PHIL1005 Lecture Notes - Lecture 10: Existential Quantification, Universal Quantification, Hertz
10.1 PREDICATE LOGIC TIPS AND TRICKS
Choosing which connective to use after a quantifier
- ‘All frogs are green’
o∀x (Fx &Gx)
oUniversal quantifier going with hook
o∀x (Fx⊃Gx)
- For this to be false there needs to be a thing that is F but not G.
- ‘There is a green frog’
ox(Fx Ǝ⊃ Gx)
ox(Fx Ǝ& Gx)
oExistential quantifier going with & (conjunction)
oFor this to be false there has to be nothing that is both F and G.
‘At least’ goes with conjunction
- ‘Charlie loves at least two things’
- e: Charlie
- xLy: x loves y
- x y((cLx & cLy) &(xy)Ǝ Ǝ
‘At most’ goes with hook
- ‘Charlie loves at most two things.’
o∀x ∀y ∀z (((cLx & cLy) & cLz) ⊃ (z = x v z = y)))
- ‘Charlie loves exactly two things’
ox y(((cLx & cLy) &(xy) & Ǝ Ǝ ∀z (cLz ⊃ (z = x v z = y)))
oLower bound Upper bound
Identifying the Antecedent
-∀x ((Fx & Gx) ⊃ Hx)
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- To find a situation in which this conditional claim is false, we look for an object that
satisfies the antecedent but not the consequent.
- In this case the antecedent is a conjunction.
- UNDERLINE THE ANTECEDENT.
-∀x (Fx ⊃ y (xRy & xy))Ǝ
oTo find a situation in which the above is false, we need to find an F (so that it
satisfieds the antecedent) that does not R something distinct from itself (so it
does not satisfy the consequent).
oIn this situation there is such an object so the formula is false.
-∀x (Fx ⊃ y (xy & xRy))Ǝ
oIn this situation there is no F that does not R something other than itself
oSo in this scenario the above formula is
true.
Simplifying Transformations
-∀x ((Fx & Gx) ⊃ Hx)
oA ⊃ B
o
oB v not A
-∀x ((Fx & Gx) ⊃ Hx) ≡∀y (Hy v ~(Fy & Gy))
Exercises
-∃x (Fx & ∀y (Fy ⊃ x=y))
-∀x (Gx ⊃ ~Hx)
- ~∀x (Gx v ~Hx)
-∀x ∃y (Gy & xRy)
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Document Summary
Choosing which connective to use after a quantifier. All frogs are green": x (fx &gx, universal quantifier going with hook, x (fx gx) For this to be false there needs to be a thing that is f but not g. & gx: existential quantifier going with & (conjunction, for this to be false there has to be nothing that is both f and g. Charlie loves at least two things" e: charlie xly: x loves y x y((clx & cly) &(x y) Charlie loves at most two things. ": x y z (((clx & cly) & clz) (z = x v z = y))) Charlie loves exactly two things" o x y(((clx & cly) &(x y) & Z (clz (z = x v z = y)): lower bound. To find a situation in which this conditional claim is false, we look for an object that satisfies the antecedent but not the consequent. In this case the antecedent is a conjunction.
