Creating Logic, Part 2: Expanding Logic Reading Notes

Creating Logic, Part 2: Expanding Logic Reading Notes

Logic is the study of the way humans think and argue, and in particular, it deals with the basic concepts of propositions and reasoning.

The basic concepts of logic include propositions, propositional logic, and the symbolic system of logic. A proposition is a statement that is determined to be true or false. Propositional logic is the field of logic that deals with propositions and uses logical operators to represent logical relations. The symbol system of logic is a system that defines symbols to express logical operations, and in propositional logic, logical operators such as “negation,” “logical conjunction,” and “logical disjunction” are used.

In addition to propositional logic, there are other fields of logic such as first-order predicate logic and higher-order logic. First-order predicate logic is a type of propositional logic that can express more complex logical relations by introducing predicates and variables. Higher-order logic deals with logic involving mathematical concepts such as sets and functions.

Logic also includes the study of methods for determining the truth or falsehood of propositions and for making correct inferences. For example, there are methods of determining the truth or falsity of propositions, such as truth tables and proofs. The truth table is a method of determining the truth of a proposition by testing all possible combinations of truth and falsehood for the proposition. Proof is the logical procedure for showing that a proposition is true, using axioms and rules of inference to derive the truth of the proposition.

Logic is also closely related to philosophy, mathematics, and artificial intelligence. In philosophy, the concepts of logic are used to make logical arguments, and in mathematics, logic is used to construct the basic system for deriving mathematical propositions. In artificial intelligence, it is the basis for reasoning and knowledge information processing techniques.

Mathematical theories on which logic is based include set theory, as described in “Overview of Set Theory and Reference Books,” algebra, as described in “Structures, Algorithms, and Functions,” and formal linguistics and semantics, as described in “Formal Languages and Mathematical Logic,” which can be found at See also.

we present reading notes from “Making Logic,” a well-known textbook on mathematical logic.

Following the previous Part I, Beginning Logic, I will now discuss my reading notes for Part II, Expanding Logic.

Part II: Expanding Logic

Chapter 5: Expanding the Target Language of Logic
5.1 Why do we need to expand the language?

Arguments whose correctness cannot be characterized by symbolization with an artificial language L
Octopuses are cephalopods
Cephalopods are parasitic organisms from Mars
Octopuses are parasitic organisms from Mars
The above is a valid argument.
If we try to symbolize this argument with L, we will find that the three propositions are simple propositions
It can only be analyzed in the form “P, Q, therefore R.”
How can we guarantee validity?
Validity can be guaranteed by the following form
Fat is ◆
◆ is □
Fat is □.
It seems that “Fat, ◆, and □ appear in common in each proposition” is the basis of validity.
We need to analyze the internal structure of the simple propositions so far.
The logic of questioning the internal structure of propositions
Logic that does not question the internal structure of simple propositions

5.2 Symbolization of Propositions in Predicate Logic

5.2.1 What is the internal structure of a proposition?
Let’s assume that a proposition has the structure “Fat is □”.
Fat is a subject
individual indicative expression (singular term)
general term
□ is a predicate
5.2.2 Symbolization of Sentences with Individual Indicative Expressions as Subjects
5.2.3 Variety of individual-directed expressions
There is a wide variety of expressions that support individuals.
The following three types are typical
(1) Proper name
Tank Yoshida”, “Robert Rodriguez”, “Shinano River”.
(2) Definite description
An expression in which multiple nouns, adjectives, and verbs are combined to form a single entity.
The longest bridge in Japan, director of the Hollywood version of “Godzilla
In English, “the planet nearest sun,” “the author of Neuromancer,” etc., begin with the definite article “the.
(3) Demonstrative
(3) Demonstrative: “this,” “that,” “I,” “you
The challenge of definite descriptive phrases
Flexible and infinite combinations can be created, but
Can create names for things that don’t exist
Example
The planet between Mercury and Venus is not more massive than the Earth.
A proposition and its negative proposition can both be false
We have to accept a third value as truth value that is neither true nor righteous, or we have to accept a proposition that has no truth value (truth value gap)
The challenge with directives
The target of support differs depending on who is using the word and where.
The situation in which the word is used must be taken into account.
As individual indicative expressions
Consider as typical only those expressions that always have an object of instruction and that refer to the same individual regardless of the context.
Symbolization of propositions with proper names as subjects
Example
Steven Spielberg is a film director.
A rule that replaces the components of this proposition with symbols in a logical language
Write “Steven Spielberg is a film director” as Ds
Lower case letters correspond to individual indicative expressions
Capital letters correspond to predicates
View predicates as propositional functions
A predicate is a function from a set of individuals to a set of propositions.
5.2.3 Symbolizing Propositions with Common Nouns as Subjects
Symbolizing Propositions with Common Nouns as Subjects
Example
“Film directors have a good sense of management.”
[Interpretation] Mx:x is rich in managerial sense d:film director
What we want to do here is
Steven Spielberg is a film director Ds film director has a strong sense of management Md ———————————————————— Steven Spielberg has a strong sense of management Ms
Ds and ds need to be connected to get a good grasp of correctness
Can we symbolize “filmmaker” in d by representing it in d?
No matter what x we take (Dx→Mx)
D:x is a film director
M:x is a manager
Quantization and Individual Variable Terms
We call “tribe” an omnisymmetric symbol, an omnisymmetric quantifier (limit quantum), or a universal quantifier (limit quantum).
The “x” does not refer to a specific individual, as in the individual definite term “s”, but rather to all persons or individuals without determination.
5.2.4 Another quantifier
Differences in the position of negation
The scope of the individual variant x is a group of people.
[interpretation] Gx:x is good
All humans are good
Not all humans are good
For every human x, x is not good
Existential quantifier
Some thing x exists ~ (there exists (at least one) thing x such that ~ is)
Relation between ∃ and ∀
¬∀x¬Px ⊨⫤ ∃xPx
∀x¬Px ⊨⫤ ¬∃∃Px
de Morgan’s Law
∀x¬Px ⊨⫤ ∃xPx
¬∀xPx ⊨⫤ ∃x¬Px
Some and Some
There exists P
Some thing is P
We don’t often talk about “some things” without any limitations at all.
Usually the proposition is “Some S’s are P’s” or “Some S’s are P’s”.
Example
An artist is a genius.
There is such a thing as a genius and an artist.
[Interpretation] Mx:x is an artist Gx:x is a genius
Summary.
Defensive use of expressions made with quantifiers and negation.
The [1] and [3] groups are negations of the other.
[2] and [4] are also negations of each other
The [1] group is completely contained within the [2] group.
If [1] is true, then [2] is also true.
The formula of the [1] group logically implies the formula of the [2] group.
The formula of [4] group logically implies the formula of [3] group.
5.2.5 Get plenty of translation practice!
Tips for translation
Everything is P = Everything is P ✒︎ ∀xPx
There is something that is P = Something is P ✒︎ ∃xPx
Every P is Q = Every P is Q✒︎ ∀x(Px→Qx)
Some P is Q = Something P is Q✒︎ ∃x(PxΛQx)
But and Only
But”, “only”, and “only” are in some sense the reverse of “all”.
Example
Only Japanese people were taken hostage.
Only Japanese people are being held hostage.
[Interpretation] Hx:x is a hostage Jx:x is a Japanese
Translation tips: only and only
“Only B is A.”
“Only B is A.” “Only B is not A.”
“All that is A is B.”
All A’s are B’s.
Caution
If every and some are different, not only are ∀ and ∃ different, but so are → and Λ in parentheses.
“Every P is Q”
“Some P is Q”
Why can’t they both be Λ or →?
∀x(PΛQ)
Example
P: is a bird
Q: born from an egg
Everything in this world is a bird born from an egg
∀x(P→Q)
Every bird is born from an egg.
∃x(P→Q)
∃x(Px→Qx) ⊨⫤ ∃x¬(PxΛ¬Qx)
Example
P:is a jellyfish
Q: has a spine
There exists such a thing that is not an invertebrate jellyfish

5.3 Creating a Language for Predicate Logic

5.3.1 Definition of the language MPL
The language we are going to define is called MPL.
Vocabulary of MPL
The vocabulary used in MPL can be divided into four groups
(1) Term
Individual definite terms
a,b,c,…
Individual variable terms
x,y,z,…
(2) Predicate symbols
P,Q,R,S,…
(3) Logical definite term
Combinators
→, Λ, V, ¬
Quantifiers
∀, ∃
(4) Auxiliary symbols
(, )
Definition of MPL grammar, i.e. logical formula
Definition
(1) A logical formula is one in which a single term is placed behind a single predicate symbol. This is called an atomic formula.
(2) If A and B are logical formulas, then (AΛB), (AⅤB), (A→B), and (¬A) are logical formulas.
(3) If A is a logical formula and 𝛏 is an individual variable, then ∀𝛏A and ∃𝛏A are each logical formulas.
(4) Only formulas that are considered logical by (1), (2), and (3) are logical formulas.
How to use the graphic characters
(1) “𝛏” is the Greek letter “xi”.
(1) “𝛏” is the Greek letter “xi”, and “ζ” is the Greek letter “zeta”.
𝛏 can be any of x,y,z,… without specifying which one.
(2) For individual definite terms a, b, etc., use the first letters of the Greek lowercase letters “α” and “β” as graphic symbols.
(3) Individual variable terms and individual definite terms together are called terms.
The Greek letter “τ” (tau) is used as a graphic character to represent an arbitrary term.
Tree of Formation
For any logical formula, there is only one tree of formation
∀x(Dx→Mx)
(∀xDx→Mx)
(∀xDx→∀xMx)
Unique readability theorem holds for the definition of logical formulas in MPL
Arrangement for omission of parentheses
Arrangements for omitting extra parentheses
(∀xPx→Qx) becomes ∀xPx→Qx
∀x(¬Px) becomes ∀ x¬Px
5.3.2 Free and bounded manifestations of quantifier domains and varieties
The “unnatural” logical formula
∀xPy
∃yQa
∃x∀xPx
Approaches
It’s okay to mix in a little extra unintelligible stuff into a logical formula.
Modify the definition to eliminate these symbolic sequences from the logical formula.
The definitions get complicated.
What structural features are common to unnatural formulas?
Free or bounded appearance of variable terms
By definition, there are also formulas such as Px,(PaΛQx) that leave bare individual variable terms unaccompanied by quantifiers.
For example, “~ is an anime geek” or “Tarantino is a film director and ~ is an actress.”
The truth is not determined.
The individual variable x is free to appear (occur free) or free variable.
Logical expressions such as ∀xPx, ∃y(PyΛQy) are translated into Japanese propositions without spaces.
Individual variable terms are said to appear as bound (occur bound).
When x appears in several places in one logical expression
Depending on where it appears, it may be a free variant or a bound variant.
Example
∃xPxΛQx
∃xPx and Qx attached by Λ.
x in Px is a bound variant
x in Qx is a free variable
∃xPxΛQx is a bounded variational term
We need to properly define free and bound manifestations.
(1) Manifestations of the individual variable 𝛏
The same individual variant appears in several places, such as ∃xPxΛQx and ∃(PxΛQx).
Each occurrence is called an occurrence of the individual variable x.
(2) Action regions of quantifiers
The scope of a quantifier is
A quantifier’s scope is the sub-logic to which the quantifier is bound in the tree of formation.
Example
The scope of ∀ in the tree of formation of the logical formula ∀xDx→Mx is
Dx, not
(Dx→Mx).
(3) A quantifier that binds 𝛏
If the individual variable term immediately following a quantifier is 𝛏, such as ∀𝛏 or ∃𝛏
We say that the quantifier is a quantifier that binds 𝛏.
∃ of ∃x is a quantifier that binds x
The ∀ of ∀y is the quantifier that binds y.
(4) Free Manifestations of Individual Variables
The flow of individual variables 𝛏 is free.
⇔
The manifestation is not in the middle of the quantifier’s domain that binds 𝛏.
When a manifestation is not free, it is said to be bound.
(5) Closed and open formulas
A closed formula (or closed formula) is
A closed formula is a formula that does not contain any free manifestation of an individual variable.
A formula that does not is called an open formula.
Some people refer to a closed formula as a sentence.
We do not use this term in this book.
(6) Vacuous occurrence of quantifiers
A quantifier whose occurrence ∀𝛏 or ∃𝛏 is vacuous.
⇔
The variable 𝛏 is not free to appear in the action region of the quantifier. Translated with www.DeepL.com/Translator (free version)

5.4 Extending the method of tableau

5.4.1 Taking cues from concrete examples
Applying Tablau’s Method to Predicate Logic
Extending Tablau’s Method
Make rules for ∀ and ∃ so that the criteria are equivalent to those for logical propositions
Take a set of formulas that are intuitively obvious to be inconsistent
Make rules for expanding tableaux so that they can be judged as contradictory by the same criteria as before.
Create a rule for expanding tableaux
The new expansion rules are
∀Tribe and ∃, and their negative forms.
Example
Anyone who collects 10 service stamps will receive a ramen ticket.
There is a person who collected 10 service stamps.
No one received a ramen ticket.
Rules for Ontogeny Quantifiers
Derive the tableau
Arrange the three logical formulas vertically
Let’s start with the second logical formula
Assign an individual definite term “a” to x such that Px is true
a-kun” who has collected 10 service stamps
∃𝛏A ↓………. [∃] A[α/𝛏] Rules for universal quantifiers
Notice the first line.
Px→Qx holds for all x.
It must also hold for a
Substitute a for x in Px→Qx
Rules for negation + quantifiers
Notice the third line where we haven’t applied any rules yet
What is the expression that is true when ¬∃xQx is true?
De Morgan’s rule
Substitution of rules (tableau is closed)
5.4.2 Expansion Rules and How to Use Them
Is it safe to say that [∀] and [∃] are the same rule?
There are constraints on the expansion rules for existential quantifiers
The set {∃xPx, ∃x¬Px} is satisfiable
Bad tableau
Originally.
For ∃, there are constraints on the individual definite terms that can be substituted
If you use ∃ first, you can use anything, because at that point no individual definite term appears in the path
Expansion rules for ∃
[EI] ∃𝛏A ↓ where α is an individual fixed term that has not appeared in the path so far A[α/𝛏].
Difference between universal exemplification and existence exemplification
Expansion rules for ∀.
[UI] ∀𝛏A ↓ where α is an arbitrary individual fixed term A[α/𝛏].
Universalization and existence exemplification are the same operation of assigning an individual definite term to a free manifestation of 𝛏.
The difference is in the proviso about what kind of individual definite terms can be assigned.
Universalization is applied repeatedly
Consider the argument A, “∀x(Px→Qx), Pa, ¬Qb, so ¬(¬QaⅤPb)”.
Step 1 Do not close yet
Apply UI to the first line
Not closed yet
Since the first line asserts that Px→Qx holds for everything, it holds for b, not just a
Closed.
UI is applied over and over again in a secondary way
[UI] must in principle be applied repeatedly to the number of individual definite terms that appear
Strategy
Apply [EI] before [UI] as much as possible.
If you do so, you can avoid introducing too many individual definite terms.

Chapter 6. Oh, I didn’t know the semantics of predicate logic yet.
6.1 We must create a semantics for predicate logic

Introduction
Up to the previous section, we have developed tableaux for use with MPL.
If we can show the reliability of the tableau, it will be complete.
If the tableau is reliable, then
An open tableau arises from Γ
⇔
Γ is satisfiable
What does it mean for a set of formulas written in predicate logic to be inconsistent or satisfiable?
{∀x(Px→Qx), ¬∃xQx, ∃xPx} is a contradiction.
By inconsistent, do you mean that there is no truth-value assignment that would make all three expressions true at once?
What is the truth-value assignment for predicate logic expressions anyway?
In the case of propositional logic
You said, “PΛ¬Q becomes 1 under ethic assignment by assigning 1 to atomic formula P and 0 to atomic formula Q.”
Since the atomic formulas P and Q are the smallest units, he did not ask further, “When does P become 1?
In the case of predicate logic
We can say, “¬∃xQx is 1 when ∃xQx is 0.
Since ∃xQx is not the smallest unit, we have to answer when it is zero.
6.1.1 The truth of a logical expression is determined by its interpretation
What do the predicates P and Q mean?
Example
[Interpretation] Px:x is greater than 1 Qx:x is greater than 0
I don’t understand “what range of x is the range of consideration?
False for natural numbers
True for rational numbers
To determine the truth or falsity of a predicate logic formula, we need to determine the following two points
(1) What range of individual variable terms are covered by the predicate?
(2) the meaning of the predicate symbols (or individual definite terms if they are included) that appear there.
Interpretation” should not be a translation.
It is a very bad idea to try to capture the truth of a logical formula by translating it into a natural language.
6.1.2 From translation to direct description of the world
Stop interpreting MPL in Japanese.
The meaning of a language is
Not by translation into other languages.
The meaning of a language is given by the way the language describes the world
Direct mapping of MPL to the world it describes
Example
A large square is the end of the world
is a person, a molecule, or some other individual in the world
An ellipse containing a ● means that the individual contained in the ellipse has the property P
What happens to the truth value of ∀xPx→∃xPx in the above world?
In world A, ∀xPx is false
Therefore, ∀xPx→∃xPx is true.
In world B, ∀xPx is true
Therefore, ∃xPx will also be true.
In World C, ∀xPx is false.
Therefore, ∀xPx→∃xPx is true.
In any world, ∀xPx→∃xPx is true.
No matter how you interpret the symbols.
We can say “in any world” instead. Translated with www.DeepL.com/Translator (free version)

6.2 Semantics and Models

6.2.1 Formalization with the help of mathematics
Think of the world as a set
(1) Decide on some set as the domain of discussion.
If this were the world, we would name this set D.
(2) For the descriptor P, draw a small ellipse in the world and assign a subset of D corresponding to it.
It is equivalent to assigning a subset of D
(3) We can assign an element of D to an individual definite term.
Interpretation as a function
What is formalizing semantics?
Semantics-specific concepts such as “implies” and “true” are all defined using the mathematical tool of sets.
Replace the world described by MPL with a set
The mapping that “assigns” something in the world to various symbols in MPL is called
function, a set-theoretic way of speaking.
The flavoring of the symbols of MPL is the following function V
(1) V assigns a subset of D to the predicate symbol P1
(2) V assigns an element of D to the individual definite term a1
6.2.2 Model
The pair D and V <D,V> itself is an “interpretation
Example of a model
<Model M1> (1) D=(■, ●, ◆) (2) Appended value for predicate symbol VM1(P)={■, ◆} VM1(Q)={●, ◆} (3) Appended value for individual definite term a VM1(a)=■ VM1(b)=●
<Model M2> (1) D=N (set of natural numbers) (2) Attached value for predicate symbol VM2(P)=set of even numbers VM2(Q)=set of odd numbers (3) Attached value for individual fixed term a VM2(a)=7 VM2(b)=8
Closed logic formulas have truth values that depend on the model
∃x(PxΛQx) is true in model M1, but false in model M2
Pb is false in model M1, but true in model M2
Px is true or not in model M1
True if x is ■.
False if x is ●.
Conditions to be satisfied by the model
(1) D can be any set.
It can be a finite set or an infinite set, but it must not be the empty set Φ
(2) The set corresponding to the predicate symbol can be an empty set.
(3) Not all individuals in an argument domain need to have a name in MPL (individual definite term).
In standard predicate logic
(4) An individual definite term must be assigned to some individual.
6.2.3 Define the truth of logical expressions under model M
The only meaningful notion of truth is relative to the model “true under any model”.
What does it mean to say, “Under the model <D,V>, the logical formula A is true”?
When the model is fixed to one, we write V(A)=1
Determining the starting point: defining atomic formulas
Only closed logic formulas can be true or false under a given model.
Only those with an individual definite term after the predicate symbol
Various predicate symbols (P,Q,R…) (P,Q,R…) are collectively represented by the graphic character Φ
[T1] VM(Φα)=1 ⇔ VM(α)∈VM(Φ)
The fact that the atomic formula Φα is true under the model M means that
VM(α) is the one that the additive function VM of that model assigns to the individual definite term α.
contained as an element in the set VM(Φ) that VM assigns to the predicate Φ.
Let’s define the truth of a compound logical formula
Formulate a rule to determine inductively how the truth of a compound logical formula depends on the truth of its sub-formulas.
Operations to create a compound logical expression
(1) Connecting with a joiner
Defined by propositional logic
(1) When A is BΛC, VM(A)=1 ⇔ VM(B)=1 and VM(C)=1 (2) When A is BⅤC, VM(A)=1 ⇔ VM(B)=1 or VM(C)=1 (3) When A is B→C, VM(A)=1 ⇔ VM(B)=1 or VM(C)=1 (4) When A is ¬B VM(A)=1 ⇔ VM(B)=1 not
(2) Quantifiers in ∀ and ∃.
It is true that the logical formula ∀𝛏B𝛏
B is true for everything that belongs to the argument domain.
Everything that belongs to the argument domain belongs to the set assigned to B.
The fact that the logical formula ∃𝛏B𝛏 is true means that
Any individual belongs to the set assigned to B.
[T2.9] (1) If A is ∀𝛏B𝛏, then VM(A)=1 ⇔ M for all individuals i in region D. (2) If A is ∃𝛏B𝛏, then VM(A)=1 ⇔ M. i∈VM(B) for at least one individual i belonging to region D.
It won’t work.
Logic formulas such as ∃x(PxΛQx)
Substituting into (2) of [T2.9], we get
PxΛQx is not a predicate symbol
since it is missing in [T3] [T2.9].
On quantifiers
Where did we go wrong?
It is difficult to reconcile the following two requirements
(1) What we want to define inductively is “truth under model M”, which is meaningful only for closed logical formulas.
(2) However, in the inductive definition of logical formulae, closed logical formulae are not made from closed logical formulae.
Policy for rework
[Policy S].
Attempt an inductive definition for the entire logical formula, including the open formula.
We can no longer define “true” to be meaningful only for closed logical expressions.
Consider some broader notion of “being xx.
Define “is xx” inductively over all logical expressions
Then define “true” as a special case that is true only for closed logical formulas of “xx
The method used by Alfred Tarski.
[Policy T].
Do not abandon the policy of defining psychological concepts inductively only for closed logical formulas
Even for formulas such as ∃x(PxΛQx)
what these truths mean.
We will define these truths by referring to the truth of some simple closed logic formula (such as PaΛQa).
The resulting definition will be simpler if we follow policy T.
6.2.4 Define “truth under M” properly according to policy T
Use [T1], [T2] as is.
We will rework [T2.9].
If ∀x(PxΛQx) is true, then
If ∀x(PxΛQx) is true, then any closed logic formula of the form PaΛQa obtained by substituting any individual definite term for any free manifestation of x in (PxΛQx) will be true.
If we assume that all individuals in model M are assigned some individual fixed term, then
the reverse of the above also holds
If all closed logic formulas of the form PaΛQa are true, then
Then ∀x(PxΛQx) is true.
After assigning an individual resistance to each of all individuals in the argument domain D of M, we have
[T2.99] (1) If A is ∀𝛏B, then VM(A)=1 ⇔ VM(B[α/𝛏])=1 for all individual definite terms α. (2) If A is ∃𝛏B, then VM(A)=1 ⇔ VM( B[α/𝛏])=1
6.2.5 Properly define “truth under M” based on policy S
6.2.6 Definition of truth in policy S
6.2.7 Contradiction, logical consequence, and valid formula
6.2.8 Disproof model

6.3 Existence Positives and Conversational Implications

6.3.1 Suspicion that the same argument may or may not be valid
Just because some Japanese people accept corporal punishment does not mean that all Japanese people accept corporal punishment.
On the contrary
[Argument 1] The fact that all Japanese approve of corporal punishment does not mean that
It logically follows that there are Japanese who accept corporal punishment.
[Domain] Japanese [Interpretation] Px: x condones corporal punishment.
[Argument 2] ∀xPx, therefore ∃xPx.
[Domain] Human [Interpretation] Px: x tolerates corporal punishment Jx: x is Japanese
[Argument 3] ∀x(Jx→Px), hence ∃x(JxΛPx)
Checking using tableau, it comes out as not valid.
6.3.2 Where do the discrepancies lie?
In general
When you hear or assert the proposition “100 is a △” in a serious way
Unless it is very unlikely that there is such a thing, we talk about it as if it exists.
We make up for it as a hidden assumption.
6.3.3 Conversational Implications
Existential specification is a kind of what is called “conversational implicature”.
It is not included as a logical consequence of the claim, but
It is not included as a logical consequence of the claim, but it is taken into account as a kind of common understanding for the claim to be legitimate. It is not false, but it is extremely misleading.

6.4 A Little Bit of Traditional Logic

6.4.1 The four basic forms and their interrelationships
Contradiction vs. equivalence
Large and small paradoxes
Opposites vs.
6.4.2 How did traditional logic deal with argumentation?
The three-stage argument
Large Concept, Middle Concept, Small Concept
The Four Cases of Three-Stage Argumentation
How many three-stage arguments are there, and how many of them are valid?

Chapter 7: Extending the Language of Logic Further
7.1 Limitations of MPL

Introduction
Further Extensions to MPL
Because some arguments do not work well in MPL
7.1.1 Difficulties for MPL
Arguments that have plagued traditional logic
Conundrum 1
<argument 1>.
Someone is loved by everyone. Therefore, everyone loves someone else” is valid, but
<argument 2
“Everyone loves someone else. Therefore, someone is loved by everyone. is not valid.
Conundrum 2
<argument 3
All horses are animals. Therefore, the head of every horse is the head of an animal” is not valid.
<argument 4
Lisa: Marimba is percussion. So a marimba player is a percussionist? Why is this correct as an argument?
7.1.2 Where are the limits of MPL?
If we try to symbolize “everyone loves someone” and “someone is loved by everyone” in MPL, we get
The above two (∀xPx, ∃xQx) cannot be logically related symbolically
Symbolization of Argument 4
[Interpretation] Px:x is a marimba.　Qx:x is percussion. Rx:x is a marimba player.　Sx:x is a percussionist.
Since there is no common predicate between premise and conclusion
Cannot be validated
You can’t write “x is a marimba player” as a single predicate like Rx.
It needs to be paraphrased as “x is a person who plays something that is a marimba”.
Limitations of MPL
The phrase “~ is a person who plays …. MPL can’t handle the expression “to play”.
MPL can only contain predicates with a single empty space, such as Px and Qx.
Individuals such as “~ plays …. MPL cannot handle expressions that express relationships between individuals, such as “~ plays
Predicates with two or more blanks cannot be handled.
7.1.3 How to express the relationship between individuals
Predicates as functions
View predicates as propositional functions
In modern logic
Predicate as a function
In modern logic, predicates are regarded as functions that extract individual indicative expressions from sentences and leave them behind.
Example
“Spielberg is a film director.
The predicate is the function “~ is a film director” from which the individual indicative expression “Spielberg” is extracted
Indicates an attribute of an individual
Example
“Spielberg respects Akira Kurosawa.
The function “~ is a film director” with the two individual indicative expressions removed is the predicate. The predicate is “I respect Akira Kurosawa.
Indicates a binary relation
Rxy
Example
Spielberg is younger than Akira Kurosawa and older than Tarantino.
With the three individual indicative expressions removed, “~ is … is younger than and older than —-“.
Indicates a ternary relation
Rxyz
…..
N-place predicate
Indicates an n-ary relation.
Sx1x2…. .xn
7.1.4 Multiplexing
To symbolize the above proposition
Every insect has a natural enemy.
There is a natural enemy
There exists such a thing that is a natural enemy of x
∀x(Ix→∃yNyx)
Two quantifiers appear in one expression
Two different quantifiers bound each of the different individual variable terms following the same single predicate symbol
Overlapping quantification (multiple qualification)
The only thing that matters in multiple qualification is which quantifier binds which gap
Try to rewrite ∀x(Ix→∃yNyx) in Japanese.
∀x(x is an insect -> ∃y(y is the natural enemy of x))
∀x(x is an insect -> there is such a thing as a natural enemy of x)
Every insect has a natural enemy.
Points to note
As long as the original logical formula is a closed formula
There are no individual variable terms x or y in the Japanese version of it
Either ∀z(Iz→∃yNyz) or ∀z(Iz→∃wNwz) will result in “every animal has a natural enemy”.
x, y, and z only need to be distinct from each other, it doesn’t matter what you use
The role of these symbols
All they do is indicate which predicate and which space is bound to which quantifier
If the pattern of binding is the same
If the pattern of binding is the same, the two logical formulas are the same, no matter what individual variable (x,y,z…) is used. (x,y,z…).
Let’s symbolize “marimba player.
The ability to systematically deal with logical formulas involving multiple weightings is a major milestone in modern logic.
The marimba is a percussion instrument.
Mx : x is a marimba
Px : x is a percussionist
A marimba player is a percussionist
x is a marimba player
Hx : x is a person
x is a person who plays the marimba
There exists such a thing that x plays and is also a marimba player
Qxy : x is a person who plays y
x is a marimba player
x is a percussionist
The marimba player is a percussionist
<argument 4>.
∀x(Mx→Px) —————— ∀x(HxΛ∃y(QxyΛMy) → HxΛ∃y(QxyΛPy))
7.1.5 Ah-re, too, love, this is love too – The Logic of Love
Consider the hard problem 1
Suppose Lxy : x loves y.
There are 8 patterns
(!) (1) ∀x∀yLxy
(2) ∀y∀xLxy
(3) ∃x∃yLxy
(4) ∃y∃xLxy
(5) ∃x∀yLxy
(6) ∀y∃xLxy
(7) ∃y∀xLxy
(8) ∀x∃yLxy
When only quantifiers of the same kind are involved
(1) ∀x∀yLxy ≈ ∀x(∀yLxy) ≈ ∀x(x loves everybody) ≈ Everybody loves everybody
(2) ∀y∀xLxy ≈ ∀y(Everybody loves y) ≈ Everybody loves everybody
(3) and (4) are “Somebody loves somebody”.
When quantifiers of the same type overlap, the order of the quantifiers is irrelevant.
When two types of quantifiers appear mixed together
Cases where two quantifiers appear mixed together (mixed multiple quantification)
Not easy
Case (5)
∃x∀yLxy ≈ ∃x(x loves everyone) ≈ Someone loves everyone
Case (6)
∀y∃xLxy ≈ ∀x(there is someone who loves y) ≈ Everyone is loved by somebody
The remaining two logical formulas are all different in what they express.
The other two formulas are all different in what they express: “existentially” and “existentially”.
Organize the four cases where quantifiers are mixed
(5) ∃x∀yLxy ≈ There is such a person who loves all people.
(7) ∃y∀xLxy ≈ There is someone who is loved by everyone
(6) ∀y∃xLxy ≈ For every person there is someone who will love that person
(8) ∀x∃yLxy ≈ Everyone has someone who loves them.
(6)(8) is “exist accordingly”
(5)(7) is “edge existence
7.1.6 Definition of the language PPL
The definition of PPL (Polyadic Predicate Logic) is almost the same as the definition of MPL (polyadic means multinomial, M in MPL is monodic)
Vocabulary of PPL
The vocabulary of MPL is used as it is.
However, only the predicate symbols need to be changed to allow multinomial predicates.
Predicate symbols are classified by the number of empty spaces
(1) A unary predicate = a predicate with a single empty space, such as Px or Qx
(2) Binary predicates = predicates with two empty spaces, such as Lxy
(3) n-term predicate = a predicate with n empty places
Write as Pni
Use Φn as a graphic character to represent any n-term predicate without specifying any of them.
<Vocabulary of PPL
(1) Term Same as MPL
(2) Predicate symbols P11,P12,P13…. P21,P22,P23,… …. Pn1,Pn2,Pn3,…
(3) (4) Logical definite terms and auxiliary symbols are the same as in MPL
Grammar of PPL, i.e. definition of logical expressions
Very few changes
With the introduction of polynomial predicate symbols, the definition of an atomic formula changes to the following
<definition
(1) An n-term predicate symbol with n terms after it is a logical formula.
This is called an atomic formula.
(2) Same as MPL from now on.

7.2 Semantics of PPL

Introduction.
Extending the Model to Allow Defining Truth and Falsehood for Boolean Expressions Containing Multinomial Predicates
7.2.1 The World of Love
Predicates with more than two terms have corresponding relations in the world
Consider a world with only three people in it
7.2.2 Redefining the model
What shall we assign to the multinomial predicates?
Assign to the predicate the set of all individuals in the discussion domain to which the predicate applies.
Example
The predicate “~ loves …. What does the predicate “~ loves ” apply to?
What is represented by an arrow?
Instead of an arrow, “let some set-theoretic object take the place of the arrow”.
Order pair
■→●, ●→◆, ◆→■
<■, ●>, <●, ◆>, <◆, ■>
<Model M1> D={■, ●, ◆} V(L)= {<■, ●>, <●, ◆>, <◆, ■>} V(a)= ● V(b)= ◆ V(c)= ■
Lab must be true in this model
That there is an arrow in the world going from V(a)=● to V(b)=◆
For the binary predicate, we assign a subset of D2, the set of ordered pairs of two elements of D.
A ternary predicate is assigned a subset of D3, the set of ordered pairs of three elements of D.
An N-term predicate is assigned a subset of the set Dn of ordered pairs of n elements of D.
Definition of the model
Definition of the model that summarizes the above considerations
(1) The set D is not empty.
(2) V assigns to the predicate symbol Φn a subset of Dn V(Φn) ⊆ Dn
(3) V assigns an element of D to an individual term α. V(α) ∈ D
7.2.3 Tampering with the definition of truth
The definition of truth (or sufficiency relation) is also extended for atomic formulas to include multinomial predicates
[T1′] VM(Φnα1α2. .αn)=1 ⇔ <V(α1), V(α2), … V(αn)>∈V(Φn) [S1′} VM,σ(Φnτ1τ2. .τn)=1 ⇔ <σ(τ1), σ(τ2), … σ(τn)>∈V(Φn)
Keep [T2][T3], [S2][S3].
The new definition of truth when [T1′][T2][T3] takes the policy T
New definition of the sufficiency relation when [S1′][S2][S3] takes the policy S
Is it really safe?
Checking with examples
[Example] Define <model M> as follows D={◇, ◆, ☆, ◎} V(P)={◇, ◆} V(Q)={◆, ☆} V(L)={<◇, ◆>, <◆, ◎>, <☆, ◎>, <◎, ◇>} V(C)={<◇, ◆, ◇>, <◎, ◎, ◇>} V(a)=☆ V(b)=◎
(1) Is ∃xCbbx true in model M?
(2) Is ∀x∃xLxy true in model M?
Even if we allow for multiple weightings, the definitions for logical consequences, valid formulas, contradictions and satisfiability of sets of formulas, and logical equivalence hold in exactly the same form as in MPL.
It means that we can assign the same thing to different variants.
If we follow the definition of truth
The “some x and some y” of ∃x∃yLxy need not be two different things.
7.2.4 Some Theorems about the Semantics of Predicate Logic
At this stage of the development of the semantics of predicate logic
We present some facts about the semantics of predicate logic.
Theorem 29.
If two models M1 and M2 have the same domain of argument, and if all the individual definite terms and predicate symbols appearing in a closed formula A are assigned the same ones, then VM1(A)=VM2(A) (where VM1(A)=VM2(A) is an abbreviation for VM1(A)=1 ⇔ VM2(A)=1).
Theorem 30
Suppose that in model M, V(α)=V(β), that is, this model assigns the same to the individual fixed terms α and β. In this case, VM(A(α))=VM(A(β)).
In other words, even if we replace some of the α’s in the expression A(α) with β’s, the resulting expression A(β) will have the same truth value as the original expression.
If the psychology of the whole proposition remains the same even if we replace some of the indicative expressions α in a proposition with other individual-supporting expressions β that support the same individual
Not all propositions are extrapolative.
For contexts such as the one formed by “I know,” replacing the coextensive expression with one that supports the same individual will change the psychology of the proposition as a whole.
Intensional context
Other theorems
Theorem 31

7.3 Using Tableaux in PPL

7.3.1 Tackling the Marimba
7.3.2 Using Tableau in the Science of Relationships
Classifying Relationships
The relationship between “being a parent” and “being an older person” is similar in some ways and not in others
a is not a parent of a itself, and a is not an elder of a itself
A parent of a parent is not a parent (grandparent) and a is older than a is older than a
Definition.
The relation R is reflexive ⇔ ∀xRxx holds The relation R is symmetric ⇔ ∀xy(Rxy→Ryx) holds The relation R is transitive ⇔ ∀xyz(RxyΛRyz)→Rxz holds The relation R is equivalence relation ⇔ R is reflexive, symmetric, and transitive
Definition
The relation R is irreflexive ⇔ ∀x¬Rxx holds (1) The relation R is nonreflexive ⇔ ∃xRxxΛ∃x¬Rxx holds (2)
X<y and “I love you” are both non-reflexive, but in different ways
(1) X<y is not reflexive because no number can be less than itself.
(2) “I love you” is not reflexive because some people love themselves and some people don’t.
It is not reflexive because not everyone loves themselves.
Definition.
The relation R is asynmetric ⇔ ∀xy(Rxy→¬Ryx) holds The relation R is nonsynmetric ⇔ ∃xy(RxyΛRyx)Λxy(RxyΛ¬Ryx) holds
Definition
The relation R is intransitive ⇔ ∀xyz((RxyΛRyz)→¬Rxz) holds The relation R is nontransitive ⇔ ∃xyz(RxyΛRyzΛRxz)Λ∃xyz(RxyΛRyzΛ¬Rxz) holds
7.3.3 Reliability of Tablo
The tableau never stops.
Checking if {∀x∃xPxy} is a contradiction
∀x∃yPxy ∃yPay Pab ∃yPby Pbc ∃yPcy ….
The tableau never ends.
Extending to predicate logics that admit multiple weightings, tableaux are “not decidable.”
Even if it is undecidable, the result it produces is reliable
In the case of PPL, tableaux can become unstoppable
The fact that we can make a decision at any time is not the same as the fact that we can trust the decision when we make it.
Theorem 33.
Γ is inconsistent ⇔ a tableau starting from Γ is a closed tableau
(1) If the tableau configuration stops and a decision is made, then the decision is reliable.
(2) When the tableau becomes a closed tableau, the tableau composition is finished, so
The fact that the tableau starting from Γ does not end means that
No closed tableau arises from Γ
A tableau composition can only become unstoppable if the Γ put to the test is a satisfiable set of expressions
7.3.4 Proof of “If a closed tableau arises, it is a contradiction”
Theorem 33-a.
A tableau starting from Γ is a closed tableau ⇔ Γ is a contradiction
Theorem 33-b
Γ is a contradiction ⇔ a tableau starting from Γ is a closed tableau
This means that the tableau does not stop.
There is at least one path that is not closed forever
7.3.5 Proof of “A tableau starting from a contradictory case is a closed tableau”.
Hinticka Set
Definition
Theorem 34.
Any Hinticka set is satisfiable.
Proof that the open path of a tableau is a Hinticka set Translated with www.DeepL.com/Translator (free version)

7.4 Don’t Blame the Logicians — Decision Problems and Theory of Computation

7.4.1 Undecidability of PPL
It’s not just the tableau that’s bad.
Given any finite set of PPLs.
Given any finite set of PPLs, there can be no mechanical method that can determine whether they are contradictory or satisfiable in a finite number of steps.
As long as it is a mechanical method, there will always be cases where it cannot make a decision in a finite number of steps.
Decidability” should not be said about individual methods, such as tableau methods.
What should be said about PPL itself
Undecidability
Theorem 35
There is no general method (algorithm/deterministic procedure) for determining the following by a finite application of a mechanical procedure
Given a finite set of arbitrary logical formulas of PPL
Whether they are contradictory or satisfiable
Given an arbitrary argument of PPL
whether that argument is valid.
7.4.2 Decision Problems
What is a decision problem?
A question with the following form is called a “decision problem
[Given an arbitrary { }, is there an algorithm that always outputs { } in a finite number of steps?
Is there an algorithm that, given any { finite set of PPL formulas}, always outputs { the result of deciding whether it is a contradiction or not} in a finite number of steps?
According to Theorem 35, there is no such algorithm.
If you can create a concrete algorithm, even just one
Positively solved decision problem
Examples of Positively Solved Decision Problems
(1) Given any finite set of logical formulas of L, is there an algorithm that always outputs {the result of determining whether it is a contradiction} in a finite number of steps?
Algorithms that require truth tables or tableaux
(2) Is there an algorithm that always outputs {the result of judging whether it is a contradiction} in a finite step when inputting any finite set of logical formulas in MPL?
Tablo
(3) Is there an algorithm that, given any {finite set of ∀expressions}, always outputs {the result of determining whether it is a contradiction} in a finite step?
(4) Is there an algorithm that, given any {argument whose premises are ∀∃ and whose conclusion is ∀∃}, will always output {the result of determining whether it is valid} in a finite number of steps?
(5) Is there an algorithm that, given any {two pairs of natural numbers}, always outputs {the greatest common divisor of these natural numbers} in a finite number of steps?
Euclid’s Reciprocal Division Method
Negatively Solved Decision Problems
Is there an algorithm that, given any {program P and any pair of inputs i}, always outputs {the result of a decision about whether or not to stop in finite time when i is input to P and it is executed} in finite steps?
7.4.3 Concept of Algorithm and Church’s Proposal
What does it mean to “solve a decision problem negatively”?
To solve a decision problem positively.
Create an algorithm that is required by the decision problem anyway
To solve a decision problem negatively
We need to prove that the algorithm does not exist.
¬∃x is the same as ∀x¬.
In order for the proof of the absence of an algorithm to be meaningful
We need to be clear about the scope of “all algorithms.”
What are the common characteristics of all algorithms?
How do we draw the line between what is an algorithm and what is not an algorithm?
Clarifying the New Year for algorithms and Church’s proposal
Explanation of Algorithms
Procedures need to be clear
Procedures must be generic
Procedures must be finite
It is like describing a logical formula as “a set of symbols arranged in such a way that they make sense,” but it is not a rigorous description.
Notion of algorithm
Church’s “λ-definability”.
Turing’s “Turing machine computability”.
Inductive functions” by Gödel and Kurini
Others
All of the above are equivalent to each other
Church’s thesis.
We think of an algorithm as being computable by a Turing machine.
Turing machine
Mathematically rigorously defined, abstract, fictional computer
Typical way to solve a decision problem negatively
Church’s thesis (proposal) clarifies what it means to solve a decision problem negatively.
What does it mean to solve a decision problem negatively?
Proving that it is not possible to program a Turing machine to perform the required computation.
It is proved using the method of backtracking.
Assuming that a Turing machine could be programmed to perform the computation required by the decision problem
Derive a contradiction from it
Use another decision problem that has already been solved negatively.
Assume that the stopping problem has already been solved negatively
To prove the undecidability of PPL
Under the process that if we had an algorithm that could determine the inconsistency/sufficiency of any set of equations in PPL
We will prove that the algorithm can be converted to the kind of algorithm required by the halting problem by changing it slightly.
Since there is no such algorithm as the stopping problem requires
There is no algorithm that can determine the inconsistency/sufficiency of an arbitrary set of formulas in PPL.

Chapter 8: Extending the Logical Language Further and Further
8.1 Predicate Logic IPL with Identities

8.1.1 Introducing the Identity Symbol
“Billy hates all people.”
In order for ∀xHbx to be true, Hbx must also be satisfied when x is assigned Billy himself
∀xHbx logically implies that Billy hates himself (Hbb)
Billy hates everyone who is not him.
Everyone who is not Billy is hated by Billy
∀Tribe x (x is not Billy → Hbx)
To symbolize the previous case of →, we can use
To symbolize the antecedent of “~ is… In order to symbolize the previous case of →, we need a word that states that the two individuals are identical.
The word “=” is subject-neutral, meaning that it can appear in any conversation.
∀x(¬(x=b) → Hbx)
∀x(x≠b)→Hbx)
8.1.2 Predicate Logic that recognizes “=” as a logical definite term
8.1.2 Tweaking the definition and semantics of logical expressions
Changes in the definition of logical expressions
Since we have added a new logical definite term, we will change the definition of logical expressions and the definition of models.
(1) First, add an entry called “identity symbol” to the MPL vocabulary, and register “=” to it.
(2) In the definition of logical expressions, add the following definition to the item of primitive expressions
<definition>.
When τ1 and τ2 are terms, (τ1=τ2) is a logical formula.
Of course, we shall also add an arrangement for properly omitting the parentheses.
Changes in the definition of truth in the model
The symbol “=” must be given a meaning corresponding to the identity as we normally understand it.
If we take policy T, we can add (1) to the definition of truth for atomic formulas in MPL [T1′].
(1) For any individual definite terms α and β, VM(α=β)=1 ⇔ V(α)=V(β)
If we take the policy S, we can add (2) to the definition of the sufficiency relation [S1′].
(2) For any terms τ1 and τ2, VM,σ(τ1=τ2)=1 ⇔ σ(τ1)=σ(τ2)
Distinguish between the two “=”.
The three “=” in definitions (1) and (2) may look the same, but they are different symbols in Uranugi.
The “=” appearing in the parentheses following the VM is the same symbol that is a component of a logical expression.
Symbols included in the IPL vocabulary
The “=” that appears on the right side of the ⇔ and the “=” in “=1” on the left side
The attached value function V assigns the same individual to α and β.
A logical formula of the form α=β is true under the model M
The “=” in “=1” is a symbol in a meta-language used to shorten the Japanese phrase “α=β”.
the one belonging to the IPL vocabulary as “≈”.
VM(α≈β)=1 ⇔ V(α)=V(β)
Semantic theorems about identity symbols
The definitions remain unchanged when viewed as valid expressions, contradictions, and logical induction.
Theorem 36
(1) ⊨α=α, i.e., a formula of the form α=α is a valid formula (2) ⊨α=β → β=α (3) ⊨(α=βΛβ=γ) → α=γ (if we take the policy S, it holds for any term, not for an individual definite term) (4) ⊨∀𝛏(𝛏=𝛏), i.e., the identity relation is reflexive (5) ⊨∀𝛏∀𝛇( 𝛏=𝛇→𝛇=𝛇) Identity relation is synmetric (6) ⊨∀𝛏∀𝛇∀𝛇 𝛈((𝛏=𝛇Λ𝛇=𝛈) → 𝛏=𝛈) is transitive as well (7) α=β ⊨ A[α/𝛏] ↔ ↔ A[β/𝛏] (8) α≠ α ⊨ That is, an expression of the form α≠α is a contradiction
(7) is called the principle of indiscriminability of identivals or Leibnitz’s principle.
8.1.3 Treating identities with tableaux
Introducing the expansion rule for “=”
Rules for negative forms
α≠α → x
¬(α=α) → x
Expansion rules for negative logical formulas containing =.
PaΛ¬Pb a=b Pa ¬Pb

8.2 Piece Count Expressions and Identity Symbols

8.2.1 Expressing “There are many”.
Introduction
The identity symbol can be used to symbolize the phrase “there are at least n P” or “there are exactly n P” for any finite number n.
The main advantage of introducing identities
One at least
The phrase “there is at least one P” can be expressed without using the identity symbol
Just one
Just one: “There is exactly one person who fits your wishes.”
Px: fits your wishes
Since the existence of some person is asserted, ∃x(…) is of the form
What kind of person is x who is claimed to exist?
(a) He is P. (b) There is no other person who could be P besides him.
The question is: “There is no one other than x who is P.”
There is no such person as y who is P and is not x
Or logical equivalence
All people who are P are the same person as x
Only x is P
Unique existence
At most one
“There is only one person at most who fits your description.
At most one, at most one.
At most one” “At most one” “At most one” “There is just one or none”
At most one.
The negative of “two or more.
Two at least
There are at least two P’s.
Just two
“There are exactly two P’s.”
The whole thing takes the form ∃x∃y(….) The whole thing takes the form
(…) Conditions for x and y to be written in
(1) Both x and y are P. PxΛPy
(2) x and y are different. y≠x
(3) There is nothing other than x and y that can be P.
¬∃z(PzΛz≠xΛx≠y) or ∀z(Pz→(z=xⅤz=y))
From the above
∃x∃y(PxΛPyΛy≠xΛ¬∃z(PzΛz≠xΛz≠y))
8.2.2 Theory of Definite Description Phrases
What is a definite description clause?
Introduction.
“the first person who walked on the Moon”
the heaviest planet in the solar system
Expressions that refer to a single individual.
Expressions that contain predicates and are made up of them
There are challenges in translating propositions with definite descriptive phrases as their subjects into the logical language IPL.
The only survivor of the plane crash is a five-year-old child.
(1) Information in the definite descriptive phrase itself
P:the only survivor of the plane
a: was a five-year-old child
Pa.
This information can only be extracted from the existence of a five-year-old child.
(2) Definite descriptive clause with no supporting object
It is possible to specify something that does not exist.
If we assume that the non-existence is false, we may have a contradiction with the false logic formula
Clues for analysis
The cause of the difficulty is
A definite descriptive phrase, such as “the first person to walk on the sun”
The reason for the difficulty is that the definite descriptive phrase “the first man who walked on the sun” is considered to be a single expression like a proper name, and the individual term “a” is used.
Grammatical form of propositions
The grammatical form of the proposition is “The first person who walked on the sun (subject) + is a Russian (predicate)”.
If we stop analyzing in the framework of “indicative expression + predicate,” we can find a way out.
There is a person, (a) he is a five-year-old child, (b) he is a survivor of the plane, and (c) he is the only survivor of the plane.
You can paraphrase it as
A proposition that satisfies the three conditions (a), (b), and (c)
Description Theory
Symbolize based on the ideas above
∃x(x is a survivor of that plane crash Λx is a 5 year old child Λx and there are no other Λx’s in that plane)
∃x(PxΛQxΛ∀y(Py→y=x))
Bertrand Russell’s theory of description
Taste the analysis
What appeared to be a single semantic unit in the original proposition appears in discrete pieces
A definite description phrase is no longer a single expression.
Definite description clauses can be eliminated by applying logical analysis to the entire proposition containing them.
Symbols that, like proper names and predicates, cannot be taken out by themselves and asked about their meaning in isolation

Summary of Part II

In the next article, I will discuss my reading notes for Part 3: Another Look at Logic.