index 3fd046c..e971fe9 100644 (file)
@@ -150,10 +150,54 @@ in terms of negation and the other connectives.

Exercise: assume that there are two entities in the domain of
discourse, Alice and Bob.  Assume that Alice is a woman, and Bob is a
-man.  Show the following computations:
+man.  Show the following computations, where `i = (w,n,r,g)`:
+
+    1. {i}[∃x.person(x)]
+
+       = {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}[person(x)]
+       = {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}
+
+    2. {i}[∃x.man(x)]
+
+       = {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}[person(x)]
+       = {(w,n+1,r[x->n],g[n->b])}
+
+
+    3. {i}[∃x∃y.person(x) and person(y)]
+
+       = {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}[∃y.person(x) and person(y)]
+       = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])
+         }[person(x) and person(y)]
+       = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])
+         }
+
+    4. {i}[∃x∃y.x=x]
+
+       = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])
+         }[∃x∃y.x=x]
+       = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])
+         }
+
+    5. {i}[∃x∃y.x=y]
+
+       = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])
+         }[∃x∃y.x=y]
+       = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),
+          (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])
+         }

-    1. {}[∃x.person(x)]
-    2. {}[∃x.man(x)]
-    3. {}[∃x∃y.person(x) and person(y)]
-    4. {}[∃x∃y.x=x]
-    5. {}[∃x∃y.x=y]