Aspect, Tense, and Modality

Aspect, tense, and modality Commutative laws i) p ∨ q ⇔ q ∨ p ii) p ∧ q ⇔ q ∧ p Associative laws i) ((p ∨ q) ∨ r) ⇔ (p ∨ (q ∨ r)) ii) ((p ∧ q) ∨ r) ⇔ (p ∧ (q ∧ r)) Distributive laws i) (p ∨ (q ∧ r) ⇔ ((p ∨ q) ∧ (p ∨ r)) ii) (p ∧ (q ∨ r) ⇔ ((p ∧ q) ∨ (p ∧ r)) Complement law i) ¬ ¬ p ⇔ p DeMorgan’s laws i) ¬ (p ∨ q) ⇔ (¬p ∧ ¬q) ii) ¬ (p ∧ q) ⇔ (¬p ∨ ¬q) Conditional laws i) p à q ⇔ ¬p ∨ q ii) p à q ⇔ ¬p à ¬q iii)p à q ⇔ ¬(p ∧ ¬q) Biconditional laws i) p ↔ q ⇔ ((p à q) ∧ (q à p)) ii) p ↔ q ⇔ ((¬p ∧ ¬q) ∨ (p ∧ q)) I. Situation and viewpoint aspect (1) Harry was building a raft. (telic) Harry was gazing at the sea. (atelic) (2) Jim was running a race. Sally was singing a song. vs. Jim was running in the park. vs. Sally was singing (songs). (3) a. John destroyed the cart in/*for an hour. John pushed the cart *in/for an hour. b. John ran to the station in/*for an hour. John ran toward the station *in/for an hour. c. The light flashed once in/*for an hour. The light flashed constantly *in/for an hour. d. Four people died in/*for two days. People died *in/for two days. situation type state activity accomplishment semelfactive achievement (4) (5) (6) (7) (8) static + – – – – durative + + + – – She hates ice cream. The cat is watching the birds. I learned French in high school. Someone sneezed. The cease-fire began at noon yesterday. (telic) (atelic) (telic) (atelic) (telic) (atelic) (telic) (atelic) telic n/a – + – + (state) (activity (unbounded process)) (accomplishment (bounded process)) (semelfactive (instantaneous atelic event)) (achievement (instantaneous change of state)) (9) The air was smelling of jasmine. (10) L’air sentait le jasmin. II. Tense (11) a. b. c. d. Susan went to the movies yesterday. Bill gave Susan a book for her birthday. I didn’t turn off the stove. Frege was a great logician. The set of times T is ordered by the relation ≤, which is: reflexive: ∀t (t ≤ t) anti-symmetric: ∀t 1 ∀t 2 ( t 1 ≤ t2 ∧ t2 ≤ t1 ) à t1 = t2 transitive: ∀t 1 ∀t 2 ∀t 3 ( t 1 ≤ t2 ∧ t2 ≤ t3 ) à t1 ≤ t3 Restrictions on our model of time: connectedness: ∀t 1 ∀t 2 ( t 1 ≤ t2 or t2 ≤ t1 ) density: ∀t 1 ∀t 2 (t 1 < t 2 ) à ∃t 3 (t 1 < t 3 ∧ t3 < t 2 ) 1 no smallest element: no largest element: ∀t 1 ∃t 2 (t 2 < t 1 ) ∀t 1 ∃t 2 (t 1 < t 2 ) Syntax: If φ is a wff, then Pφ is a wff. If φ is a wff, then Fφ is a wff. past and future ‘it was the case that some time in the past, φ’ ‘it will be the case that some time in the future, φ’ Semantics: [[P φ]]M, g, t = 1 iff there exists a t' ∈ T such that t' < t and [[φ]]M, g, t' = 1 [[F φ]]M, g, t = 1 iff there exists a t' ∈ T such that t < t' and [[φ]]M, g, t' = 1 (12) a. b. c. d. e. John is sleeping. John was sleeping. John will be sleeping. John will have been sleeping. (Last week, Mary was told that) John would be sleeping. FP sleep(John) PF sleep(John) (13) John left at 4:00pm. [[ P John leave at 4:00pm ]] M, to, g =1 iff there is a t such that t < t0 and 4:00pm(t) and [[John leave]] M, t, g = 1 [[ P φ After P ϕ ]]M, t, g = 1 iff there is a t' such that t' < t and [[φ]]M, t', g = 1 and there is a t'' such that t'' < t' < t and [[ϕ]]M, t'', g = 1. (14) Susan arrived after John left. (15) Susan arrived yesterday morning. III. Modality and possible worlds (16) You can drive this car. a) epistemic – it is possible for you to drive the car. b) deontic – you have permission to drive the car. (17) a. b. c. d. e. The morning star is the morning star. The morning star is the evening star. The morning star is Venus. The evening star is Venus. Venus is Venus. (intension and reference) (18) a. Susan must be home. b. Susan may (possibly) be home. Syntax: necessity and possibility If φ is a wff, then ❑φ is a wff. ‘it is necessarily the case that φ’ If φ is a wff, then ◊ φ is a wff. ‘it is possibly the case that φ’ Semantics: [[❑φ]]M, w, g, t = 1 iff for all worlds w' ∈ W, [[φ]]M, w', g, t = 1 [[◊ φ ]]M, w, g, t = 1 iff for at least one world w' ∈ W, [[φ]]M, w', g, t = 1 (19)a. Susan must be home. b. Susan may (possibly) be home. c. Susan possibly will not be home. ❑ (home(Susan)) ◊ (home(Susan)) ◊ (F(¬home(Susan))) (20) a. Sally dreamt that she was married to Robert Redford. b. I think that John is smart. (world creating predicates) (21) a. Fred believes that a unicorn ate all his vegetables. b. Fred knows that a unicorn ate all his vegetables. (22) a. If I win the lottery tonight, I will quit my job. b. If I were rich, I would not be doing this right now. (counterfactuals) 2