The Clause 5 Loop
Atlas

17The tests

The proof

Every link checked by a solver

  • Logic by machine: the Z3 solver, 88 checks.
  • Each conclusion follows. Each fails when one premise is dropped. Each rival is tested for consistency with the words.
  • All 88 come back as they should.

A logic program checked every step: 88 checks, each with the expected result.

What the grades meannine grades, from the firmest to the weakest
  1. statedThe source says it in its own words, and the study quotes them.
  2. it followsLogic alone gets there from what is stated, and a computer solver checks the step.
  3. strong supportThe evidence for it far outweighs the evidence against it.
  4. moderate supportMore of the evidence points for it than against it, but real evidence points the other way.
  5. one readingThe words allow it but do not require it; another reading fits them as well.
  6. left openThe evidence does not settle it either way.
  7. weak supportSome evidence points for it, but more points against it.
  8. not shownIt is claimed, but the record does not establish it.
  9. ruled outThe words or the logic exclude it.

A chain is only as good as its weakest link: a conclusion never gets a better grade than the weakest step it needs.

Read in full1,220 words · the study as written

Every link in the study is checked by a solver, Z3, and so is every claim that a rival construction exists. The solver runs three kinds of check. A conclusion follows when its premises, joined to the conclusion’s denial, cannot all be true. A control drops one premise at a time and confirms the conclusion then fails, so no premise listed is idle. A rival is consistent with the words when the solver finds them true together, and that is how the claim of completeness is tested. All 88 checks give the result expected of them.

What each conclusion needs

Conclusion What it needs (routes joined by or)
AC1 (A4, AU4)
AC2 (AC1, AU5, AU7, A3)
AC3 (A1, A2, AU2)
AC4 (AU9, AU10, W-orig, F-orig)
AC4-refuted (N4c, W-orig, F-orig)
AC5 (AC3, AU11, A8)
BC1 (B6, B7)
BC2 (BU5, F-cpi)
BC3 (BC2, BU6, BU1, B8)
binary (A4, AU4, AU5, AU7) or (AU12)
BC4 (BU2, F-debt) or (BU3, BU8)
BC5 (BC3) or (BC4) or (BU7)

The checks, one by one

Check Result Expected
AL1 route 1: the conclusion follows from A4, AU4 true true
AL1 route 1: control, without A4 false false
AL1 route 1: control, without AU4 false false
AL1 A5 (a purely nominal increase is no accession) follows from AU4 and the definition true true
AL1 with the rival measure (an accession counted in dollars), the same part is income true true
AL1 A4 alone, with the definition, decides neither way (not income) false false
AL1 A4 alone, with the definition, decides neither way (income) false false
AL2 route 1: the conclusion follows from AC1, AU5, AU7, A3 true true
AL2 route 1: control, without AC1 false false
AL2 route 1: control, without AU5 false false
AL2 route 1: control, without AU7 false false
AL2 route 1: control, without A3 false false
AL2 K-A4 (an excise on the sale, valid without the Amendment) is consistent with AC1 and AU5 when AU7 is not entered true true
AL2 K-A4 contradicts AC1 and AU5 once AU7 is entered false false
AL2 Q5, a mixed gain: with AC1, the real part r taxable and the inflation part not (K-A5) true true
AL2 Q5: without AU5, “deficiency = zero” on the whole gain does not follow from AC1 false false
AL3 route 1: the conclusion follows from A1, A2, AU2 true true
AL3 route 1: control, without A1 false false
AL3 route 1: control, without A2 false false
AL3 route 1: control, without AU2 false false
AL3 without AU2, the conclusion stands with an actor that is not the House acting by a Bill (stated by 12 U.S.C. § 263) true true
AL4 route 1: the conclusion follows from AU9, AU10, W-orig, F-orig true true
AL4 route 1: control, without AU9 false false
AL4 route 1: control, without AU10 false false
AL4 route 1: control, without W-orig false false
AL4 route 1: control, without F-orig false false
AN4 route 1: the conclusion follows from N4c, W-orig, F-orig true true
AN4 route 1: control, without N4c false false
AN4 route 1: control, without W-orig false false
AN4 route 1: control, without F-orig false false
AN4 AU10 contradicts the canon’s definition of surplusage, given the clause, the facts and AU9 false false
AN4 control: without AU10, the canon, the clause, the facts and AU9 are consistent true true
AN4 AU9 contradicts the 1828 meaning of revenue (the produce of the tax that levies it), given who levies false false
AN4 control: without the 1828 meaning, AU9 is consistent with who levies true true
AN4 the clause as written does not entail AU8 (that whatever raises revenue must be a House Bill) false false
AN4 with AU8 entered, the facts and AU9 contradict it: AU8 would make the monetary action a violation false false
AN4 the third construction (the law raises the increment) is consistent with the clause as written true true
AL5 route 1: the conclusion follows from AC3, AU11, A8 true true
AL5 route 1: control, without AC3 false false
AL5 route 1: control, without AU11 false false
AL5 route 1: control, without A8 false false
AL5 without AU11, a statute can authorize it (the reading against: Hampton, Consumers’ Research) true true
AL6 route 1: the conclusion follows from A4, AU4, AU5, AU7 true true
AL6 route 1: control, without A4 false false
AL6 route 1: control, without AU4 false false
AL6 route 1: control, without AU5 false false
AL6 route 1: control, without AU7 false false
AL6 route 2: the conclusion follows from AU12 true true
AL6 route 2: control, without AU12 false false
AL6 A9 tested: the third construction K-A3 is consistent with the words entered as written (A4 granted) true true
AL6 with AU4, AU5 and AU7 entered, K-A3 is excluded false false
AL6 with AU12 entered, K-A3 is excluded false false
AL6 control: without AU4 (and without AU12), K-A3 survives true true
AL6 control: without AU5 (and without AU12), K-A3 survives true true
AL6 control: without AU7 (and without AU12), K-A3 survives true true
AL6 the words permit K-A1 (enforce) true true
AL6 the words permit K-A2 (double extraction) true true
AL6 the words permit K-A3 (the unit of account) true true
AL6 the words permit K-A5 (the real part only: the whole gain is not the inflation part) true true
BL1 route 1: the conclusion follows from B6, B7 true true
BL1 route 1: control, without B6 false false
BL1 route 1: control, without B7 false false
BL2 route 1: the conclusion follows from BU5, F-cpi true true
BL2 route 1: control, without BU5 false false
BL2 route 1: control, without F-cpi false false
BL3 route 1: the conclusion follows from BC2, BU6, BU1, B8 true true
BL3 route 1: control, without BC2 false false
BL3 route 1: control, without BU6 false false
BL3 route 1: control, without BU1 false false
BL3 route 1: control, without B8 false false
BL3 K-B2 (inflation is not a tax) is consistent with the clause, B8 and the transfer true true
BL4 route 1: the conclusion follows from BU2, F-debt true true
BL4 route 1: control, without BU2 false false
BL4 route 1: control, without F-debt false false
BL4 route 2: the conclusion follows from BU3, BU8 true true
BL4 route 2: control, without BU3 false false
BL4 route 2: control, without BU8 false false
BL4 BU2 contradicts the borrowing clause (“To borrow Money”), given that the debt is met by borrowing false false
BL4 control: without BU2, the borrowing clause and the facts are consistent true true
BL5 route 1: the conclusion follows from BC3 true true
BL5 route 1: control, without BC3 false false
BL5 route 2: the conclusion follows from BC4 true true
BL5 route 2: control, without BC4 false false
BL5 route 3: the conclusion follows from BU7 true true
BL5 route 3: control, without BU7 false false
BL6 the words (borrowing granted; no new Bill; the transfer; the House’s consent to the framework) permit K-B2: inflation not a tax, no bypass true true
BL6 with K-B2 the practice is authorized true true
BL6 with BU6 and BU1 entered, K-B2 is excluded false false