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In the zero-based introduction to logic, there are three modules that must be mastered. These three modules are:
▶Inference Connectives
▶Three Major Rules of Inference for Conditional Propositions
▶Equivalence Transformation of Disjunctive and Conditional Propositions
These three modules are extremely important, akin to the “Nine-Character Multiplication Table” in mathematics, serving as a crucial foundation for advancing to higher levels of logic!
Inference Connectives
The everyday expressions for A→B include:
·····is·····
As long as (if, once)·····then·····
If······then······
If·····then·······
In order to·····must······
All······are·······
Cannot be separated from
Practice:
(1) If it rains, then the ground is wet. (It rains → the ground is wet)
(2) If you go, I will also go. (You go → I go)
(3) If we win, we can have a happy life. (Win → Happy life)
(4) If heaven has feelings, it will also grow old. (Heaven has feelings → Heaven grows old)
(5) As long as you work hard, you will definitely succeed. (Work hard → Success)
(6) All merchants are deceitful. (Merchants → Deceitful)
(7) If you don’t reach the Great Wall, you are not a true hero. (Don’t reach the Great Wall → Not a true hero)
(8) The harder you work for the management exam, the luckier you will be. (The harder you work → The luckier you will be)
(9) Arrogant soldiers must lose. (Arrogant soldiers → Lose)
(10) If you study hard, you will achieve good results and get into college. (Study hard → Good results and get into college) = (Study hard → Good results and get into college)
The everyday expressions for B→A include:
·····only (then)·······
Unless·····only·······
Must······only·······
Not·······not······
No·····then no·····
······is·····indispensable (essential, necessary, foundational, prerequisite)
Practice:
(1) Only when you are 18 years old do you have the right to vote. (Right to vote → 18 years old)
(2) Studying hard is the foundation/prerequisite/necessary condition for getting into college. (Getting into college → Studying hard)
(3) Unless we are humble, we will not make progress. (Not humble → Will not make progress)
(4) It is difficult for you to get into college unless you listen carefully. (Not listening carefully → Difficult to get into college)
(5) You must follow my instructions, otherwise, there will be no salary. (Not following my instructions → No salary)
(6) Unless you go, I will go. (I go → You go)
Note: Unless you go, I will go = Only if you go, I will go.
(7) If the mountains do not have ridges and the waters do not flow, then I dare to sever ties with you. (Only if the mountains do not have ridges and the waters do not flow, then I dare to sever ties with you = Sever ties with you → (Mountains do not have ridges and waters do not flow).
The Three Major Rules of Inference for Conditional Propositions
Contrapositive Rule:
A→B=B→¬B→¬A
For example:
If it rains, then the ground is wet (It rains → the ground is wet);
If the ground is not wet, then it must not have rained (¬Ground wet → ¬It rains).
Transitive Rule:
A→B, B→C, C→D = A→B→C→D
For example:
If there is no sky, there is no earth; if there is no earth, there is no home.
No sky → No earth, No earth → No home.
No sky → No earth → No home
Contrapositive + Transitive Rule:
¬C→¬B→¬A = A→B→C
For example:
No sky → No earth → No home = If there is a home, there is earth, and there is a sky.
Equivalence Transformation of Disjunctive and Conditional Propositions
The equivalence replacement of “or” and “then” is the equivalence transformation of disjunctive and conditional propositions, indicating that the following equivalences hold:
(A∨B) = (¬A→B)
(A→B) = (¬A∨B)
Practice:
(1) At least one of Li and Zhao will be promoted.Li∨Zhao=¬Li→Zhao
(2) At most one of Li and Zhao will be promoted.¬Li∨¬Zhao=Li→¬Zhao
(3) Unless Li is not promoted, Zhao will be promoted.Li→Zhao=¬Li∨Zhao
(4) If Li is promoted, then Zhao cannot be promoted.Li→¬Zhao=¬Li∨¬Zhao
(5) Only if Li is promoted, will Zhao be promoted.Zhao→Li=¬Zhao∨Li
(6) Unless both Li and Zhao are promoted, Zhang will not be promoted.Zhang→(Li∧Zhao)=¬Zhang∨(Li∧Zhao)
The blessing of students in WWII
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