1,277
edits
Changes
all truth tables inserted...
(a) George will cut down a tree . (George will marry Martha ∨ George will die a bachelor)
(b) (Abe will not become mayor . city will not prosper) ∨ (Abe will become mayor . Ava will become head of chamber of commerce)
(c) (Steve escapes the country ∨ Steve befriends Sally)⊃ Steve will be safe(d) (-Jerry stays in town . -Joan reappears) ⊃ Jan will triumph . Jan will convince Joe(e) (-French object to pact . -Belgians object to pact) . Italian forces withdraw from Spain . attacks on British ships cease ⊃ Italo-British pact will take effect(f) (-Mail-order campaign breaks Dripsweet monopoly . -mail-order campaign restores competition) ⊃ Jones will mortgage his home ∨ (Jones will sell his car . Jones will sell his boat) 5.(a){|! p || q || r || p.-r || p.q || p.q ⊃ r || p.-r ∨ (p.q ⊃ r)|-|T || T || T || F || T || T || T|-|T || T || F || T || T || F || T|-|T || F || T || F || F || T || T|-|T || F || F || T || F || T || T|-|F || T || T || F || F || T || T|-|F || T || F || F || F || T || T|-|F || F || T || F || F || T || T|-|F || F || F || F || F || T || T|}(b){|! p || q || r || s || -p.-q || q.-s || p.-r.s || -p.-q ∨ q.-s ∨ p.-r.s|-| T || T || T || T || F || F || F || F|-| T || T || T || F || F || T || F || T|-| T || T || F || T || F || F || T || T|-| T || T || F || F || F || T || F || T|-| T || F || T || T || F || F || F || F|-| T || F || T || F || F || F || F || F|-| T || F || F || T || F || F || T || T|-| T || F || F || F || F || F || F || F|-| F || T || T || T || F || F || F || F|-| F || T || T || F || F || T || F || T|-| F || T || F || T || F || F || F || F|-| F || T || F || F || F || T || F || T|-| F || F || T || T || T || F || F || T|-| F || F || T || F || T || F || F || T|-| F || F || F || T || T || F || F || T|-| F || F || F || F || T || F || F || T|}(c){|! p || q || r || (p∨q)≡-r || p∨(-q ⊃ r) || (p∨q)≡-r . p∨(-q ⊃ r)|-|T || T || T || F || T || F|-|T || T || F || T || T || T|-|T || F || T || F || T || F|-|T || F || F || T || T || T|-|F || T || T || F || T || F|-|F || T || F || T || T || T|-|F || F || T || T || T || T|-|F || F || F || F || F || F|} 6.(a){|! p || p || p$p|-|T || T || F|-|F || F || F|}(b){|! q || p || q$p|-|T || T || |-|T || F || |-|F || T || |-|F || F || |}(c){|! p || q || p$q || (p$q)$p|-|T || T || |-|T || F || |-|F || T || |-|F || F || |}(d){|! p || q || r || (p$q)$r|-| T || T || T || |-| T || T || F || |-| T || F || T || |-| T || F || F || |-| F || T || T || |-| F || T || F || |-| F || F || T || |-| F || F || F || |}(e){|! p || q || r || (p⊃r) || (p⊃r)$q|-| T || T || T || |-| T || T || F || |-| T || F || T || |-| T || F || F || |-| F || T || T || |-| F || T || F || |-| F || F || T || |-| F || F || F || |} 7. (a){|! p || p || r || (p⊃r) || (p⊃p) . (p⊃r)|-| T || T || T || |-| T || T || F || |-| F || F || T || |-| F || F || F || |}(b){|! p || r || q || (r⊃q) || (p⊃q) || (r⊃q) . (p⊃q)|-| T || T || T || |-| T || T || F || |-| T || F || T || |-| T || F || F || |-| F || T || T || |-| F || T || F || |-| F || F || T || |-| F || F || F || |}(c){|! p || r || q || (p⊃f(p,q,p)) || (q⊃f(p,q,p)) || (p⊃f(p,q,p)) . (q⊃f(p,q,p))|-| T || T || T || |-| T || T || F || |-| T || F || T || |-| T || F || F || |-| F || T || T || |-| F || T || F || |-| F || F || T || |-| F || F || F || |}(d){|! p || r || q || ((p⊃q)⊃r) || ((q⊃p)⊃r) || ((p⊃q)⊃r) . ((q⊃p)⊃r)|-| T || T || T || |-| T || T || F || |-| T || F || T || |-| T || F || F || |-| F || T || T || |-| F || T || F || |-| F || F || T || |-| F || F || F || |} 8.(a) Yes, as the truth-values for s#-t and -s#t are the same.(b){|! s || t || u || (s#t) || (s#t)#u || t#u || s#(t#u)|-| T || T || T || |-| T || T || F || |-| T || F || T || |-| T || F || F || |-| F || T || T || |-| F || T || F || |-| F || F || T || |-| F || F || F || |} 9. (a) Compare the values of the 2nd and 3rd rows; if they are equivalent, then # is commutative. Otherwise, it isn't.(b) Say P and Q are equivalent, i.e. both true. Then, for a connective to be anti-commutative, that means that P#Q is equivalent to -(Q#P). However, this is impossible as P#Q and Q#P are equivalent, so the negation of the latter cannot be equivalent to the former. 10.(a) No. Say p = "God exists", but there is actually no God. Many people believe God exists is true. Say p = "oceans are blue", which they are. Many people believe oceans are blue is true. Thus the truth value of the statement depends on more than just the truth-value of its single component.(b) No. Say p = "the sun's gravitational field exerts an attractive force on all objects", q = "some frogs are green"; then the whole statement is false. However, say p remains the same and q = "the planets orbit around the sun"; then, the whole statement is true. Thus the whole statement can be both true and false if p & q are both true, and it is not truth-functional as the truth-values of p & q alone do not determine its truth value.(c) Yes; this is simply the "not" connective, whose output depends solely on the truth value of the input.
