# Syllogism

THEOREM (Modus Barbara)
 $$\forall x : M(x) \Rightarrow P(x)$$ All $$M$$ are $$P$$, $$\forall x : S(x) \Rightarrow M(x)$$ and all $$S$$ are $$M$$; $$\forall x : S(x) \Rightarrow P(x)$$ thus all $$S$$ are $$P$$.
Proof.
THEOREM (Modus Celarent)
 $$\forall x : M(x) \Rightarrow \neg P(x)$$ No $$M$$ is $$P$$, $$\forall x : S(x) \Rightarrow M(x)$$ and all $$S$$ are $$M$$; $$\forall x : S(x) \Rightarrow \neg P(x)$$ thus no $$S$$ is $$P$$.
Proof.
THEOREM (Modus Darii)
 $$\forall x : M(x) \Rightarrow P(x)$$ All $$M$$ are $$P$$, $$\exists x : S(x) \wedge M(x)$$ and some $$S$$ are $$M$$; $$\exists x : S(x) \wedge P(x)$$ thus some $$S$$ are $$P$$.
Proof.
THEOREM (Modus Ferioque)
 $$\forall x : M(x) \Rightarrow \neg P(x)$$ No $$M$$ is $$P$$, $$\exists x : S(x) \wedge M(x)$$ and some $$S$$ are $$M$$; $$\exists x : S(x) \wedge \neg P(x)$$ thus some $$S$$ are not $$P$$.
Proof.
THEOREM (Modus Baroco)
 $$\forall x : P(x) \Rightarrow M(x)$$ All $$P$$ are $$M$$, $$\exists x : S(x) \wedge \neg M(x)$$ and some $$S$$ are not $$M$$; $$\exists x : S(x) \wedge \neg P(x)$$ thus some $$S$$ are not $$P$$.
Proof.
THEOREM (Modus Bocardo)
 $$\exists x : M(x) \wedge \neg P(x)$$ Some $$M$$ are not $$P$$, $$\forall x : M(x) \Rightarrow S(x)$$ and all $$M$$ are $$S$$; $$\exists x : S(x) \wedge \neg P(x)$$ thus some $$S$$ are not $$P$$.
Proof.
THEOREM (Modus Barbari)
 $$\forall x : M(x) \Rightarrow P(x)$$ All $$M$$ are $$P$$, $$\forall x : S(x) \Rightarrow M(x)$$ and all $$S$$ are $$M$$, $$\exists x : S(x)$$ and some $$S$$ exist; $$\exists x : S(x) \wedge P(x)$$ thus some $$S$$ are $$P$$.
Proof.
THEOREM (Modus Celaront)
 $$\exists x : M(x) \wedge \neg P(x)$$ No $$M$$ is $$P$$, $$\forall x : S(x) \Rightarrow M(x)$$ and all $$S$$ are $$M$$, $$\exists x : S(x)$$ and some $$S$$ exist; $$\exists x : S(x) \wedge \neg P(x)$$ thus some $$S$$ are not $$P$$.
Proof.
THEOREM (Modus Camestros)
 $$\forall x : P(x) \Rightarrow M(x)$$ All $$P$$ are $$M$$, $$\forall x : S(x) \Rightarrow \neg M(x)$$ and no $$S$$ is $$M$$, $$\exists x : S(x)$$ and some $$S$$ exist; $$\exists x : S(x) \wedge \neg P(x)$$ thus some $$S$$ are not $$P$$.
Proof.
THEOREM (Modus Felapton)
 $$\forall x : M(x) \Rightarrow \neg P(x)$$ No $$M$$ is $$P$$, $$\forall x : M(x) \Rightarrow S(x)$$ and all $$M$$ are $$S$$, $$\exists x : M(x)$$ and some $$M$$ exist; $$\exists x : S(x) \wedge \neg P(x)$$ thus some $$S$$ are not $$P$$.
Proof.
THEOREM (Modus Darapti)
 $$\forall x : M(x) \Rightarrow P(x)$$ All $$M$$ are $$P$$, $$\forall x : M(x) \Rightarrow S(x)$$ and all $$M$$ are $$S$$, $$\exists x : M(x)$$ and some $$M$$ exist; $$\exists x : S(x) \wedge P(x)$$ thus some $$S$$ are $$P$$.
Proof.