1 / 13

Inference Rules for Quantified Propositions

Inference Rules for Quantified Propositions. Universal Specification. If a statement of the form  x , P( x ) is true, then P( c ) is true for arbitrary c in the universe of discourse. This can be written:  x , P( x ) ________  P( c ) for all c.

lreginald
Download Presentation

Inference Rules for Quantified Propositions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Inference Rules for Quantified Propositions

  2. Universal Specification • If a statement of the form x, P(x) is true, then P(c) is true for arbitrary c in the universe of discourse. • This can be written: x, P(x) ________  P(c) for all c. • Example: M(x): x is mortal. From x, M(x), we infer that “Socrates is mortal”.

  3. Universal Generalization • If a statement P(c) is true for each element c of the universe, thenx, P(x). • This can be written: P(c) for all c ________  x, P(x).

  4. Example of Universal Generalization Prove: if R is an antisymmetric relation, so is R-1. • Let xRy be an arbitrary element of R where x  y. • yR-1x. (Defn. of inverse relation.) • (y,x)  R. (R is antisymmetric) • (x,y)  R-1. (Step 3 & defn. of inverse relation.) • (x,y), (xR-1y  yR-1x)  x = y. (UG) • R-1 is antisymmetric. (Step 5 & defn. of antisymmetric)

  5. Existential Specification • Ifx, P(x) is true then there is an element c such that P(c) is true. • This can be written: x, P(x) ________  P(c), for some c. • Element c is not arbitrary. • We know only that some c satisfies P. • We do not necessarily know which one (e.g., from a non-constructive proof).

  6. Existential Generalization • If P(c) is true for some c, then x, P(x). • This can be written: P(c), for some c ________  x, P(x).

  7. Protocol To infer from quantified premises: • Properly remove quantifiers. • Use existential or universal specification • Argue with the resulting propositions. • Properly prefix the correct quantifiers. • Use existential & universal generalization

  8. Example arguments • All humans are fallible. • All government agents are human. • Therefore, all government agents are fallible. ______________________ • H(x): x is a human. • F(x): x is fallible. • G(x): x is a government agent.

  9. x, ( H(x)  F(x) ). • x, ( G(x)  H(x) ). _________________ • x, ( G(x)  F(x) ).

  10. Proof: 1. x, ( H(x)  F(x) ) [premise 1] 2. H(c)  F(c) [ step 1 & U.S.] 3. x, ( G(x)  H(x) ). [ premise 2] 4. G(c)  H(c) [ step 3 & U.S.] 5. G(c)  F(c) [steps 2, 4, & transitivity] 6. x, ( G(x)  F(x) ). [step 5 & U.G.]

  11. In English: • All CCS classes are easy. • This is a CCS class. • Therefore, this class is easy. • A more compact representation: • x, C(x)  E(x). • C(CCS CS 2). • Therefore, E(CCS CS 2).

  12. Proof 1. x,C(x) E(x)[premise 1] 2. C(CCS CS 2) E(CCS CS 2) [step 1, U.S.] 3. C(CCS CS 2) [premise 2] 4. E(CCS CS 2) [step 2, 3, & modus ponens]

  13. Characters •     •        •    •        •          • ALL:                      

More Related