Temporal Connective Truth-Condition Examples #
@cite{heinamaki-1974} @cite{rett-2020} @cite{karttunen-1974}
Concrete scenarios verifying that the Anscombe, Rett, and Karttunen formalizations produce correct truth-value judgments.
Scenarios 1–6 verify @cite{rett-2020} Table 1 for before/after. Scenarios 7–10 verify @cite{heinamaki-1974} Chs. 6, 8, 9 for since, by, till.
Scenarios (ℕ time points) #
| # | ME | EE | Connective | Result |
|---|---|---|---|---|
| 1 | point(1) | stative [5,10] | before | True |
| 2 | point(12) | stative [5,10] | after | True |
| 3 | point(1) | accomplishment [3,8] | before | True |
| 4 | point(12) | accomplishment [3,8] | after | True |
| 5 | point(7) | stative [5,10] | before | False |
| 6 | point(7) | stative [5,10] | after | False |
| 7 | stative[5,10] | point(5) | since | True |
| 8 | point(1) | point(3) | by | True |
| 9 | point(3) | point(3) | by | True |
| 9' | point(3) | point(3) | before | False |
| 10 | stative[5,10] | point(5) | till | True |
ME: "John left" — punctual event at time 1 (early).
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ME: "John left" — punctual event at time 12 (late).
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ME: punctual event at time 7 (inside the stative EE).
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EE: "she was president" — stative, running [5, 10].
Equations
- Phenomena.TemporalConnectives.Examples.ee_stative = { start := 5, finish := 10, valid := Phenomena.TemporalConnectives.Examples.ee_stative._proof_2 }
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EE: "she climbed the mountain" — accomplishment, [3, 8].
Equations
- Phenomena.TemporalConnectives.Examples.ee_accomplishment = { start := 3, finish := 8, valid := Phenomena.TemporalConnectives.Examples.ee_accomplishment._proof_2 }
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"John left₁ before she was president₅₋₁₀" — True under Anscombe. Witness: t = 1 ∈ ME, and 1 < all t' ∈ [5, 10].
"John left₁ before she was president₅₋₁₀" — True under Rett. Witness: t = 1, m = 5 (the GLB of [5, 10] on the ≺ scale).
"John left₁₂ after she was president₅₋₁₀" — True under Anscombe. Witness: t = 12, t' = 5 (any point in EE suffices).
"John left₁₂ after she was president₅₋₁₀" — True under Rett. Witness: t = 12, m = 10 (the LUB of [5, 10] on the ≻ scale).
"John met Mary₁ before she climbed the mountain₃₋₈" — True under Anscombe. Witness: t = 1, and 1 < all t' in [3, 8].
"John met Mary₁ before she climbed the mountain₃₋₈" — True under Rett. Witness: t = 1, m = 3 (the GLB of [3, 8] on the ≺ scale).
"John met Mary₁₂ after she climbed the mountain₃₋₈" — True under Anscombe. Witness: t = 12, t' = 3.
"John met Mary₁₂ after she climbed the mountain₃₋₈" — True under Rett. Witness: t = 12, m = 8 (the LUB of [3, 8] on the ≻ scale).
"John left₇ before she was president₅₋₁₀" — False under Anscombe. Any witness t from ME (t=7) fails: t'=7 ∈ EE and ¬(7 < 7).
"John left₇ after she was president₅₋₁₀" — False under Rett. The max on the ≻ scale is 10, and ¬(7 > 10).
Both theories agree on scenario 1 (ME before stative EE).
Both theories agree on scenario 2 (ME after stative EE).
Both theories agree on scenario 3 (ME before accomplishment EE).
Both theories agree on scenario 4 (ME after accomplishment EE).
INCHOAT of "she was president₅₋₁₀" = {point(5)}. Verifies that inchoative coercion extracts the onset.
COMPLET of "she climbed the mountain₃₋₈" = {point(8)}. Verifies that completive coercion extracts the telos.
ME: stative [5, 10] — "He has been happy."
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EE: punctual event at time 5 — "she arrived."
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ME: punctual event at time 3 — "arrived at 3pm" (for by coincidence case).
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EE: punctual event at time 3 — "3pm" (deadline).
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"He has been happy₅₋₁₀ since she arrived₅" — True under Karttunen.since.
Witness: t = 5 ∈ B, and ∀t' ∈ A (i.e., 5 ≤ t' ≤ 10), 5 ≤ t'.
Since is veridical for its complement in scenario: she arrived.
"He arrived₁ by 3pm₃" — True under Karttunen.by_.
Witness: t = 1 ∈ A, and ∀t' ∈ B (t' = 3), 1 ≤ 3.
"He arrived₃ by 3pm₃" — True under Karttunen.by_.
Witness: t = 3 ∈ A, 3 ≤ 3 (coincidence allowed).
"He arrived₃ before 3pm₃" — FALSE under Anscombe.before.
Need 3 < 3, which fails. Shows by ⊋ before.
By but not before on the same scenario: demonstrates the strict
weakening from before_implies_by.
"He slept₅₋₁₀ till she arrived₅" — True under Karttunen.till.
Witness: t = 5 ∈ both time traces (overlap).
Till agrees with until on the same scenario (definitional).
Not...until scenarios #
Karttunen's identity: punctual until = ¬before (eq. 33). We verify this on concrete time points.
| # | ME | EE | Construction | Result |
|---|---|---|---|---|
| 11 | point(3) | point(5) | not...until | True |
| 12 | point(3) | point(5) | presup + assert | when |
| 13 | point(7) | point(5) | not...until | False |
ME: "The princess woke up" — punctual event at time 3 (early).
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EE: "The prince kissed her" — punctual event at time 5.
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Scenario 11: "The princess woke up₃ before the prince kissed her₅" — TRUE. 3 < 5. This is the base before that gets negated in punctual until.
Scenario 11': "The princess didn't wake up₃ until the prince kissed her₅" — FALSE. NOT(BEFORE) = NOT(wake₃ before kiss₅) = ¬(3 < 5) = False. The princess DID wake up before the kiss, so "not until" is false.
ME: "The princess woke up" — punctual event at time 5 (AT the kiss).
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Scenario 12: "The princess didn't wake up₅ until the prince kissed her₅" — TRUE. NOT(BEFORE) = NOT(wake₅ before kiss₅) = ¬(5 < 5) = True. She woke up at exactly the time of the kiss.
Scenario 12': Presupposition (wake₅ before kiss₅ ∨ wake₅ when kiss₅) is satisfied: the waking happens at the kiss time (when), so the left disjunct is false but the right is true.
Scenario 12'': Presupposition + assertion → when (disjunctive syllogism). This is Karttunen's key result: "not until" + presupposition derives "when".
ME: "The princess woke up" — punctual event at time 7 (AFTER the kiss).
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Scenario 13: "The princess didn't wake up₇ until the prince kissed her₅" — TRUE. NOT(BEFORE) = NOT(wake₇ before kiss₅) = ¬(7 < 5) = True. She woke up after the kiss, so she didn't wake up before it.
Scenario 13': wake₇ is NOT when kiss₅ (no overlap at any time point).
Scenario 13'': The presupposition (before ∨ when) is NOT satisfied, so not...until is vacuously true but pragmatically infelicitous. This models why "She didn't wake up until the prince kissed her" is odd when she woke up AFTER the kiss — the presupposition fails.