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Mindmap-Galerie Introduction to Logic (Chen Bo)
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Introduction to Logic (Chen Bo)
Have completed self-study: first-order logic and informal logic, logic is the science of reasoning and argumentation (the discipline that studies reasoning), This map is one of my arsenal of tools for my use
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Introduction to Logic (Chen Bo) Mainly formal logic
Chapter 1 Logic is the science of reasoning and argumentation
Section 1 The etymology and meaning of “logic”
1. The ancient Greek etymology of “logic”
English logos can be traced back to the Greek word "logos"
Polysemy, main meaning
General laws, principles, and rules
Speech, propositions, explanations, explanations and arguments
Rationality, reasoning, the ability to reason, abstract theory as opposed to experience, and methodical reasoning as opposed to intuition
Scale, relationship, proportion and ratio, etc.
2. History and Current Situation of Logic
Representatives of formal logic in ancient Greece (mainstream)
Aristotle's lexical logic
syllogism
Stoic propositional logic
Divide propositions into atomic propositions and compound propositions around "implication", give four metalogical rules and use them to prove many theorems
There was a disruption, it didn’t enter the mainstream
Famous dialectics in China's pre-Qin period
Mohist logic has the highest achievement
ancient indian logic
Because it clearly refers to the knowledge of reasoning, Buddhist logic
status quo
basic logic
Classical logic and non-classical logic (formal logic and informal logic)
metalogic and inductive logic
Apply logic
general logic
Intersection with various disciplines
3. Objects of Logic: Reasoning and Demonstration
What is logic?
Is the science of reasoning and argumentation (the study of reasoning)
main mission
Provides criteria for identifying valid reasoning, argumentation, and invalid reasoning and argumentation
Teach people to reason and argue correctly
Teach people to identify, expose and refute faulty reasoning and arguments
reasoning
The thinking process or form of thinking that leads to a new proposition (conclusion) from one or some known propositions (premises)
deductive reasoning
Generally recommend individual
Inevitability: True or False Absolute
efficient
invalid
inductive reasoning
Generally recommended individually
Probability: strong or weak possibility
Strong induction
weak induction
Argument
The process or language form of using certain reasons to support or refute a point of view
Section 2 Propositional Analysis and Logical Types
1. Sentences, propositions, statements, judgments and truth values
In the broad sense, all statements are true or false, while in the narrow sense, only propositions are true or false. A proposition that is asserted (true or false) is a judgment.
Propositions refer to sentences that express judgments. Those that do not express judgments are not propositions (such as interrogative sentences, imperative sentences, and exclamatory sentences). Big Dictionary p348
2. Compound propositions and propositional logic
Compound propositions are composed of connectives and simple propositions (atomic propositions)
various connectives
Couple (conjunction)
Disjunction (disjunction)
compatible
incompatible
either ... or
Hypothesis (condition)
if then
Only talent, unless
if and only if (if then and only if)
negative
Symbols represent propositions
Constant items
∧, ∨, →, ←→, ┓
variables
p, q, r, s, t, etc.
3. Categorical propositions and lexical logic
A categorical proposition asserts that the object S has a certain property P, also called a property proposition.
Possess subject, predicate, joint, and quantity terms
If all S is P
4. Individual words, predicates and quantitative logic (predicate logic)
Possess individual words, predicates, quantifiers, connectives, etc.
Individual words (indicated by lowercase letters)
Constant items
Specific proper noun abc represents
variables
Uncertain individual xyz representation
Predicate (indicated by capital letters)
Represents the properties of individuals in the domain of discourse and the relationship between individuals
For example, F(x) is a one-element predicate symbol, R(x,y) is a binary predicate symbol, and so on. There are several individuals.
quantifier
Full name∀
∀xF(x)
It reads that for all x, x is F
Existence∃
∃xR(x,y)
Read as there is x such that x has an R relation with y
For example, R means >, x>y
5. Mutation logic, expansion logic and meta-logic
It belongs to modern logic and is different from traditional logic.
Section 3 Reasoning Forms and Their Validity
1. The formal structure of reasoning
The model or framework that retains the specific content of a proposition
For example: If it rains tomorrow, then Xiao Ming will not come to school. It will rain the next day, so Xiao Ming will not come to school.
If p, then q p, so q
2. The validity of the reasoning form (is it relevant? Irrelevant? Relevant? Conclusion reached?)
Effective reasoning can lead to true conclusions from true premises, but cannot lead to false conclusions.
Not a single special case leads to a false conclusion
Invalid reasoning may also lead to true conclusions from true premises
There are other special cases that lead to false conclusions (common in lexical logic (syllogism))
For a reasoning or argument to reach a true conclusion and be convincing, it needs to satisfy
true premise
The reasoning form is valid
On the contrary, what are the ways to refute or weaken a conclusion?
Directly refute the conclusion
Refuting the premise (argument)
rebuttal reasoning form
3. Reasoning and argumentation in daily thinking
What is it used for?
exchange of ideas
How to spot logic errors
What to do if you find out
"You mean" test
Section 4 Basic Laws of Logic
Logic is the cultivation and training of the rational spirit
They constitute the most basic premises and presuppositions of rational thinking, and are the minimum prerequisites for rational dialogue and conversation to continue.
What will happen if you fail to comply?
There may be logical errors and perceptual emotions.
You may get into an argument or be unable to continue the conversation.
1. Law of Identity
A is A
In the same thinking process, all thoughts (including concepts and propositions) must remain identical with themselves
The same form of expression (speech, etc.) or thought cannot be confused with multiple meanings unless specifically stated.
Fallacies that may arise if it is violated
Confusing concepts (unintentional)
Stealing the concept (intentional violation)
Transfer topic ()
secretly change the topic
2. The law of contradiction (the law of non-contradiction)
Not (A and not A)
Two contradictory propositions cannot be both true or false
Derive lexical logic: Two mutually opposing propositions cannot be both true, but they can both be false.
A Venn diagram can visually represent
3. The law of excluded middle
A or not A
Two contradictory propositions must be one true and one false
Derive lexical logic: Two mutually opposed particular propositions cannot be both false, but they can be both true.
4. The law of sufficient reason (Brainitz)
A,A logically deduce B┣B
If you want to prove that B is true, you must first prove that A is true, and prove that B can be logically deduced from A.
Here "┣" means "launch"
In mathematics books, "=>" also means "introduction": A==>B represents a sufficient condition. When A is established, B is also established.
Specific requirements
1. Reasons must be given for the point of view to be argued.
2. The reasons given must be true
3. The arguments to be argued must be inferred from the reasons given.
If you fail to meet the requirements, you will make the mistakes of "no reason", "false reason" and "cannot be deduced".
Argument should be based on careful and detailed thinking, test the thinking process, and finally decide whether to accept (believe) the idea or viewpoint
Counterexample: Some ideas and opinions may be very pleasant and reasonable in general terms, but they cannot withstand rigorous and precise analysis and testing.
summary
What is the difference and connection between reasoning and argumentation?
The difference is that reasoning can start from false premises, while argumentation needs to start from true premises or premises that are commonly accepted by everyone.
What is logic? Purpose?
Logic is the science of reasoning and argumentation
This book refers to formal logic
Purpose
Recognize whether reasoning and arguments are valid or invalid
Teach people how to reason and demonstrate correctly
Identify, expose, and refute faulty reasoning and arguments
Analysis of propositions from different angles leads to differences in logical theories
propositional logic
lexical logic
predicate logic
Can be used for both of the above, with a wider range
Chapter 2 Propositional Logic (Connective Logic, expressing the relationship between propositions)
Section 1 Daily Connectives and Compound Propositions
1. Simple propositions and compound propositions
Simple propositions are divided into different terms and cannot be further divided into propositions. They are also called atomic propositions.
A compound proposition is a proposition that contains other propositions. It is formed by connecting other propositions with certain connectives.
For example: It’s not raining today
Compound proposition classification
2. Joint proposition
And: a proposition that asserts the simultaneous existence of several things.
∧ (conjunction)
and, and, and, and then etc.
The branch proposition of a couplet is called a "link". Sometimes the subject or predicate of a couplet can be omitted.
Examples of provincial subjects
Examples of predicate terms
Three valid forms
Synthetic formula
decomposition
negative
3. Disjunctive proposition
Or: Conclude that at least one of several things exists.
∨ (disjunction)
Or, either, or, if not, just wait.
"Disjunctive branch" "Disjunctive branch"
If a disjunctive proposition exhausts all disjunctive components, then this disjunctive proposition must be true
Types and valid expressions
Compatible (can be true at the same time)
negative affirmative
positive affirmative
Incompatible (cannot be true at the same time)
negative affirmative
affirmative negative
4. Hypothetical proposition
Conditional proposition: asserts a certain conditional relationship between the antecedent and the consequent
→(implies)
A branch statement (antecedent and consequent) has one condition and one result.
Sufficient conditions (false if the first part is true and the second part is false)
If, then
affirmative antecedent
Negative postcondition
Necessary conditions (false if the former is false and the latter is true)
Only, only
negative antecedent
Only p, only q Not p So non-q
affirmative postpartum
Necessary and Sufficient Condition
if and only if
p and q are both true and false
5. Negative proposition
Not
┓
Section 2 Truth Value Connectors Truth Value Form
1. From daily connectives to truth-value connectives
Propositional connectives are also called propositional constants (they have only a fixed meaning and will not change)
A propositional connective that connects several propositions is a several-element connective.
Problems with daily connectives in logic
imprecise
Contains a lot of illogical content
Such as juxtaposition, succession, progression, transition, contrast, etc.
Rules and conventions for omitting parentheses
(1) The outermost parentheses of the formula can always be omitted
(2) Like arithmetic, when there are no parentheses, multiply and divide first and then add and subtract: priority is high to low ┓, ∧, ∨, →, ←→
(3) It is agreed that (A∧B)∧C can be written as A∧B∧C, and the same is true for ∨, but A→(B→C) is written as A→B→C.
2. Truth value form assignment and assignment
┓p, (p∧q), (p∨q), (p→q), (p←→q) are negation, conjunction, disjunction, implication and equality respectively.
Let p be true/false, this is called assigning a truth value, and the meaning of the truth connective is called interpretation (truth function)
A set of truth assignments and an interpretation (a truth function) constitute a truth assignment.
If p→q, let p be true and q be false, then p→q be false
A formula containing n propositional variables has 2ⁿ possible truth value combinations.
Formula = truth form = truth function
p and q are equivalent to x (independent variable) and y (dependent variable) in the function
3. Negation
4. Conjunction
Both p and q are true
5. Disjunction
Compatible: p and q are true if at least one of them is true
Incompatible: If any one of the alternatives is true, the other alternatives must be false.
6. Implication
The premise is true, and the conclusion is false only if it is false (cannot be generalized using if-then)
So a true proposition can be entailed by any proposition (true consequent)
A fact can be entailed by any proposition, that is, it happened no matter what.
Substantive implication conflicts with the daily connective "if then". When two implication symbols appear, it becomes awkward and counterintuitive.
When one accuses substantive entailment, it also logically leads to accusing the understanding of the remaining ┓∨∧ truth connectives
Either p or q
p or q
Not p → q
┓p∨q
┓┓p→q
p→q
Two expressions can be considered equivalent if their truth tables are consistent.
7. Equivalence
The antecedent and the consequent are both true and false, otherwise the equation is false
8. Symbolization of compound propositions in natural language
First determine which proposition the natural language belongs to
Analyze the meaning and which proposition it is equivalent to
For example, "Want (p)" in Example 2 means that you want to achieve a certain result, which is a necessary condition hypothesis proposition q→p
Only p is equivalent to q
if q then p
The language is unnatural, awkward and weird (reasons for discarding meaning and content)
Only p is q
if not p then not q
if p then q
Only q is p
If p then q is equivalent to "p only if q"
If p then q is equivalent to "not p unless q", or "not p unless q"
p→q is equivalent to ┓q→┓p
if p then q else r
(p→q)∧(┓p→r)
q unless p
¬p→q
¬q→p
p, otherwise q
Same as above
p unless q
¬q→p
subtopic
Section 3 Tautologies and their determination methods
truth form
Tautology (valid, satisfiable)
True value is always true
Contradictory form (invalid form, satisfiable form)
permanent holiday
Even true form (not valid form)
Some are true and some are false
1. Tautology
The aim of propositional logic is to find the set of all tautologies
Determination procedure
1 Each step of the program is specified by a set of rules given in advance.
2The program can end in finite steps
3. It can give the only certain result for the judged object.
Common tautology doubts
Peirce's Law
A∨B→((A→B)→B)
The law of excluded middle B∨┓B is omitted, that is, the law of excluded middle can be replaced by only the implication symbol
Reinforced front piece
(A→B)→(A∧C→B)
It is controversial. If the properties of C can achieve non-B, the result B may not be obtained.
A∧┓B→B is even true, which is not true when A is true and B is false.
The law of identity, the law of contradiction, and the law of excluded middle in propositional logic: A→A. A∨┓A .┓(A∧┓A) are only the spiritual embodiment of the three basic laws of logic and are not identical.
2. Truth table method
A formula containing n propositional variables has 2ⁿ possible truth value combinations.
Use a list to list the truth value combinations of all proposition variables, list the truth values of all sub-formulas from simple to complex, and finally get all the truth value situations of the formula
Advantages: mechanical, simple to operate, intuitive and clear at a glance, the most reliable
Disadvantages: For formulas with many proposition variables, the workload is too large and takes a lot of time.
3. Reductio ad absurdum assignment method
If the assignment is false, if there is a contradiction, it is not a contradiction.
Advantages: Simplification of truth table
Disadvantages: Multiple assignments may be required, which is not intuitive and easy to make mistakes.
4. Tree diagram method
reductio ad absurdum assignment method
Agreed rules: five truth-value connectives, a total of 9 rules
A branch represents a truth value combination
Bifurcation represents several situations
To determine formula A, let A be false, then ┓A is true and then start drawing the tree diagram of ┓A
A is a tautology if and only if ┓Every branch (truth value combination) of the tree diagram of ┓A is closed (marked ×)
As long as there is a branch without an ×, A is not a tautology
When encountering sub-formulas that are bifurcated and non-bifurcated, draw the non-bifurcated ones first, otherwise it will be repeated and the workload will be heavy.
Section 4 Tautological Implications and Tautological Equivalences
1. The formal structure of reasoning: tautological implication
Look up
truth connectives for the outermost level
Implication FAQ
Default conditions
circular argument
On whether God is omnipotent
2. Tautology of Equivalence and Substitution Rules
These rules serve as tools
It can be used for any connective word, as long as the replaced formula is equivalent to the replacing formula
Section 5 Natural Reasoning of Propositional Logic
1. PN (Propositional Logic Deduction Rule System)
Theorem: It is a formula derived using Pᴺ deduction rules without any premises or assumptions.
Can be used directly for inference without proof
The rules of analogy deduction do not require proof
When Γ├A (Γ is a certain formula set or hypothesis), and Γ = ∮ (the empty set), then A is a provable formula of Pᴺ, referred to as the theorem
Pᴺ deduction rules
conjunction
1～Elimination Rule∧⁻
A can be derived from A∧B; B can be derived from A∧B
A∧B├A;A∧B├B (simplification)
2~Introduction of rules ∧⁺
From A and B, we can deduce A∧B
A,B├A∧B (merge)
Extract
3 Elimination Rules∨⁻
A∨B,A→C,B→C├C (difficult reasoning simple formula)
4Introduction of rules ∨⁺
A├A∨B;B├A∨B (additional law)
implication
5→⁻
A→B, A├B (confirmed)
(Must be used to deduce the unpremised axiom) 6→⁺
If Γ, A├B, then Γ├A→B (introducing a hypothesis, which itself can be assumed to be false by contradiction, and must be true)
Proof by contradiction: A formula set Γ is false if and only if the premises are true and the conclusion is false
If there is nothing missing in the kitchen and there are ingredients, you can cook
Equivalent to: There is no shortage of anything in the kitchen. If there are ingredients, you can cook.
Commonly used implication expressions of Pᴺ theorem
¬A→(A→B), B can have any formula
Used for proof by contradiction, introducing the contradictory formula to obtain the original formula
Similarly A→(¬A→B)
equivalent
7←→⁻
A←→B├A→B;A←→B├B→A
8←→⁺
A→B, B←A├A←→B
negative
9 ┓⁻
If Γ, ┓A├B, ┓B then Γ├A (proof by contradiction)
10┓⁺
If Γ, A├B, ┓B then Γ├┓A (reductio ad absurdum)
self-introduction rules
11∈
If Ai∈Γ, then Γ├Ai (assuming a set of premises is equivalent to assuming each premise)
Any hypothesis can be deduced from a set of hypotheses
Pᴺ theorem and its proof or deduction method
Writing convention (the purpose is to establish a seamless or flawless chain of reasoning)
① List all given premises in separate lines at the beginning, and indicate the premise to the right of each premise formula
② If you want to introduce assumptions, in the same way as ①, it is best to list all the assumptions at the beginning and mark the assumptions one by one.
③Every time a hypothesis is listed, move it one space to the right of the formula above.
Show that this is an assumption based on the previous assumption
④Every time a formula is listed, indicate the formula and deduction rules it relies on to the right of the formula.
⑤The formulas obtained from ∨⁻∨⁺∧⁻∧⁺→⁻ ←→⁻←→⁺∈ under an assumption are all aligned with this assumption, indicating that these formulas all depend on this assumption and previous assumptions.
⑥If a formula is obtained based on →⁺┓⁻┓⁻, then let it be carried and aligned with the assumptions above, indicating that it relies on the assumptions above and before, and the assumptions and formulas in this building are lifted and cannot be used.
⑦Draw a vertical line after the step number of the deduction to indicate the start and end of the deduction; if it is a hypothesis, add a small circle at the top
Meta-theorem, the proof process is very complicated
A chain of reasoning with gaps
Leibniz proved that 2 2 = 4
Adding the associative law of addition without any premise
Don't write according to the rules, jump too fast to be clever
It’s better to walk slowly and steadily
Strictly precise but also somewhat technical (how to get the right from the left, or how to get the left from the right)
Assume possible conditions for all branches, like the violent solution of Sudoku
If the right-hand side is disjunctive, assume each separately, and if the right-hand side is conjunctive, both will be obtained.
Use of proven theorems and derived rules (to simplify the proof process)
The Pᴺ theorem is a tautology. Similarly, the PR equivalent permutation is a tautology and can be quoted directly.
Chapter 3 Term Logic (split the proposition to express the nature of each component within the proposition)
1. Blunt proposition
basic structure
(Quantity term) Subject term (Co-term) Predicate term
Extra content incorporates it into the structure, ignoring its (propositional logic) relationships
The positive couplet can be omitted, but the negative couplet cannot be omitted.
Classify propositions according to quantities
full name proposition
special proposition
There is a proposition, there is at least one individual
Therefore, if S is P, it cannot be inferred that S is not P.
From the weak principle
At least one, at most all
singular proposition
Refers to a proper noun or description, meaning "this, that"
Classification
The full name is affirmative proposition SAP (A)
All S's are P's
Full name Negative SEP (E)
All S's are not P's
Special name affirmation SIP (I)
Some S is P
Specially called negative SOP (O)
Some S is not P
singular
Singular affirmative SaP(a)
a is P
Singular negation SeP(e)
a is not P
Treated as a special case of universal proposition, it is easy to commit the fallacy of confusing concepts.
subject-predicate relationship
The subject and predicate terms only consider the denotation (the object, collection, or category referred to by the term) and do not study the connotation (the content and meaning expressed by the term).
Example: people
Connotation: Animals capable of thinking activities
Denotation: all people who have ever existed
The essence is the relationship between two non-empty sets. To deal with the relationship, you must first find the extension.
The reason for not considering the connotation: everyone has different understandings, different opinions, troublesome
denotative relationship
same relationship
S equals P
inclusion relationship
S contains P
included in
S is included in P (there is S in P)
cross
Some S are P, some S are not P
Completely different
The relationship between subject and predicate relative to the third concept (the three together are the complete set) can be divided into
contradictory relationship
opposition relationship
right relationship
opposition relationship
A and E
Can't be the same as true, can be the same as false
contradictory relationship
A and O
E and I
True and false cannot be the same, one must be true and the other false
SAP←→┓SOP
The same is true for the following
Differential relationship (subordinate relationship)
A and I
E and O
The universal true implies the specific true, the specific false implies the universal false
Lower opposition relationship
I and O
Can be both true and false
ductility
definition
Whether the given categorical proposition asserts (involves) all extensional properties of the subject or predicate
It is concluded that all extensions are distributed, otherwise they are not distributed.
4 types of proposition spread situation
A
Main week means not week
For example: All people are animals, so all animals are people
It does not appeal to all animals, but only to the part of all animals that are human beings.
E
Lord Zhou means Zhou
I
If the Lord is not Zhou, it is called Bu Zhou.
Some extensions of S and P are not mentioned.
O
If the Lord is not Zhou, then Zhou is called Zhou
Some people are not Peking University students, and some Peking University students are not human beings.
It did not sue whether the Peking University students were anything other than people. It only sued all the Peking University students and some people.
generalize
The full name is Zhuzhou, the special name is Zhubuzhou, it is definitely called Buzhou, and it is negatively called Zhou.
Personally think the importance
In daily language, the reasons for refuting the other party are mentioned or not.
2. Direct reasoning
definition
Inference that starts from a categorical proposition (premise) and derives another categorical proposition as a conclusion
Notice
Distinguish between P and ┓p
terms and propositions respectively
method
Substitution ("in other words")
Definition: Change a categorical proposition from affirmation to negation (qualitative), or negation to affirmation, and change the predicate to its contradictory concept (complement) to obtain an equivalent categorical proposition.
Features
The subject term remains unchanged, and the quantity term (full name, special term, singular term) remains unchanged.
Co-terms (yes, no, both, neither) and predicate terms become their own contradictory concepts
P changes to P, that is, the set of P becomes the complement set
The obtained new categorical proposition has the same truth value as the original categorical proposition
It cannot be simply represented by AEIO.
SAP←→SEP
All people are animals ←→ All people are not non-animals
SEP←→SAP
All people are not animals ←→ All people are non-animals
SIP←→SOP
Some S are P←→Some S are not non-P
SOP←→SIP
Some S are not P←→Some S are not P
transposition method
Definition: A new categorical proposition (conclusion) is obtained by exchanging the subject and predicate terms of a categorical proposition, keeping the quality unchanged, and changing the quantity terms.
If the items in the premise are not distributed, the conclusion must not be distributed.
Characteristics: The premise and conclusion are not necessarily equivalent, but the premise must not be less than the conclusion. That is, if the premise is not spread, the conclusion cannot be spread.
SAP→PIS
All S is P→Some P is S
SEP→PES
All S is not P → All P is not S
SIP→PIS
Some S is P→Some P is S
SOP cannot be transposed
Some S is not P → Some P is not S
After changing, they are not talking about the same thing, that is, the subject and predicate are interchangeable.
Some people are not college students → Some college students are not people ×
transposition method
Definition: First change the quality and then change the position to obtain a new categorical proposition
SAP→SEP→PES
SEP→SAP→PIS
SIP cannot change the quality position
SIP→SOP, SOP cannot be transposed
SOP→SIP→PIS
Substitution method (not necessarily equivalent)
SAP→SEP→PES→PAS
Where there is smoke, there is fire → Where there is smoke, there is fire → Where there is no fire, there is smoke → Where there is no fire, there is no smoke
There must be death
No death, no life
PAS→PES
SOP cannot be changed violently
PIS→POS
Thinking about switching from SAP to SOP, the premise is not distributed, but the conclusion is distributed. What's going on?
Correspondence reasoning
against relational reasoning
SAP→┓SEP
SEP→┓SAP
differential relation reasoning
SAP→SIP
SEP→SOP
┓SIP→┓SAP
┓SOP→┓SEP
contradictory relationship reasoning
SAP→┓SOP
SEP→┓SIP
SIP→┓SEP
SOP→┓SAP
┓SAP→SOP
┓SEP→SIP
┓SIP→SEP
┓SOP→SAP
Reasoning against relation
┓SIP→SOP
┓SOP→SIP
Reasoning about singular propositions
SAP→a is P
Be careful not to confuse concepts
All Chinese people are hardworking (people)
I am Chinese
I am hardworking (person)
Chinese people (collective concept) are hardworking
I'm not necessarily hardworking
a is P→SIP
Three, syllogism
definition
A syllogism is a reasoning in which two categorical propositions are connected by a common term and a new categorical proposition is drawn as a conclusion.
Composition (omitting joint terms, quantity terms, and case)
major premise
P (major term) common term M (middle term)
minor premise
S (small term) middle term M
in conclusion
Subject term S (minor term) Predicate term P (major term)
Usually the major premise involves the most content among the three. The conclusion of valid reasoning must not involve any more than the previous claim.
grid
Definition: Based on the different positions of the middle term in the premise, as well as the major premise above and the minor premise below, syllogisms are divided into four different types.
first grid
MP SM SP
The minor premise must be affirmed The major premise must be called in full
The middle letter can only be A or I, and the first letter can only be A or E.
AAA⁻1,AAI-1,AII-1,EAE-1,EAO-1,EIO-1
second grid
PM SM SP
Two premises must have one or not The major premise must be called in full
M are all predicates, so there must be a no, the conclusion is no, P is extended, and the major premise must be full The conclusion must be negative
AEE-2, AEO-2, AOO-2, EAE-2, EAO-2, EIO-2
third grid
MP MS SP
The minor premise must be affirmed The conclusion must be specific
Assume that the minor premise is false, then the conclusion is false and P weeks, then the major premise is P weeks, then the major premise is false, and both are false, then the minor premise must be affirmative Then the conclusion S is incomplete and must be specifically called
AAI-3,AII-3,EAO-3,EIO-3,IAI-3,OAO-3
fourth grid
PM MS SP
If the major premise is certain
then the minor premise must be fully named
If the minor premise is certain
then the conclusion must be specifically
If a premise is denied
Then the major premise must be fully named
EAO-4
AEO-4
If the major premise is special
then IAI-4 is required
If the minor premise is special
then EIO-4 is required
AAI-4,AEE-4,AEO-4,EAO-4,EIO-4,IAI-4
6 in each grid, a total of 24 valid equations, 9 of which contain valid equations with assumptions, and 6 difference equations (the conclusion can be universal but the specific one is obtained)
Mode
The total amount
4*4*4*4 cells=256
Definition: Syllogisms are divided into different types based on the quality and quantity of the three categorical propositions that constitute the syllogism.
Valid form
Measure to judge
rule
Illustration
Venn diagram or Euler diagram
axiomatic deduction
Based on the major premise (most situations) and the minor premise (specific situations), infer the conclusion (a small number of new situations)
rule
General rule (sufficient for all syllogisms)
Rule 1
In a syllogism, there are and can only be three different terms
The term "four concept errors" with more than three terms has multiple meanings
For example, Chinese universities are spread all over the country. Peking University is a university in China, so Peking University is spread all over the country.
Confusing the concept of the whole (collective) and the concept of the individual
Less than three terms "disguised syllogism"
It may be impossible to reason, the conclusion depends on the truth value of the premises
Rule 2
The middle term is extended at least once in the premise
As a bridge and medium, the middle term should create a certain relationship between the major premise and the minor premise to produce an inevitable result (conclusion). It is necessary that one of the two premises is a total relationship (distribution, this relationship occurs in any situation), and the other is a whole or partial relationship.
Error violating rule 2
The middle term is not spread out twice
The premise is true and the conclusion is true
By chance, the conclusion is true, but the reasoning form is invalid, which is not logical fidelity (the process is wrong and the result is right)
Conclusion false
Rule 3
Items that are not distributed in the premises must not be distributed in the conclusion.
Rule 3 violation error
Improper planning of major projects
Improper spread of minor items
Rule 4
Two negative premises cannot lead to any definite (inevitable) conclusion
There are many uncertain situations
Rule 5
If one of the premises is negative, the conclusion is negative If the conclusion is negative, then one of the premises must be negative
Rule 5 violation
The conclusion is in conflict with the premise The premise is yes and no, the conclusion is yes The premise is both yes, the conclusion is no
Derivation rules (for easy recognition and convenience)
Rule 6
Both premises cannot be specific
II,IO,OI,OO
Rule 7
If the premise has a special name, the conclusion must have a special name.
According to rule 6, one must be complete and one special
theorem
A correct syllogism with a complete conclusion in which a term cannot be extended twice
One word summary rebuttal: not necessarily
everyday language syllogism
standard form
First convert all premises and conclusions into standard form categorical propositions
Use contradictory relationships to deal with "not"
Note that double negatives express the affirmation "none...is not" and "all...are"
Distinguish between conclusion, major and minor premises, and middle term
The conclusion does not contain the middle term, please pay attention to the four-term error
Write in format
Determine whether the syllogism is valid
non-standard form
Elliptical form
Provincial major premise
minor premise
in conclusion
Completion
Compound
The syllogism contained in the premise needs to be sorted out and supplemented.
chained syllogism
Contains various syllogisms, the premise may be omitted from the middle conclusion
Many synonyms
Can be deformed by time, place and other parameters
4. The issue of the existential meaning of categorical propositions
SAP→SEP→PES→PAS→SIP→SOP
Violation of the rule that items in the premises should not be distributed and the conclusion should not be distributed
Reason: The logical implication of the term "existence hypothesis" is established, that is, the universal name can definitely deduce the specific name, which presupposes the existence of the subject (non-empty and non-complete set)
If the meaning of existence is removed, then the relationship between AEIO and Dang is no longer established.
A and E are no longer superiorly opposed. If S does not exist, A and E can be equally true (false premises imply any conclusion)
I and O no longer object. If S does not exist, they can be both false.
The restricted transposition method and restricted transposition method involved in A change to I are no longer valid.
In the syllogism, the 9 valid formulas that lead to the particular conclusion from the two universal premises are no longer valid.
The lexical logic categorical proposition AEIO contains S.
5. Graphical judgment of validity of syllogism
method
Euler diagram judgment method
No restrictions
Venn diagram judgment method
It is not assumed that the subject exists, and the 9 valid expressions of the universal name and the special name are invalid here.
If it is assumed that the subject exists, draw ⊕ to indicate non-empty
The three circles represent the subject, predicate, and middle terms respectively. Draw all the contents mentioned in the premise.
Give priority to drawing the full name of the proposition, and draw the shadow for the domain of discussion without the subject.
Special propositions are represented by " ". If you are not sure which side of the line to place, just draw " " on the line.
Chapter 4 Predicate Logic
derived reasons
Make up for the limitations of propositional logic and lexical logic, and be able to handle relational propositions and their reasoning, property propositions containing connectives in quantifiers and their reasoning (can handle properties and relationships)
area of research
Inferences based on connectives
Inferences based on quantifiers
Inferences based on both connectives and quantifiers
All propositions can be reasoned with predicate logic
Section 1 Individual words, property predicates, quantifiers and formulas
Proposition splitting in predicate logic
individual words
Symbols representing individuals in the object domain
individual variables
xyz, etc. represent an uncertain object within a specific range (domain of discourse or individual domain)
An n-element function containing n elements represents the relationship between individual variables.
For example, G (x, y) means that the relationship between x and y has G properties, and the binary function
individual constant
abc, etc.~determined objects
something with a proper name
The capital of a certain country F(x) The capital of China F (xᵃ)
Domain of discourse (individual domain)
Generally refers to the whole domain, that is, things that can be thought of and talked about in the world
Talk about everything in daily conversation, not a specific scope
If the domain of discourse is D, Vx is expressed as all the values of x in the domain of discourse D
predicate
Unary predicate (property predicate)
Predicate symbols, represented by uppercase letters
Represents the nature of an individual, with only one term
Two terms or more represent the relationship between them, n-ary predicate
Atomic formula
For example, F(a), G(x) means that a is F and x is G.
Multiple predicates (relational predicates)
Involving n objects, n>1
quantifier
Full name V
VxF(x) is read as "for all x, x is F"
∀xAx：Ax¹∧Ax²∧……∧Axⁿ∧……
All individuals with a certain attribute (F) in the domain of discourse
Existence∃
∃xF(x) is read as "x exists such that x is F"
∃xAx：Ax¹∨Ax²∨……∨Axⁿ∨……
There are individuals with certain attributes in the domain of discourse
connective word
Jurisdiction
Quantitative formula
Such as Vx(F(x)→G(x)) ∃xF(x)∧VyH(y)
The scope of quantifiers
If there are parentheses, take care of what is inside the parentheses. If there are no parentheses, just ignore the shortest formula next to it.
Such as VxF(x)∧G(x)
The scope of the quantifier Vx is F(x)
constraint variables
Formulas that appear with constraints
constraint appears
A certain occurrence of a variable is governed by a quantifier, that is, it appears within the scope
free variables
There are formulas that appear freely
Open formula
A formula containing at least one free variable whose true value cannot be determined
closed formula
A formula without free variables, determined by the interpreted truth value of a given universe of discourse and predicate symbols and constants
Individual variables can be both constrained and free at the same time
Symbolization of qualitative propositions in natural language
6 types of categorical propositions
Full name definitely SAP
Vx(S(x)→P(x))
subset relationship
SEP
Vx(S(x)→┓P(x))
SIP
彐x(S(x)∧P(x))
There exists x, x is S and x is P
intersection relationship
SOP
彐x（S（x）∧┓P（x））
There exists x, x is S and x is not P
a is P
P(a)
example
F(x): x’s father G(x): author of x Q(x): x is from the Qing Dynasty P (x, y): x is a textile official of y a: Cao Xueqin b: "A Dream of Red Mansions" c: Jiangning
The author of Dream of Red Mansions was from the Qing Dynasty
Q(G(b))
Cao Xueqin’s grandfather was an official in the Textile Administration of Jiangning
P(F(F(a)),c)
a is not P
┓a
┓P(a)
If the domain of discussion is determined to be a specific range, then we can only talk about the properties of individuals within the scope of the domain of discourse.
If everyone is not a plant, the domain of discussion is human beings
Vx┓S(x)
SEP abbreviation
For all individuals, if the individual is a human, then the individual is not a plant
Section 2: Relational predicates, overlapping quantification, properties of binary relationships
relational proposition
Conclude that there is a certain relationship between individuals
elements
individual words
relational predicate
Involving two or more individuals, more than two dyads
quantifier
First-order language L (first-order predicate logic language)
second-order predicate logic
Quantifier scope affects predicates, not just individuals
composition
initial symbol
individual variables
individual constant
predicate symbol
quantifier
connective word
auxiliary symbol
form rules
If A is a formula, A can be preceded by a quantifier Or A can be a quantifier (null constraint) Or if A contains a quantifier, it can be followed by a quantifier (repetition constraint)
VxA, 彐xA, A can be any formula
overlapping quantifiers
There are also quantifiers in the scope of quantifiers
Repeat bound individual words
Formulas containing overlapping quantifiers are called overlapping quantification formulas
Pay attention to distinguish between repeated quantization, overlapping quantization and null constraints
Repeated quantification means that multiple quantifiers constrain the same object (individual), only the latter one takes effect.
If 彐xVx彐xF(x) is equal to 彐xF(x)
An empty constraint means that the quantifier has no constraint object, which means it has no effect.
If VxF(y) is equal to F(y)
Vx彐yA cannot be changed to 彐yVxA
The jurisdiction has changed
Symbolization of relational propositions in natural language
For example, there is no largest natural number (referring to 0, 1, 2, 3...)
It is best to translate it into a formula that has no negative symbols and whose scope is clear at a glance.
It can be understood as "There is always a natural number larger than any natural number"
For any x, if x is a natural number, then there exists y such that y is a natural number and y is greater than x
The literal translation is that there is no largest natural number
everyone has parents
Everyone has people like their parents
Everyone has a father and a mother
Bad translation that does not express relationship: Vx (Hx→Px)
If John has a donkey, then John likes it
For any individual, if it is a donkey and is owned by John (a), then a likes it
Vx（Dx∧Hax→Lax）
Translated into existence, it implies free variables (which can be anything in the domain of discourse). It is inappropriate.
It is also inappropriate to translate existence as implying existence, indicating that the antecedent and the consequent are not related, and the donkey in the antecedent is not necessarily the donkey in the consequent.
Pay attention to express the relationship between the predicates, that is, expand the predicate symbols and write
Individual quantity problem
Quantifiers such as at least, exactly, at most, etc.
Use s≠t to express ¬ (s=t)
Logical properties of binary relationships Sorting problems
Different relationships of different natures
reflexive
x has an R relationship with itself x
Symmetrical
xy position can be changed
The relation R is symmetric if and only if, VxVy(R(x,y)→R(y,x))
transitive
There can be an R relationship between xyz and xyz
Section 3 Model and Assignment Universal Valid Formulas
L obtains meaning and truth value through M and assignment
Model M
Individual domain D
Given a non-empty set composed of individuals with certain properties
If the individual domain D is the global domain, then x is anything
An interpretive function I on D
I interprets the individual constant c in L (first-order language) as a specific individual I(c) in D, and the predicate symbol is interpreted as a set of individuals with certain properties in D
For example, in σ(F(t1,t2,t3...)), F represents the set of individual words in the following brackets
A closed formula (a formula without free variables) only has things (predicate symbols, quantifiers, constraint variables, individual constants), and the meaning and truth value are determined.
Assign a value to σ (only two values, true and false, T and F can be chosen)
Assign ρ: assign individuals in D to all free variables in L at once
(Specify who to send for what purpose)
Like Li Bai, it’s impossible to judge what Li Bai is without assigning him
σ＝<M,ρ>
Various formulas are true under σ if and only if
F(t¹t²…)
is true under σ if and only if t¹t²… does have an F relation (belongs to the set F)
σ<t¹t²…>∈σ(F)
VxA (think of formula A as a set)
A is always true after interpreting a freely occurring x in A as every individual word in the individual domain D
彐xA
Interpreting a freely occurring x in A as following an individual word in D makes A true
┓∧∨→←→The truth conditions are the same as propositional logic
Universally valid formulas (laws of predicate logic, also called often true formulas)
Give an example and try to explain
∀xF(x)→F(y)
F(y)→∃xF(x)
∀x(F(x)∨¬F(x))
¬∃x(F(x)∧¬F(x))
∀xFx↔¬∃x¬Fx
∃xFx↔¬∀¬Fx
∀x(Fx→Gx)→(∀xFx→∀xGx)
Why can't it be ↔?
If ten people pass the exam, we will invite them all to a meal (the requirements are more stringent). If you cannot decide who has passed the exam, we will invite them to a meal (the requirements are more relaxed).
In the latter case, the coach's promise to treat all ten people to dinner will be fulfilled only after all ten people have passed the exam. As long as one of them fails, you don’t have to cash it out. Of course, you can cash it out.
∀x（Fx∧Gx）↔（∀xFx∧∀xGx）
Why can't it be ∨?
If all people are male and female, it cannot be inferred that all of them are male or all of them are female.
∃x（Fx∨Gx）↔（∃xFx∨∃xGx）
∧?
Someone is both a boy and a girl. It happens that someone is a boy and someone is a girl. But some people are men and some people are women. It cannot be inferred that some people are both men and women.
∃x∀yRxy→∀y∃xRxy
Universal validity judgment problem
Predicate logic is undecidable. There is no universal way to determine all propositions, and it can only be determined locally.
Whether certain individuals whose causes are quantified have certain properties must be investigated one by one. If the individual domain is infinite, it will be difficult to find out unless one is found to be wrong. Overlapping quantification will be even more troublesome.
local judgment method
tree diagram
The 9 connective rules of propositional logic are still valid |The vertical bar indicates that new branches are obtained from all the branches upstairs
First use the connective rule, then use the quantifier rule with α requirement, and finally use the quantifier rule without α requirement If there is a requirement for α to be forked, it must be forked first.
Extended quantifier rules (Purpose eliminates quantifier)
∀ (Cannot tick, cannot exhaust examples, so can be used repeatedly)
: ∀xAx : | A(x/t), if t is free to substitute for x (t cannot be governed by any quantifier, If there is a quantifier in A, then t cannot be governed by A, That is, if there is a governed individual y in A, then t cannot be y)
If ∀xAx is true, then A is true for any individual in the individual domain In its individual domain, part of a group of individuals is true, several individuals are true, and a specific individual is also true.
¬∀(can be ticked, can only be used once, Examples can be found in the individual domain)
: ¬∀xAx : | ¬A (x/α) if α is a specific constant term (not sure which one it is yet) that has not appeared before in this branch (other branches are available) (to avoid the same individual being affected by multiple predicates)
If ¬∀xAx is true, then Ax is not true for at least some individuals in the individual domain (SOP)
∃⁻ (can be ticked... logic can only guarantee one example)
: ∃xAx : | A(x/α) if α is a specific constant that has not appeared before
¬∃(cannot exhaust examples)
: ¬∃xAx : | ¬A(x/t) if t substitutes freedom for x
When the tree diagram is not closed
Partial branch loop without contradiction (unary predicate)
Predictably satisfiable formula, that is, the original formula is not a universally valid formula
Not cyclic but branching infinitely (predicate of two or more elements) The original formulas that can be terminated are all valid formulas
It is impossible to judge whether there is a contradiction or not, and it is impossible to terminate the tree diagram, that is, it is not known whether the original formula is valid or not.
Explanation method (model method) with examples
An explanation is an assignment
σ: <<D, I>, ρ> That is, a model plus assignments
The proof is not universally valid and requires counterexamples (anti-models) To prove that the formula can be satisfied, you only need to give an example
To prove that it is not a universally valid expression, but that it is satisfiable, one counterexample and one positive example are required.
Proving universal validity requires that all logically possible explanations be true
The proof is not satisfiable and requires that all logically possible explanations are false
Only proof by contradiction (dendrogram available)
Assuming that the original formula is not valid/satisfiable, There is a counterexample that makes the original formula invalid It follows that there is no such counterexample
Pay attention to the points you need to make clear when explaining
Individual domain D
The meaning of constant symbols and predicate symbols I
If the open formula is involved, what free variable does ρ assign in D?
Natural deduction of predicate logic
QᴺInference Rules
is an expansion of Pᴺ
Pᴺ cannot handle the quantifier, so use Qᴺ to eliminate the quantifier, then Pᴺ handles the propositional connectives, and finally use Qᴺ to add the quantifier according to the desired look.
4 added quantifier rules
∀⁻
∀xAx┣A（x/t）
For example, ⱯxƎyRxy, t substituted into x cannot be y (restriction t is not governed)
The situation when t is not constrained
t is the individual constant
A is the atomic formula (without quantifier)
A is a content word, but x is not governed by A
A content word, and x is governed by A
t must be a variable other than the individual variables governed by A Otherwise, the substitution of t for x in A is not free (constrained)
∀
Ax┣∀xAx (x is any free variable)
If it cannot be ensured that the free variable x in the premise is arbitrary, Then you need to add a mark to x, indicating that the Ɐ rule cannot be used
Any such as Rxyz, it can be Rxxx, as long as x is not marked
What are free variables
Uncertain individual words in formulas without quantifier constraints
Like x in Rax x in Fx
Situations where free variables need to be marked
Free variables for a given premise
It is assumed that the free variables introduced
Free variables derived from premises or assumptions
There are free variables that specifically refer to constant terms as subscripts.
Without marking
Free variables obtained from Ɐ⁻
Ǝ⁻
ƎxAx┣A (x/α), α is a special constant term that has not appeared before If there is a free variable y other than x in A, mark y (same as above)
Ǝ
A(x/t)┣ƎxAx, t cannot be constrained
Note: Same as the tree diagram method, quantifiers with α requirements are deduced first, and those without α requirements are deduced later.
Deduction principle
What formula do you want to get at each step in the deduction process, and how do you get the conclusion from the premises?
The quantifier rule can only be used for the front end and the scope is the entire formula (this rule can only be used for the whole, the same as the Pᴺ rule)
Such as A→ⱯxⱯyⱯzB
Must be eliminated first→ Only when you get the necessary parts can youⱯ⁻ And it can only eliminate the outermost layerⱯ
If you want to get ⱯxⱯzB
First eliminate → then eliminate Ɐx, then eliminate Ɐy and finally introduce Ɐx
Be reliable and complete
All valid expressions ↔All Qᴺ theorems
That is to say, the formulas derived by Qᴺ are generally valid and can be used as derived rules.
Equivalent word theory and description analysis
1. Equivalent word theory
Reason for expanding L
"=" is commonly used in mathematics and natural language and is important.
Characteristic properties of words such as
reflexivity
Ɐx（x=x）
symmetry
ⱯxⱯy（x=y→y=x）
Transitivity
ⱯxⱯyⱯz（x=y∧y=z→x=z）
principle of indistinguishability
ⱯxⱯy（x=y→（Fx→Fy）)
Leibniz proposed
the principle of identity of the indistinguishable
ⱯxⱯy(（Fx↔Fy）→x=y)
Same as above
Uses of words such as
Can symbolize some natural languages
At least one x is F
ƎxFx
At least two x are F
ƎxƎy（Fx∧Fy∧¬（x=y））
There are two individuals that are F and they are different
At least three x are F
ƎxƎyƎz(Fx∧Fy∧Fz∧¬(x=z)∧¬(x=z)∧¬(y=z))
At most one x is F
ⱯxⱯy（Fx∧Fy→x＝y）
If there are two individuals, they are the same individual
at most two
ⱯxⱯyⱯz（Fx∧Fy∧Fz→（x＝y）∨（x＝z）∨（y=z））
For any z, either x is equal to y, or x is equal to y.
If there are three individuals, at least two of them are the same individual
at most n
In the same way, if there are n 1 individuals, at least two of them are the same individual
Exactly one x is F
at most one and at least one
ƎxFx∧ⱯxⱯy（Fx∧Fy→x=y）
Ǝx（Fx∧Ɐy（Fy→x＝y）) abbreviation
Exactly n
at most n and at least n
Li Qian has a pair of children
Li Qian: α Sxα: son of α Dyα: α’s female
ƎxƎy（Sxα∧Dyα∧Ɐz（Szα∨Dzα→（z＝x）∨（z＝y））)
There is such an individual x individual y, x is the son of α and y is the daughter of α, and for all z, if z is the son or daughter of α, then z and x are the same individual or z and y are the same individual
Chapter 5 Inductive Logic
definition
A knowledge system with inductive reasoning and inductive methods as its basic content
Compared
deductive reasoning
Fidelity and inevitability reasoning The conclusion concludes no more than the premises
There are supporting premises for inductive reasoning
inductive reasoning
probabilistic reasoning The conclusion asserts more than the premises
Classification
traditional inductive logic
individual experience rises to general knowledge of universal necessity
modern inductive logic
credibility, probability statistics
significance
Inspire people to boldly explore from the known to the unknown. Creation, invention, discovery, etc. are inseparable from inductive logic.
reasoning method
simple enumeration method
Definition: The part of an object that has been observed to have a certain property and no counterexamples have been encountered This leads to the conclusion that all objects of this type have this attribute.
Reliability requirements
The number of objects to be inspected must be sufficient
broad enough
The gap between objects is large enough
The very unreliable simple enumeration method is called
Oversimplification and hasty generalization
In essence, inductive reasoning is based on partial generalizations.
scientific induction
Observation plus scientific research is a deformation of simple enumeration.
Individual differences exist between scientific research and scientific research
Even if it sounds ugly, it can be divided into grades, depending on how scientific it is.
expression formula
All S observed so far are P, and scientific research shows that there is an inevitable connection between S and P Therefore, all S, whether observed or not, is P
complete induction
Investigate the quantity and distribution of simple enumeration methods to the extreme
Small range of application but reliable enough
Observed all S All S is P without counterexamples So all S are P
exclusionary induction
Ways to find cause-and-effect relationships (Designed based on the characteristics of causal relationship)
Seek common ground
Some phenomena appear sometimes and sometimes not. Due to their universality, cause and effect always accompany them. These phenomena are certainly not the causes of the phenomena under study
formula
Occasion 1 has the antecedent phenomenon ABC and the studied phenomenon a Occasion 2 has ABD, a 3 has ACE, a So A (probably) is the cause of a
advantage
Provides ideas for finding causal relationships and has a certain degree of reliability
shortcoming
Maybe they mistook the appearance for the cause and failed to discover the real cause behind it.
If it’s insomnia, look for the cause and the common ground I found someone who took a shower every day but things were different every day, but I ignored the excitement caused by various things.
How to avoid insomnia avoid or stop excitement
Find a different method
Occasion 1 There are ABCD and a Occasion 2 has BCD but no a So A is the cause of a
Commonly used in controlled experiments
Seek common ground and seek differences
Combining the above two, the two premises are put together to draw a conclusion.
Head situations (for example, there is an A) Tail situations (there is no A)
Covariation method (control variable method)
If both A and a change to a certain extent following one of them, there may be a causal relationship.
residual method
There is ABCDabcd Aa has a causal relationship Bb cc So Dd has a causal relationship
Characteristics of causal relationships
universality
coexistence
sequence
The cause is always first, the effect is always last. But it’s not necessarily the reason before, there may be other reasons Easy to confuse
How to avoid confusion
"Is this really the case? Is it possible? That’s it for the time being, but it’s hard to say in the future.”
complex diversity
There are multiple causes and one effect, one cause and one effect, one cause and many effects, etc. There are also primary causes and secondary causes, distal causes and proximate causes (direct causes, fundamental causes)
reasoning by analogy
A has attribute abcd Babc So B has d
Can make people draw inferences from one instance and gain inspiration or inspiration
Like Luban invented the saw
Very unreliable analogical reasoning is called
mechanical analogy Ridiculous analogy
Simulation method
model, modeling
comparison method
Compare the lists and find similarities and differences
Common mistakes
forced comparison, deceptive comparison Fake comparison, no comparison at all
hypothetical deduction
step
1. Starting point: problems and dilemmas
2. Forming a hypothesis: abductive reasoning
Phenomenon to be explained e if h, then e So h
e If h1 or h2 or...hn, then e Not h2 Not h3… So h1
3. Deducing observations from hypotheses
4. Testing hypotheses: confirmation and falsification
evaluation standard
conservatism
universality
simplicity
rebuttability
There must be empirical evidence and be in line with the world
Metaphysics has no empirical evidence
Modesty
Accuracy
After continuous confirmation or falsification, discard or modify
Credibility is getting higher and higher
Hume's induction problem
Is inductive reasoning sound?