PHILOSOPHY 330 ADVANCED SYMBOLIC LOGIC

H. Hamner HIll

Southeast Missouri State University

HOMEWORK ASSIGNMENTS

All homework assignments due at the start of class on the Thursday of the Unit of the assignment.

Unit # Assignment

#1 p. 20, #2

p. 32, #3 (1st part); also for (p v q)(** **
↔
(r v p));

#4;

#5 (1st 2 parts);

symbolize as a logical schema in dot notation:

If if if p then q then p then p.

#2 p. 40, #1;

p. 45, #1 (except lower left part);

#3 (optional);

Give a truth-value analysis of:

-(pq):q→r.↔.** **
** **v

#3 p. 52, #2;

#5;

#7 (schemata are on p. 307).

Use fell swoop wherever you can on these problems.

#4 p. 59, #1;

#2;

p. 67, #2;

#4 (3rd pair only);

#8 (concisely, but cogently).

#5 p. 74, #1 (2nd and 3rd schemata only), and apply the same instructions to:

-(pq)→:**
**** **v .↔
;

Develop into developed alternational normal form:

pv r;

p. 74, #3 (concisely).

Investigate for redundant clauses and literals:

pr v qr v ps v ps v s;

Transform into a conjunctional normal schema ( v. p. 83):

v r v pq (dualize, distribute, simplify, dualize)

In Chapter 13's axiom system give a proof of:

→ .→.q→p:→:p→.q→p

#6 P. 113, #1 (except second part), #3.

#7 p. 141, #2 (for the last 3 parts of ex. #1 in Ch. 17 and for ex. # 4 of Ch. 19, giving schemata in all cases)

p. 141, #3

#8 p. 147, #1 for all 4 schemata (but limit universe to just {a,b,c})

p. 154, #4, #5 (for # 4 only)

p. 154, #2; also put 'Fx ↔.∀xGx q∃xHx' into prenex form

#9 Use an appropriate argument to determine the validity or nonvalidity of

'∃y(Fy ↔∀xFx)' and '∀y(Fy ↔∃xFx)'.

Give schemata for the premises of p. 159 #4 (v.p. 312).

Give a __falsifying__ interpretation
for each of the following in as small a non-empty universe as possible:

∃x(Fx↔-Gx).∀x-Fx.→ ∀xGx;

∀x(Fx → Gx).∃xFx.∃x(-Gx.Hx).∃y(-Hy.-Fy). → ∀x(Gx → .Fxq-Hx);

-∃xHx v ∀xHx v ∃x(Fx → -Gx);

Give a __satisfying__ interpretation
for each of the following in as small a non-empty universe as possible:

∀x∀y∀z(Fxy.Fyz.→Fxz).∃x∀yFxy.∃x∀yFyx.∃x-Fxx;

∀x∀y(Fxy→-Fyx).∀x∃yFxy.

#10 pp. 173-74, #3, #5, #7 (last three parts only);

supplement #5 by treating the following schemata as it instructs:

∃x∃y(Fxy.-Fyx), ∀xFxx → ∀x∃yFxy

∃x∀yFxy, ∀x(Fxx → ∃yFxy)

Give quantifiational schemata for the following sentences:

(a) Every logician quotes a logician.

(b) Not all logicians quote themselves.

(c) Some, but not all, dodos are logicians.

(d) No dodo who quotes every dodo is a logician.

(e) A dodo is a logician if he or she is quoted by a logician.

(f) Any dodo who quotes all dodos who quote him (her) quotes himself (herself).

#11 Using the method of pure existentials, show the validity of ∀u∀v∃x∃y(Fxu.Fvy.→.Fxx q Fyy). Also, by the same method, determine whether ∀y(∃x∀zFzxy → ∃zFzyy). If non valid, give a falsifying interpretation.

#12 Using the Main Method, do problems 5, 6, and 7 from p. 189. Also, by the same method, show that ∀y∃x(Fx → Gy) and ∀x∃y(Fx.-Gy) are inconsistent.

By either the method of pure existentials or the main method, show that ∀xFxy→∃xGx and ∀y-Gy together imply ∃z-Fzy. Also, do problem #4 p. 203 (symbolized on p. 315, but try to nail it down yourself), and show, by the main method, that ∀y∃z∀x(Fx.Gxy.→.Fy q Gzy) is valid.

#13 (1) Give an interpretation that falsifies the schema: ∃x∀y∃z(Fyz → Fxz. → .Fxx → Fyx).

(2) Show that "Everybody loves a lover," on its most likely construal, together with "Bob loves Carol" as a further premise implies that "Ted loves Alice." You may use either the main method or the method of pure existentials (Boolos).

(3) Use the main method, plus rules of passage (9) and (10) but no others, to show that ∃x∃u∀y∀v(Gy → Fx.Fu → Gv) implies ∀y∃x(Fx ↔ Gy).

(4) Show that "Everybody loves my baby but my baby loves nobody but me" implies "I am my baby." To simplify things a little, you may symbolize the conclusion as if it read "My baby is me." (Cartwright)

(5) Use the main method, plus the axioms of identity, to show that ∃xFx and ∃y-Fy imply ∀x∃y(x≠y).

(6) Show that "There is a dodo who admires every dodo," "Dodos all admire whatever admire them," and "A dodo who admires any dodo admires every dodo," imply "Each dodo admires every dodo." (Koopman)

#14 (1) Show that "I like anyone who laughs at himself but like no one who laughs at all his friends" implies "Anyone who laughs at himself has a friend other than himself. (Solomon)

(2) Use the main method, plus the axioms of identity, to show that ∀x∀y(Dx.Dy. → x=y) and ∃x∀y(Dy → y=x) imply each other.

(3) Determine whether "Any fish can swim faster than any smaller fish" implies "If there is a largest fish, then there is a fastest fish."

#15 (1) Express the following schema in the clearest English you can, letting 'Qxy' = 'x bears Q to y' and 'Rxy'='x bears R to y'.

(a) ∀x∀y(Qxy ↔ .∀z(Rzx → Rzy))

(2) Using the predicates 'Q' and 'R' symbolize the following:

(i) If there is a thing to which everything bears R and a thing to which everything bears Q then everything bears R to everything.

(ii) Nothing is borne R by everything unless something is borne Q by everything.

(iii) Everything bears Q to something or other.

(iv) Anything that is borne Q is borne R.

(v) Nothing that is not borne R is borne Q by anything other than itself.

(3) Determine which of (i) –(v) are implied by (a). No justification is necessary.

(4) Show, by an appropriate formal argument, that (a) implies one of (i)-(v).

(5) Show that (a) implies ∀x∀y∀z(Qxy.Qyz. → Qxz).