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Logic in Predicate Statements: Analysing Relationships & Conclusions

Understand logic in predicates, translating statements, negation, and reasoning to draw sound conclusions in language expressions.

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Logic in Predicate Statements: Analysing Relationships & Conclusions

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  1. 321 Section, Week 2 Natalie Linnell

  2. Extra Credit problem

  3. All lions are fierceSome lions do not drink coffeeSome fierce creatures do not drink coffee • Translate into logic (provide defs for predicates) Ax(P(x)->Q(x)), Ex(P(x)/\-R(x)), Ex(Q(x)/\-R(x)

  4. Discussion Discuss: Why A->? Why not A/\? Why E/\? Why not E->?

  5. All lions are fierceSome lions do not drink coffeeSome fierce creatures do not drink coffee • Negate all the statements Ex(P(x)/\-Q(x)), Ax(-P(x)vR(x)), Ax(-Q(x)vR(x))

  6. All lions are fierceSome lions do not drink coffeeSome fierce creatures do not drink coffee • Argue whether the reasoning to conclude the third statement from the first two is sound Yes; 2existential instationiation;1 universal instantiation

  7. All hummingbirds are richly coloredNo large birds live on honeyBirds that do not live on honey are dull in colorHummingbirds are small • Translate into logic (define any predicates used) Ax(P(x)->S(x)), -Ex(Q(x)/\R(x)), Ax(-R(x)->-S(x)), Ax(P(x)->-Q(x))

  8. All hummingbirds are richly coloredNo large birds live on honeyBirds that do not live on honey are dull in colorHummingbirds are small • Negate all the statements Ex(P(x)/\-S(x)), Ex(Q(x)/\R(x)),Ex(-R(x)/\S(x)), Ex(P(x)/\Q(x))

  9. All hummingbirds are richly coloredNo large birds live on honeyBirds that do not live on honey are dull in colorHummingbirds are small • Argue whether the fourth statement follows from the first 3 Yes; 1, 3 contrapositive and modus ponens, 2(modus ponens if in ->; else -/\ means if one true other false)

  10. Show that xP(x)  xQ(x) is not equivalent to x(P(x)  Q(x)) • Let P(x) and Q(x) be statements from math or the world to illustrate this. P(x)=x is positive, Q(x)=x is negative; any two contradictory statements would do

  11. There is a student in this class who has been in every room of at least one building on campus • Translate into logic (define any predicates) ExEyAz(P(z,y)->Q(x,y))

  12. Every student in this class has been in at least one room of every building on campus • Translate into logic (define any predicates) AxAyEz(P(z,y)/\Q(x, z))

  13. xy (xy = y) • Domain is real numbers – what concept does this capture? Multiplicative identity

  14. xy (((x < 0)  (y<0))  (xy>0)) • Domain is real numbers – what concept does this capture? Product of two negative integers is positive.

  15. Everyone has exactly one best friend • Translate into logic – do not use uniqueness quantifier AxEy(B(x,y)/\Az(x!=y)->-B(x,z)

  16. It is not sunny this afternoon and it is colder than yesterdayWe will go swimming only if it’s sunnyIf we do not go swimming, then we will take a canoe tripIf we take a canoe trip, then we will be home by sunset • Show that “We will be home by sunset” follows 1.5 eg 6

  17. Negate xy P(x, y) v xy Q(x, y)

  18. Negate xy (P(x, y)  Q(x,y))

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