Lecture Notes, 6 Oct 2010
Last Update: 6 October 2010
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§1.5: Rules of Inference (cont'd)
- Rules of Inference for Quantifiers:
- Let A(t) represent any proposition or propositional function
one with a multiple-place predicate)that contains at least one
occurrence of the term t.
- Let A(t1 := t2) represent
any proposition or propositional function
in which all free occurrences of term t1
replaced by (i.e., are assigned the value) term t2.
- E.g., if A(x) is: (P(x) → Q(x,y)),
then A(x := c) is: (P(c) → Q(c,y))
- The rules:
- Universal Instantiation (UI)
A(x := c) i.e., A(c)
where c could name any object in the domain
- Universal Generalization (UG)
∀xA(c := x) i.e.,
where c must name an arbitrary object in the
not any particular or special one
(cf. proofs in geometry)
- Existential Instantiation (EI)
A(x := c) i.e., A(c)
where c must name an arbitrary object
in the domain,
not any particular one
i.e., c must be a brand-new name.
- Existential Generalization (EG)
∃xA(c := x) i.e.,
where c names some actual object in the domain
that is known to satisfy A.
- Sample Proof of Validity:
P1: Every student in 191 is a freshman.
P2: Every freshman will pass the midterm.
P3: Fred is a student in 191.
C : ∴ Some student in 191 will pass the midterm.
- Syntax & Semantics of Representation:
191(x) = x is a student
Frosh(x) = x is a freshman
Pass(x) = x will pass the midterm
fred = Fred
P1: ∀x[191(x) → Frosh(x)]
P2: ∀x[Frosh(x) → Pass(x)]
C : ∴ ∃x[191(x) ∧ Pass(x)]
Want to show C (i.e., ∃x[191(x) ∧ Pass(x)]
- Can show C if can find (remember: ∃ is a search)
a 191 student who passes
- We know about one individual: Fred
- ∴ try to show (191(fred) ∧ Pass(fred)):
- to do that, need to show 191(fred) and need to show
- To show 191(fred) is easy: It's P3
- To show Pass(fred):
use UI (x := fred) on P2, then MP if can show
Can show Frosh(fred) from P1 by UI (x := fred)
then MP if can show 191(fred)
But 191(fred) is easy: It's P3 (again)
- Proof of validity:
|1 ||∀x[191(x) → Frosh(x)]|| : ||P1|
|2 ||(191(fred) → Frosh(fred))|| : ||1; UI (x := fred)|
|3 ||191(fred)|| : ||P3|
|4 ||Frosh(fred)|| : ||2,3: MP|
|5 ||∀x[Frosh(x) → Pass(x)]|| : ||P2|
|6 ||(Frosh(fred) → Pass(fred))|| : ||5; UI (x := fred)|
|7 ||Pass(fred)|| : ||4,6; MP|
|8 ||(191(fred) ∧ Pass(fred))|| : ||3,7; Conj|
|9 ||∃x[191(x) ∧ Pass(x)]|| : ||8, EG (fred := x)|
- Understanding & Creating Proofs
- (click on link & read the .ppt, .pdf, or .html file)
- (you might also find the article by Bowling (1977)
- (we'll look at "Proof Strategies" next time)
Text copyright © 2010 by William J. Rapaport
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