First-Order Logic (FOL or FOPC) Syntax
User defines these primitives:
Constant symbols (i.e., the "individuals" in the world) E.g., Mary, 3
Function symbols (mapping individuals to individuals) E.g., father-of(Mary) = John, color-of(Sky) = Blue
Predicate symbols (mapping from individuals to truth values) E.g., greater(5,3), green(Grass), color(Grass, Green)
FOL supplies these primitives:
Variable symbols. E.g., x, y
Connectives. Same as in PL: not (~), and (^), or (v), implies (=>), if and only if (<=>)
Quantifiers: Universal (A) and Existential (E)
Universal quantification corresponds to conjunction ("and") in that (Ax)P(x) means that P holds for all values of x in the domain associated with that variable. E.g., (Ax) dolphin(x) => mammal(x)
Existential quantification corresponds to disjunction ("or") in that (Ex)P(x) means that P holds for some value of x in the domain associated with that variable. E.g., (Ex) mammal(x) ^ lays-eggs(x)
Universal quantifiers usually used with "implies" to form "if-then rules." E.g., (Ax) cs540-student(x) => smart(x) means "All cs540 students are smart." You rarely use universal quantification to make blanket statements about every individual in the world: (Ax)cs540-student(x) ^ smart(x) meaning that everyone in the world is a cs540 student and is smart.

Existential quantifiers usually used with "and" to specify a list of properties or facts about an individual. E.g., (Ex) cs540-student(x) ^ smart(x) means "there is a cs540 student who is smart." A common mistake is to represent this English sentence as the FOL sentence: (Ex) cs540-student(x) => smart(x) But consider what happens when there is a person who is NOT a cs540-student.
Switching the order of universal quantifiers does not change the meaning: (Ax)(Ay)P(x,y) is logically equivalent to (Ay)(Ax)P(x,y). Similarly, you can switch the order of existential quantifiers.
Switching the order of universals and existentials does change meaning:
Everyone likes someone: (Ax)(Ey)likes(x,y)
Someone is liked by everyone: (Ey)(Ax)likes(x,y)
Sentences are built up from terms and atoms:

A term (denoting a real-world individual) is a constant symbol, a variable symbol, or an n-place function of n terms. For example, x and f(x1, ..., xn) are terms, where each xi is a term.
An atom (which has value true or false) is either an n-place predicate of n terms, or, if P and Q are atoms, then ~P, P V Q, P ^ Q, P => Q, P <=> Q are atoms
A sentence is an atom, or, if P is a sentence and x is a variable, then (Ax)P and (Ex)P are sentences
A well-formed formula (wff) is a sentence containing no "free" variables. I.e., all variables are "bound" by universal or existential quantifiers. E.g., (Ax)P(x,y) has x bound as a universally quantified variable, but y is free.
Translating English to FOL
Every gardener likes the sun.(Ax) gardener(x) => likes(x,Sun)
You can fool some of the people all of the time.(Ex) (person(x) ^ (At)(time(t) => can-fool(x,t)))
You can fool all of the people some of the time.(Ax) (person(x) => (Et) (time(t) ^ can-fool(x,t)))
All purple mushrooms are poisonous.(Ax) (mushroom(x) ^ purple(x)) => poisonous(x)
No purple mushroom is poisonous.~(Ex) purple(x) ^ mushroom(x) ^ poisonous(x) or, equivalently,(Ax) (mushroom(x) ^ purple(x)) => ~poisonous(x)
There are exactly two purple mushrooms.(Ex)(Ey) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ~(x=y) ^ (Az) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z))
Deb is not tall.~tall(Deb)
X is above Y if X is on directly on top of Y or else there is a pile of one or more other objects directly on top of one another starting with X and ending with Y.(Ax)(Ay) above(x,y) <=> (on(x,y) v (Ez) (on(x,z) ^ above(z,y)))
Inference Rules for FOL
Inference rules for PL apply to FOL as well. For example, Modus Ponens, And-Introduction, And-Elimination, etc.
New (sound) inference rules for use with quantifiers:
Universal EliminationIf (Ax)P(x) is true, then P(c) is true, where c is a constant in the domain of x. For example, from (Ax)eats(Ziggy, x) we can infer eats(Ziggy, IceCream). The variable symbol can be replaced by any ground term, i.e., any constant symbol or function symbol applied to ground terms only.
Existential IntroductionIf P(c) is true, then (Ex)P(x) is inferred. For example, from eats(Ziggy, IceCream) we can infer (Ex)eats(Ziggy, x). All instances of the given constant symbol are replaced by the new variable symbol. Note that the variable symbol cannot already exist anywhere in the expression.
Existential EliminationFrom (Ex)P(x) infer P(c). For example, from (Ex)eats(Ziggy, x) infer eats(Ziggy, Cheese). Note that the variable is replaced by a brand new constant that does not occur in this or any other sentence in the Knowledge Base. In other words, we don't want to accidentally draw other inferences about it by introducing the constant. All we know is there must be some constant that makes this true, so we can introduce a brand new one to stand in for that (unknown) constant.
Paramodulation
Given two sentences (P1 v ... v PN) and (t=s v Q1 v ... v QM) where each Pi and Qi is a literal (see definition below) and Pj contains a term t, derive new sentence (P1 v ... v Pj-1 v Pj[s] v Pj+1 v ... v PN v Q1 v ... v QM) where Pj[s] means a single occurrence of the term t is replaced by the term s in Pj
Example: From P(a) and a=b derive P(b)
Generalized Modus Ponens (GMP)
Combines And-Introduction, Universal-Elimination, and Modus Ponens
Example: from P(c), Q(c), and (Ax)(P(x) ^ Q(x)) => R(x), derive R(c)
In general, given atomic sentences P1, P2, ..., PN, and implication sentence (Q1 ^ Q2 ^ ... ^ QN) => R, where Q1, ..., QN and R are atomic sentences, and subst(Theta, Pi) = subst(Theta, Qi) for i=1,...,N, derive new sentence: subst(Theta, R)
subst(Theta, alpha) denotes the result of applying a set of substitutions defined by Theta to the sentence alpha
A substitution list Theta = {v1/t1, v2/t2, ..., vn/tn} means to replace all occurrences of variable symbol vi by term ti. Substitutions are made in left-to-right order in the list. Example: subst({x/IceCream, y/Ziggy}, eats(y,x)) = eats(Ziggy, IceCream)