Predicate Logic Hurley 8.1 Translation

Capital Letters for Predicates English Predicate: ___ is a Rabbit ___ is gigantic ___ is a doctor ___ is helpless Symbolic Predicate: R ___ G ___ D ___ H ___

Singular Statements Statement: Socrates is mortal Tokyo is populous The Sun-Times is a newspaper King Lear is not a fairy tale Berloiz was not a German Symbolic Translation: Sm Pt Ns ~Fk ~Gb

Compound Statements Symbolic Translation: Bp → Ta Di v Li Ew → Cw (Cn v Ds) → Pg Ri = (Sc • Sm) Statement: If Paris is beautiful, then Andre told the truth. Irene is either a doctor or a lawyer. Sentor Wilkins will be elected only if he campaigns. General Motors will prosper if either Nissan is crippled by a strike or Subaru declares bankruptcy. Indianapolis gets rain if and only if Chicago and Milwaukee get snow.

Universal Statements We use what is called the ‘Boolean’ interpretation of universal statements. Statement Form: All S are P. No S are P. Boolean Interpretation: If anything is an S, then it is a P. If anything is an S, then it is not a P.

Universal Statements (cont.) Statement Form: All S are P. No S are P. Symbolic Translation: (x)(Sx → Px) (x)(Sx → ~Px) Verbal Meaning: For any x, if x is an S, then x is a P For any x, if x is an S, then x is not a P.

Universal Statements (cont.) Statement: All skyscrapers are tall. No frogs are birds. All ambassadors are statesmen. No diamonds are rubies. Symbolic Translation: (x)(Sx → Tx) (x)(Fx → ~Bx) (x)(Ax → Sx) (x)(Dx → ~Rx) Important: (x) and (Ex) have the same scope as the tilde … it applies only to what immediately follows it. Eg, in this statement, (x)Rx v Bx, the universal quantifier does not apply to Bx.

Particular Statements We use what is called the ‘Boolean’ interpretation of particular statements. Statement Form: Some S are P. Some S are not P. Boolean Interpretation: At least one thing is an S and it is also a P. At least one thing is an S and it is not a P.

Particular Statements (cont.) Statement Form: Some S are P. Some S are not P. Symbolic Translation: (Ex)(Sx • Px) (Ex)(Sx • ~Px) Verbal Meaning: There exists an x such that x is an S and x is a P. There exists an x such that x is an S and x is not a P.

Particular Statements (cont.) Statement: Some men are paupers. Some diseases are not contageous. Some jobs are boring. Some vehicles are not motorcycles. Symbolic Translation: (Ex)(Mx • Px) (Ex)(Dx • ~Cx) (Ex)(Jx • Bx) (Ex)(Vx • ~Mx)

Translation Examples: Singular, Universal, & Particular Statement: There are happy marriages Every pediatrician loses sleep. Animals exist. Unicorns do not exist. Anything is conceivable. Sea Lions are mammals. Sea Lions live in these caves. Egomaniacs are not pleasant companions. A few egomaniacs did not arrive on time. Only close friends were invited to the wedding. None but citizens are eligible to vote. It is not the case that every girl scout sells cookies. Not a single psychologist attended the convention. Symbolic Translation: (Ex)(Mx • Hx) (x)(Px → Lx) (Ex)Ax ~(Ex)Ux (x)Cx (x)(Sx → Mx) (Ex)(Sx • Lx) (x)(Ex → ~Px) (Ex)(Ex • ~Ax) (x)(Ix → Cx) (x)(Ex → Cx) ~(x)(Gx → Sx) … or … (Ex)(Gx • ~Sx) (x)(Px → ~Ax) … or … ~(Ex)(Px • Ax)

More Complex Statements Statement: Only snakes and lizards thrive in the desert. Oranges and lemons are citrus fruits. Ripe apples are crunchy and delicious. Azaleas bloom if and only if they are fertilized. Peaches are edible unless they are rotten. Cats and dogs bite if they are frightened or harassed. Symbolic Translation: (x)(Tx → (Sx v Lx) (x)[(Ox v Lx) → Cx)] (x)(Ax • Rx) → (Cx • Dx) (x)[(Ax → (Bx = Fx)] (x)[(Px • Ex) → ~Rx)] (book is wrong) (x)(Cx v Dx) → [(Fx v Hx) → Bx]

Even More Complex Statements Statement: If Elizabeth is a historian, then some women are historians. If some cellists are music directors, then some orchestras are properly led. Either everything is alive or Bergson’s theory is not correct. All novels are interesting if and only if some Steinbeck novels are not romances. Green avacados are never purchased unless all the ripe ones are expensive. Symbolic Translation: He → (Ex)(Wx & Hx) (Ex)(Cx • Mx) → (Ex)(Ox • Px) (x)Ax v ~Cb (x)(Nx → Ix) = (Ex)(Sx • ~Rx) (x)[(Gx • Ax) → ~Px] v (x)[(Rx • Ax) → Ex]