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