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.    

See also

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