模态逻辑，或者叫（不很常见）内涵逻辑，是处理用模态如“可能”、“或许”、“可以”、“一定”、“必然”等限定的句子的逻辑。模态逻辑可以用语义的“内涵性”来描述其特征：复杂公式的真值不能由子公式的真值来决定的。允许这种决定性的逻辑是“外延性的”，经典逻辑就是外延性的例子。模态算子不能使用外延语义来形式化：“乔治·布什是美国总统”和“2 + 2 = 4”是真的，但是“乔治·布什必然是美国总统”是假的，而“2 + 2 = 4是必然的”是真的。

形式模态逻辑使用模态判决算子表示模态。基本的模态算子是和。（有时分别使用“L”和“M”）。它们的意义依赖于特定的模态逻辑，但它们总是以相互定义的方式来定义：

Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. Modals—words that express modalities—qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p"). Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"), deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p") and doxastic modalities, or modalities of belief ("It is believed that p").