated
"""Tangibool evaluator (decision model).

Generated deterministically from the validated Boolean model; it evaluates exactly that
model. It does not establish that the model is a complete, legal, or safe statement of a
real-world rule, and evaluating examples does not verify the rule.

Source revision: reference
Interpretation status: not stated
Exporter: 1.0.0 (format tangibool.python/1; engine 1.0.0; semantics tangibool-semantics/1)
Logic digest: 8011103ad8477802d259b49c72c4820a227eb3e9b7313de8b28e9567ca0bda95
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78

Inputs (dict keys, declared order) and generated parameter names:
    'A' -> input_A
    'B' -> input_B

Outputs (declared order) and their functions:
    'implication' -> output_implication(inputs)
    'inverse' -> output_inverse(inputs)

Assumptions (the declared scope, checked by in_scope(inputs)):
    none; every complete assignment is in scope.

Input contract:
    Every required input must be exactly True or False (type(value) is bool). Missing
    inputs, None, strings, integers (including 0 and 1), and keys that are not declared
    inputs raise InputError before anything is evaluated, including inputs that a
    short-circuiting operator would never read. Results are always bool.

Scope:
    Decision functions raise OutOfScopeError when the inputs violate an assumption; an
    out-of-scope assignment is never turned into a False result. Pass check_scope=False
    to evaluate without the scope check, and use in_scope(inputs) to test it separately.

Usage:

    import tangibool_model as model

    model.evaluate({'A': False, 'B': False})
    # -> {'implication': True, 'inverse': True}
    model.output_implication({'A': False, 'B': False})
    # -> True
    model.evaluate({'A': 1, 'B': False})
    # raises InputError: 1 is an int, not a bool
"""

import collections.abc as _abc
from typing import Dict, Mapping, Tuple

__all__ = [
    "MODEL_INFO",
    "IDENTIFIER_MAP",
    "INPUT_NAMES",
    "OUTPUT_NAMES",
    "ASSUMPTIONS",
    "InputError",
    "OutOfScopeError",
    "minterm_index",
    "in_scope",
    "evaluate",
    "output_implication",
    "output_inverse",
]

MODEL_INFO = {
    "format": "tangibool.python/1",
    "mode": "decision",
    "model": None,
    "revision": "reference",
    "interpretation": None,
    "exporter": "1.0.0",
    "engine_version": "1.0.0",
    "semantics_version": "tangibool-semantics/1",
    "logic_digest": "8011103ad8477802d259b49c72c4820a227eb3e9b7313de8b28e9567ca0bda95",
    "assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
    "primary_output": "implication",
    "unused_inputs": (),
    "subcircuits": (),
}

# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
    "v_f65bfa5ba2be": {"kind": "input", "name": "A", "identifier": "input_A"},
    "v_4000e503a585": {"kind": "input", "name": "B", "identifier": "input_B"},
    "f_7a565ef6a2fa": {"kind": "output", "name": "implication", "identifier": "output_implication"},
    "f_55527cf17f1f": {"kind": "output", "name": "inverse", "identifier": "output_inverse"},
}

INPUT_NAMES = ("A", "B")
OUTPUT_NAMES = ("implication", "inverse")
ASSUMPTIONS = ()

_DECLARED_INPUTS = frozenset(INPUT_NAMES)
_SCOPE_INPUTS = ()
_ASSUMPTION_FORMULAS = {entry["name"]: entry["formula"] for entry in ASSUMPTIONS}


class InputError(ValueError):
    """Raised when inputs are missing, undeclared, or not exactly True/False.

    Every required input must satisfy type(value) is bool. Missing inputs, None, strings,
    integers (including 0 and 1), and keys that are not declared inputs are rejected before
    anything is evaluated. A missing input is never treated as False. The individual problems
    are listed in the problems attribute.
    """

    def __init__(self, problems):
        self.problems = tuple(problems)
        super().__init__("Invalid inputs: " + "; ".join(self.problems))


class OutOfScopeError(ValueError):
    """Raised when the inputs violate the model's assumptions (its declared scope).

    Outside its assumptions the rule has no decision: this error is never a False result.
    The names of the violated assumptions are in the violated attribute.
    """

    def __init__(self, violated):
        self.violated = tuple(violated)
        details = ", ".join(name + " (" + _ASSUMPTION_FORMULAS.get(name, "?") + ")" for name in self.violated)
        super().__init__(
            "Out of scope: the inputs violate "
            + ("assumption " if len(self.violated) == 1 else "assumptions ")
            + details
            + ". The rule has no decision outside its assumptions; this is not a False result."
        )


def _short_repr(value: object) -> str:
    text = repr(value)
    return text if len(text) <= 60 else text[:57] + "..."


def _checked_inputs(inputs: Mapping[str, bool], required: Tuple[str, ...], check_scope: bool) -> Dict[str, bool]:
    """Validate inputs strictly before anything is evaluated and return a plain copy."""
    if type(check_scope) is not bool:
        raise TypeError("check_scope must be True or False")
    if not isinstance(inputs, _abc.Mapping):
        raise InputError(["inputs must be a mapping from input names to True/False, got " + type(inputs).__name__])
    problems = []
    for key in inputs:
        if type(key) is not str or key not in _DECLARED_INPUTS:
            declared = ", ".join(repr(name) for name in INPUT_NAMES) or "none"
            problems.append("undeclared input " + _short_repr(key) + " (declared inputs: " + declared + ")")
    needed = tuple(required) + (_SCOPE_INPUTS if check_scope else ())
    for name in needed:
        if name not in inputs:
            problems.append("missing required input " + repr(name))
    for key, value in inputs.items():
        if type(key) is str and key in _DECLARED_INPUTS and type(value) is not bool:
            problems.append(
                "input " + repr(key) + " must be True or False (type bool), got "
                + type(value).__name__ + " " + _short_repr(value)
            )
    if problems:
        raise InputError(list(dict.fromkeys(problems)))
    return dict(inputs.items())


def _require_bool_args(function: str, pairs: Tuple[Tuple[str, object], ...]) -> None:
    problems = [
        "argument " + repr(name) + " of " + function + " must be True or False (type bool), got "
        + type(value).__name__ + " " + _short_repr(value)
        for name, value in pairs
        if type(value) is not bool
    ]
    if problems:
        raise InputError(problems)


def minterm_index(*values: bool) -> int:
    """Row index of an assignment over a declared variable order.

    The first value is the most significant bit, so all-False is row 0. Used for indexed
    minterm (sum) and maxterm (product) forms.
    """
    index = 0
    for value in values:
        if type(value) is not bool:
            raise InputError(["minterm_index() takes True/False values, got " + type(value).__name__ + " " + _short_repr(value)])
        index = index * 2 + (1 if value else 0)
    return index


def _scope_values(values: Dict[str, bool]) -> Tuple[Tuple[str, bool], ...]:
    """Value of every enabled, confirmed assumption for validated inputs, in declared order."""
    return ()


def _require_in_scope(values: Dict[str, bool]) -> None:
    violated = tuple(name for name, holds in _scope_values(values) if not holds)
    if violated:
        raise OutOfScopeError(violated)


def in_scope(inputs: Mapping[str, bool]) -> bool:
    """True when the inputs satisfy every assumption (the model's declared scope).

    Requires the inputs the assumptions read; raises InputError for invalid inputs.
    """
    values = _checked_inputs(inputs, (), True)
    return all(holds for _, holds in _scope_values(values))


# ---- Formula bodies (inputs are validated by the public functions below) ----


def _compute_output_implication(input_A: bool, input_B: bool) -> bool:
    """Decision output 'implication': A -> B"""
    return (not input_A) or input_B


def _compute_output_inverse(input_A: bool, input_B: bool) -> bool:
    """Decision output 'inverse': NOT A -> NOT B"""
    return (not (not input_A)) or (not input_B)


# ---- Public API ----


def output_implication(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
    """Decision output 'implication' (stable ID f_7a565ef6a2fa).

    'implication' := A -> B

    Required inputs: 'A', 'B'.
    Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
    when check_scope is True and the inputs violate an assumption.
    """
    values = _checked_inputs(inputs, ("A", "B"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_implication_value = _compute_output_implication(values["A"], values["B"])
    return output_implication_value


def output_inverse(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
    """Decision output 'inverse' (stable ID f_55527cf17f1f).

    'inverse' := NOT A -> NOT B

    Required inputs: 'A', 'B'.
    Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
    when check_scope is True and the inputs violate an assumption.
    """
    values = _checked_inputs(inputs, ("A", "B"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_inverse_value = _compute_output_inverse(values["A"], values["B"])
    return output_inverse_value


def evaluate(inputs: Mapping[str, bool], *, check_scope: bool = True) -> Dict[str, bool]:
    """Evaluate every output (declared order); each output is computed once.

    Required inputs: 'A', 'B'.
    Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
    when check_scope is True and the inputs violate an assumption.
    """
    values = _checked_inputs(inputs, ("A", "B"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_implication_value = _compute_output_implication(values["A"], values["B"])
    output_inverse_value = _compute_output_inverse(values["A"], values["B"])
    return {
        "implication": output_implication_value,
        "inverse": output_inverse_value,
    }

PostgreSQL

-- Tangibool PostgreSQL export (tangibool.postgresql/1)
-- Target: PostgreSQL 16 or later with typed boolean inputs; tested with PostgreSQL 18.3 (PGlite 0.5.8).
-- Not a claim of compatibility with other SQL dialects.
-- Model: not stated
-- Source revision: reference
-- Interpretation status: not stated
-- Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1)
-- Logic digest: 8011103ad8477802d259b49c72c4820a227eb3e9b7313de8b28e9567ca0bda95
-- Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
-- Mode: decision
--
-- Read-only: this file contains only WITH / VALUES / SELECT statements. No DDL, DML,
-- procedures, credentials, network access, or database changes. The expressions are pure
-- Boolean expressions, so their result does not depend on SQL evaluation order.
--
-- NON-NULL INPUT PRECONDITION: NULL is not a truth value of this rule. Every fragment below
-- is defined only when every input column it reads is a non-NULL boolean. The guard template
-- (section 2) marks any NULL input as 'invalid_input' and any row that violates the
-- assumptions as 'out_of_scope', and fills output columns only for 'ok' rows; a NULL in an
-- output column of another row is an envelope marker, not a third truth value. Never wrap
-- the fragments in COALESCE(..., FALSE) or otherwise choose a meaning for unknown inputs.
-- This evaluates the supplied model; it does not show that the model is a complete, legal,
-- or safe statement of a real-world rule.
--
-- Input columns (stable ID: name -> column):
--   v_f65bfa5ba2be: A -> "in_A"
--   v_4000e503a585: B -> "in_B"
-- Output columns:
--   f_7a565ef6a2fa: implication -> "out_implication"
--   f_55527cf17f1f: inverse -> "out_inverse"
-- Assumptions (scope K; exported separately and never folded into an output):
--   none; every complete input row is in scope.

-- ----------------------------------------------------------------------------------------------
-- 1. Expression fragments (parenthesised; non-NULL boolean inputs required)
-- ----------------------------------------------------------------------------------------------
--
-- [out_implication] output implication (stable ID f_7a565ef6a2fa)
--   implication := A -> B
--   PRECONDITION: "in_A", "in_B" are non-NULL booleans.
--   ((NOT "in_A") OR "in_B")
--
-- [out_inverse] output inverse (stable ID f_55527cf17f1f)
--   inverse := NOT A -> NOT B
--   PRECONDITION: "in_A", "in_B" are non-NULL booleans.
--   ((NOT (NOT "in_A")) OR (NOT "in_B"))
--

-- ----------------------------------------------------------------------------------------------
-- 2. Guard template (read-only SELECT over an inputs list)
-- ----------------------------------------------------------------------------------------------
-- Replace the example rows between the markers with your own rows, or replace the inputs
-- CTE with a SELECT that provides the boolean columns listed in the header. status is
-- 'invalid_input' when any input is NULL, 'out_of_scope' when the assumptions are false, and
-- 'ok' otherwise; output columns are filled only for 'ok' rows (NULL elsewhere is an
-- envelope marker). Assumption columns are shown for rows with complete inputs.
WITH inputs ("row_id", "in_A", "in_B") AS (
  -- tangibool:example-rows:begin
  VALUES
    (1, FALSE, FALSE),
    (2, FALSE, TRUE),
    (3, TRUE, FALSE),
    (4, TRUE, TRUE),
    (5, NULL::boolean, FALSE)
  -- tangibool:example-rows:end
),
checked AS (
  SELECT
    inputs.*,
    CASE
      WHEN "in_A" IS NULL OR "in_B" IS NULL THEN 'invalid_input'
      ELSE 'ok'
    END AS "status"
  FROM inputs
)
SELECT
  "row_id",
  "in_A",
  "in_B",
  "status",
  CASE WHEN "status" = 'ok' THEN ((NOT "in_A") OR "in_B") END AS "out_implication",
  CASE WHEN "status" = 'ok' THEN ((NOT (NOT "in_A")) OR (NOT "in_B")) END AS "out_inverse"
FROM checked
ORDER BY "row_id";

SMT-LIB

; Tangibool SMT-LIB export (tangibool.smtlib/1)
; Profile: SMT-LIB 2.6 Boolean subset (QF_UF): Bool constants, define-fun, let, push/pop, assert, check-sat, echo.
; Tested with Z3 5.1.0 (z3-solver 5.2.0, WebAssembly build).
; Model: not stated
; Source revision: reference
; Interpretation status: not stated
; Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1)
; Logic digest: 8011103ad8477802d259b49c72c4820a227eb3e9b7313de8b28e9567ca0bda95
; Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
; Mode: decision
;
; Reading results: each query block states what sat and unsat mean for its question. A sat
; answer comes with a witness or counterexample (ask for it with (get-model) after sat only);
; unsat settles that scoped query and is solver-checked, not an independently checked proof
; certificate; unknown settles nothing. Blocks are scoped with push/pop, so assertions never
; accumulate across questions. This checks the supplied model, not whether it is a complete,
; legal, or safe statement of a real-world rule.
;
; Symbols (stable ID: name -> symbol):
;   v_f65bfa5ba2be: A -> in_A
;   v_4000e503a585: B -> in_B
;   f_7a565ef6a2fa: implication -> out_implication
;   f_55527cf17f1f: inverse -> out_inverse

(set-option :produce-models true)
(set-info :smt-lib-version 2.6)
(set-logic QF_UF)

; ---- Inputs (declared order)
(declare-const in_A Bool) ; A
(declare-const in_B Bool) ; B

; ---- Outputs (dependencies first)
; output implication (stable ID f_7a565ef6a2fa): A -> B
(define-fun out_implication () Bool (=> in_A in_B))
; output inverse (stable ID f_55527cf17f1f): NOT A -> NOT B
(define-fun out_inverse () Bool (=> (not in_A) (not in_B)))

; ---- Assumptions (the scope K; never folded into an output)
; K: conjunction of the enabled, confirmed assumptions (true when there are none)
(define-fun K () Bool true)

; ---- Definitions only: nothing is asserted and no check is run.
; Add (assert ...) and (check-sat) inside (push 1) ... (pop 1) to ask a question.

LaTeX

implication = A \rightarrow B
inverse = \overline{A} \rightarrow \overline{B}

The common slip

Negating a rule by negating both halves. The negation of A -> B is A AND NOT B, true on the single row the implication rejects; NOT A -> NOT B is a different implication, true on three rows. The counterexample A = FALSE, B = TRUE separates the original from its inverse.

Questions

Why do the inverse and the converse have the same truth table?

Because NOT A -> NOT B is the contrapositive of B -> A: swap and negate both sides of one and you get the other. The contrapositive page shows that operation preserves truth values, so the two columns coincide, true on rows 0, 2 and 3.

Guides

Converse, inverse and contrapositive

Operators and gates

Tools

Worked examples

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.