实质条件

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• ${\displaystyle A\to B}$
• ${\displaystyle A\subset B}$
• ${\displaystyle A\Rightarrow B}$

真值表

${\displaystyle ~A}$ ${\displaystyle ~B}$ ${\displaystyle ~A\rightarrow ~B}$（符合了“如果A为真，那么B必为真”）
F F T
F T T
T F F
T T T

形式性质

• 如果${\displaystyle \Gamma \models \psi }$${\displaystyle \emptyset \models \phi _{1}\land \dots \land \phi _{n}\rightarrow \psi }$对于某些${\displaystyle \phi _{1},\dots ,\phi _{n}\in \Gamma }$。（这是演绎定理的特定形式。）
• 上述的逆命题
• ${\displaystyle \rightarrow }$${\displaystyle \models }$而二者都是单调的；就是说如果${\displaystyle \Gamma \models \psi }$${\displaystyle \Delta \cup \Gamma \models \psi }$，并且如果${\displaystyle \phi \rightarrow \psi }$${\displaystyle (\phi \land \alpha )\rightarrow \psi }$对于任何α, Δ。（用结构规则的术语说，这叫做弱化。）

• 分配律${\displaystyle A\rightarrow (B\rightarrow C)\rightarrow ((A\rightarrow B)\rightarrow (A\rightarrow C))}$
• 传递律：(${\displaystyle A\rightarrow B)\rightarrow ((B\rightarrow C)\rightarrow (A\rightarrow C))}$
• 幂等律${\displaystyle A\rightarrow A}$
• 真理保持:在其下所有变量被指派为真值‘真’的释义生成真值‘真’作为实质蕴涵的结果。
• 交换律：(${\displaystyle A\rightarrow (B\rightarrow C))\equiv (B\rightarrow (A\rightarrow C))}$

A → B

B → A

引用

• Brown, Frank Markham（2003）, Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
• Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
• Quine, W.V.（1982）, Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
• Stalnaker, Robert. 'Indicative Conditionals'. Philosophia 5（1975）: 269–286.