Index
Symbols
CG
DG
EG
Ki
Ki operator
L[t]
L[t]u
L[u ← t]
Lc
Propset of a wff (or a set of wffs)
operator in FOL
in FOL
↓
≤s,
(A, I)
1
-operator
1
L10
¬
-consistency
(abstract) algebra
(strongly) confluent relation
modus ponens (MP)
shift-reflexive relation
3-colourability
A
abduction
absolutely consistent
abstract interpreter of LP
accessibility relation
adequate (connective set)
formal system
algorithm for semantic tableaux (PL)
ancestor (of a node)
antisymmetry
argumentation
arity
artificial intelligence
associativity without =
asymmetry
axiom
(for constructivists)
(of a theory)
of choice
(sequent calculus)
axiomatic
structure
system
axiomatico-deductive system
axiomatisation
of equality
B
backward chaining
Berry paradox
binary resolution
body of a rule
Boolean algebra
bound
occurrence of a variable
variable
branch
C
calculus
case analysis
chain (totally ordered set)
choice (axiom of)
clausal
form (FOL)
transformation
clause
(FOL)
(PL)
clauses parents
closed
branch
clause
instance
semantic tree
substitution
tableau
terms
theory
wff
closure
of an owff
operation
cnf
transformation
coherent
common knowledge
commutativity without =
compactness
of FOL
(of PL)
theorem (PL)
complementary literal
complete
complete theory
completeness
of FOL
(for refutation)
completion (Clark)
complexity
of a proof by resolution
of the resolution method
computability and Horn clauses
conclusion
conditional formula
confluent relation
confluent (convergent) relation
congruence relation
conjunct
conjunctive
clauses
normal form
consensus
consequence operation, Tarski
consequence relation
(entailment relation)
(Tarski)
consistent
(absolutely)
for negation
constant instance
constraint
projection
constructive proof
constructivist proof
continuity
contradictory
copy
counter-model (counterexample)
counterexample (counter-model)
of a set of formulas
cwff: closed wff
D
Davis and Putnam
algorithm
decidable
class (some)
decision procedure
deduction
theorem
deductive
inference
system
demodulation
density
depth of a term
descendant (of a node)
description logic
diagonalization
direct consequence
directed graph
discovery of explanatory theories
disjunct
disjunctive normal form
distributed knowledge
dnf
transformation
domain of discourse
E
e-frame
E-unsatisfiable
empty clause
eq (Prolog)
equality
axiomatization
equivalence
of formal systems
relation
Euler circles
evaluation
execution control (/)
existential quantifier
expressive power
expressivity
extension
extensional
connective
language
F
factor (of a clause)
factorization
failure node
finite
model (modal logics)
model property
set
trees
finitely
controllable class
satisfiable
first-order
logic
structure
flat terms
FOL
(first-order logic)
semantics
formal
system
system for FOL
systems approach for modal logics
theory
forward chaining
free occurrence of a variable
free variables
Frege system
fundamental theorem of arithmetic
fuzzy logic
fuzzy set
G
Gödel’s
completeness theorem
incompleteness theorem
(proof)
generalization
generalized characteristic function
Gentzen
(LK system)
system
greatest lower bound
H
halting problem
head of a rule
Herbrand
base
instance
interpretation
terms
theorem
theorem (for clauses)
universe
hierarchy (priority) of connectives
Hilbert
(proof procedure)
system
Horn clause
(computability)
I
implication
incompleteness theorem
(Gödel)
independence
induction axiom
(extensional version)
(intensional version)
induction
(proof by)
principle
inductive inference
inference
node
rule
infimum
infinite
set
tree
initial sequent
intension
interpolant
interpretation
of a formula
of a set of formulas
(in FOL)
intuitionistic proof
irreflexive relation
irreflexivity
K
K (minimal logic)
K-modal frame
König’s lemma
knowledge
(definition)
modal logic
Kripke
model
semantics
L
Löwenheim-Skolem theorem
language
lattice
least upper bound
left-hand side of a rule
Leibniz’s law
length of a clause
liar’s paradox
Lindenbaum algebra
linear
strategy
term
literal
LK (Gentzen’s system)
logic
(formal definition)
(informal definitions)
(multi-valued)
programming
logical
axiom
connective
consequence
constant
inference rule
symbols
LP abstract interpreter
M
matching
meta-language
MFOL (pure monadic class)
MFOL= (monadic first-order logic)
mgu: most general unifier
minimal logic (K)
minimally unsatisfiable
modal
frame
logics
model
(general notion)
intersection property
of a set of formulas (FOL)
of a set of formulas (PL)
monadic
class
class of FOL
first-order logic
monotonicity
monotony
more general clause
MP (modus ponens)
multi-sorted structure
multi-valued logic
N
n-valid formula
n-validity
NAF (negation as failure)
natural deduction system
negation as failure (NAF)
negative
clause
FOL
literal
normal form
nnf
non-compactness of SOL
non-logical symbols
non-rational infinite trees
non-reflexive relation
non-standard model
non-standard model
nondeterminism
normal form (Skolem)
not (Prolog)
O
ontologies
ontology
open
branch
semantic tree
tableau
wff
order relation
owff: open wff
P
p-valued logic
paradox
paramodulant
paramodulation
parent clause
partial interpretation
of a formula
partially ordered set
alternate definition
PC (propositional logic)
Peano’s axioms
PFOL (pure first-order logic)
pigeonhole principle
PL (propositional logic)
PL syntax
positive
clause
literal
possible worlds semantics
predicate logic
predicative programming
premise
prenex normal form
Presburger arithmetic
principle of mathematical induction
proof
by cases
by diagonalisation
by induction
(in LK)
procedure
system
proper
axiom
inference rule
properties of the knowledge
proposition
(formal definition)
propositional logic (PL)
provable formula
in a formal system
pure
first-order logic
literal
purity principle
Q
quantified variable
quotient algebra
R
rational infinite trees
reasoning
reductio ad absurdum
reduction
reflexivity
refutation
refutational completeness
(FOL resolution)
relational
programming
system
resolution
(in FOL)
(in PL)
method (in FOL)
(refutational completeness for FOL)
rule
resolvent
rewriting
right-hand side of a rule
rule
Russell’s paradox
S
SAT problem
satisfaction
relation (in FOL)
satisfiability
by resolution
validity (in modal logic)
satisfiable
formula in an e-frame
saturation
scope of quantifiers
sea battle argument
second-order logic
self-resolving
semantic tableaux (for p-valued logics)
(in FOL)
(PL)
(PL), algorithm
(uses)
(with unification)
semantic
trees (in FOL)
trees (in PL)
semantics of PL (of PC)
semi-decision procedure
for FOL
sequent
calculus
set (constructivist definition)
set operations fuzzy sets
set theory
sign of a literal
skolem function
normal form
skolemisation
sort reduction
sound
soundness of the resolution method
soundness of binary resolution
standard model
strategy
structural induction
structure isomorphism
sub-formula
substitution
codomain
domain
notation
subsumption
(rule)
supremum
syllogism
T
tableau
(closed)
(open)
tail of a rule
tautological
tautology
temporal logic
term
algebra
equation
replacement
theorem
(Gödel’s incompleteness)
torsion group
total order
relation
totally ordered set
(chain)
transitivity
translation approach
trees
truth
and satisfaction
functional connective
table
U
unfair strategy
unification
algorithm
unit
clause
strategy
universal quantifier
universe of discourse
unsatisfiable
upper bound
uses of semantic tableaux
V
valid
formula in an e-frame
valuation
variables
variant
Venn diagrams
very, very long proofs
W
weak equivalence
well-founded relation
wff: well-founded formula