I am playing with Sequent Calculus(Just started studying it) and I want to know how to express the Syllogism:
Socrates is a Man.
All Men are Mortal.
Therefore Socrates is a Mortal.
In terms of Sequent Calculus.
My attempt so far is as follows:
1.
There is a set Mortals and Men, so the following statement should be axiomatically true. $$ \overline{\Gamma, Mortals, Men \vdash Mortals, Men} $$
Which is the same as saying
$$ \overline{\Gamma, Mortals, Men} $$
2.
"All Men are Mortals" is a part of my assumptions, so the following statement seems to express it but I am having doubts.
$$ \overline{\Gamma, Mortals \rightarrow Men} $$
3.
"Socrates is a Mortal" is another assumption that must be a part of the environment.
$$ \overline{\Gamma, Mortals \rightarrow Socrates} $$
This looks weird, since it looks like I'm saying that all mortals are socrates, but I am trying to say that Socrates is in Mortals(by turning the turnstile into implication), but I am not sure whether $\vdash$ is the same as $\rightarrow$ except can be nested inside a sequent, or if there's another way to go around expressing this.
4. Finally, I want to say that if Socrates is in my environment, then Socrates is a Mortal,
$$ \overline{\Gamma, Socrates} $$
5.
but at this point I have no idea what's going on, but Ill try to put it together anyway:
$$ \overline{\Gamma, Socrates \vdash Mortal}\\ \overline{\Gamma, Men \rightarrow Socrates, Mortal \rightarrow Men} $$
Can somebody look over what I did and help me gain some confidence in how to express it correctly(If it is possible to express it in Sequent Calculus)?
It seems to me that we need this:
1) $Man(S) \to Man(S)$ --- top left
2) $Mortal(S) \to Mortal(S)$ --- top right
3) $Man(S) \supset Mortal(S), Man(S) \to Mortal(S)$ --- from 1) and 2) by $\supset$-left
4) $\forall x(Man(x) \supset Mortal(x)), Man(S) \to Mortal(S)$ --- from 3) by $\forall$-left
5) $\to \forall x(Man(x) \supset Mortal(x))$ --- "all men are mortals"
6) $Man(S) \to Mortal(S)$ --- from 4) and 5) by Cut
7) $\to Man(S)$ --- "Socrates is a man"