Basic atomic propositions notation and meaning
p, q, r... things that are either true or false
Propositional tableau: what follows from ¬φ∘
∘φ
Propositional tableau: what follows from ∘¬φ
φ∘
Propositional tableau: what follows from ψ∧φ∘
ψ, φ∘
Propositional tableau: what follows from ∘φ∧ψ
Propositional tableau: what follows from φ∨ψ∘
Propositional tableau: what follows from ∘φ∨ψ
∘ψ, φ
Propositional tableau: what follows from φ→ψ∘
Propositional tableau: what follows from ∘φ→ψ
φ∘ψ
Propositional tableau: what follows from φ↔ψ∘
Propositional tableau: what follows from ∘φ↔ψ
For an inference placed on the right side of a tableau to be valid, all branches must be _______
Closed
Each node of a tableau tree is called a ________
Sequent
A branch is closed if ___________________________
There's a formula that appears on both the left and right sides
is entails (⊨) a part of propositional logic?
No
Modus ponens definition
The reasoning that when the first part of an implication p->q holds (p), then the second must hold (q)
Refutation rule
Assume phi, and if after further steps you have a contradiction ⊥, phi cannot be true, therefore not phi is true.
Complete system definition
A set of rules that can be used to prove everything that is true within a certain type of logic
Three rules that make up a complete system for propositional logic
Modus ponens can also be referred to as __________
Elimination of implication
The deduction rule can also be referred to as ___________
Introduction of implication
The refutation rule can also be referred to as
Introduction/elimination of negation
Notes: conjunction rules for propositional natural deduction
Notes: disjunction rules for propositional natural deduction
Syllogism definition
An inference that has the following characteristics: two premises and one conclusion, only of one of the following forms: (All A are B), (Some A are B), (All A are not B (no A is B)), (Some A are not B (not all A are B)).
In syllogistic reasoning, A and B are _________, which are similar to the role of classes in object oriented programming.
Predicates
How many predicates does a syllogism involve?
3
What does {x | φ(x)} mean?
“the set of things x that have the property described by φ”
P\S meaning
All objects in P that are not in S
Complement of P notation
P̄
A stroked out region in a syllogistic diagram means that that region is _____
empty, no objects belong to that region
predicate logic symbols for constants
a, b, c
predicate logic symbols for variables
x, y, z
predicate logic symbols for predicates
A, B, C, .... P, Q, R
In predicate logic, a term is a ________ or a _________
variable, constant
A formula is closed if ________
Every occurence of a variable in the formula is bounded.
Substitution of a variable inside a formula only affects the ____ occurrences of the variable
free
Model notation
M = <D, I, g>
What is the D in model notation
domain
What is the I in model notation
The interpretation function, basically like variable assignment but for constants and predicates. It assigns an object or a relation in the domain to each symbol for constants and predicates, respectively.
What is the g in model notation
variable assignment, basically like the interpretation function but for variables
Notes models
More notes models
Predicate logic is an extension of propositional logic with ________, _________ propositions and ____________
structured, basic quantification
Predicate tableau: what follows from ∃xφ(x)∘
Predicate tableau: what follows from ∘∃xφ(x)
Predicate tableau: what follows from ∀xφ(x)∘
Predicate tableau: what follows from ∘∀xφ(x)
What must be done every time a new constant is introduced to a predicate tableau?
Reactivate all of the previous universal statements
Elimination of universal quantifier predicate natural deduction
Introduction of universal quantifier predicate natural deduction
Elimination of existential quantifier predicate natural deduction
Introduction of existential quantifier predicate natural deduction
Elimination/introduction of identity symbol predicate natural deduction
PDL composition of relations notation, for binary predicates Ra and Rb
Ra ∘ Rb
Composition of two relations Ra and Rb definition
Say that there are three states, x, y and z. if x is related to y via Ra and y is related to z via Rb, then x is related to z via Ra ∘ Rb
PDL union of two relations notation, for binary predicates Ra and Rb
Ra ∪ Rb
PDL repetition of a relation notation
PDL converse of a relation notation
Converse of a relation definition
Undoing an action, essentially the reversed arrow of a relation
More notation pdl
More notation pdl
Square brackets vs spiky brackets PDL
[ ] = for ALL possible executions. <> = for any (one or more) execution
PDL: what does LTS stand for
Labelled transition system
An LTS with a designated (root) state is called a ______ LTS or a _______
pointed, process graph
3 components of LTS
S (set of states), V (valuation function, which atomic propositions are true and/or false in each state), A binary relation Rₐ for every basic action a
If x = y .〈a〉.k is prefix, then we will use sˣ to denote the state __
k
More notes PDL
In hoare logic, if the execution never terminates because of an infinite while loop, the triple is considered __________
correct/partially correct
When do auxiliary variables change their value?
Never