Propositional Logic

Propositional-Logic

Declarative sentence

  • A declarative sentence (陈述句) is a sentence that aims to declare a fact.

    eg: Today is Saturday.

  • 疑问句/祈使句…(不是)

Proposition

  • A proposition is a declarative sentence that is either true or false.

Propositional Variables

  • Use variables (English letters) to represent propositions.

Truth Value

  • Assigned to propositional variables.

    True: represented by T False: represented by F

    Propositional Operators

  • Logical negation (indicated by ¬ , ! , or ⁻) 非
  • Logical conjunction (indicated by ∧, &&, or ·) 与 合取
  • Logical disjunction (indicated by ∨,   , or +) 或 析取
  • Logical exclusive disjunction (indicated by ⊕) 异或

Logical implication (indicated by →)

\[p→q\]
  • p when q img_1

p is the sufficient condition for q; q is the necessary condition for p * False always true

  • Contrapositive(质位变换命题, 逆否命题: $\neg q\rightarrow \neg p$
  • Converse (逆命题):$q\rightarrow p$
  • Inverse (否命题):$\neg p \rightarrow \neg q $

Logical bidirectional implication (indicated by ↔)

\[p ↔ q\]
  • p if and only if q
  • if p then q img_2

Compound(复合)Propositions

  • A compound proposition involves a number of propositional variables and logical operators. img_3 $(p\wedge q)\vee r$ does not mean $p \wedge (q\vee r)$

Specification Consistency

  • Equivalently, if a specification {$p_1,p_2,…p_n$} is consistent, then the compound proposition $p_1\wedge p_2\wedge…\wedge p_n$ is true for at least one assignment
  • If the specification is inconsistent, then $p_1\wedge p_2\wedge…\wedge p_n$ is always false. 1

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