summarized syllogism pretty well I think

src/smt_solvers/front.md

@@ -4,6 +4,8 @@
 Unfiltered notes below:
 
 ## 1. A History of Satisfiability
+
+### 1.1 Encountering the definition of Satisfiability
 The history of satisfiability is best understood along its logic roots.
 
 In logic, syntax and semantics are distinct.
@@ -22,17 +24,88 @@ In proof theory, validity is called derivability. Consistency is just called con
 
 Derivability: $ q $ is derivable 
 from $ p_1, \dots, p_n $ (in symbols, $ p_1, \dots, p_n \vdash q $)
-if and only if you can syntactically prove you using inference rules.there is a proof in the axiom system.
+if and only if you can syntactically transform using inference rules of the system.
 
-In model theory, validity is just called validity. Consistency is called satisfiability. $ A \vDash p $ means that $ p $
+In model theory, validity is just called validity. Consistency is called satisfiability. $ A \vDash p $ means that $p$
 is true in the structure $ A $. 
 
-Satisfiability, or semantic consistency: $ \set{p_1, \dots, p_n} $ is satisfiable
-if and only if for some $ A, A \vDash p_1, \dots, A \vDash p_n $.
+Satisfiability: $ \set{p_1, \dots, p_n} $ is satisfiable if and only if for some $ A, A \vDash p_1, \dots, A \vDash p_n $.
+
+
+The two branches of logic are systematically related. Most languages have syntactic 
+and semantic concepts that coincide exactly ($\vdash$ corresponds to $\vDash$) — such an axiom system is said to be complete.
+
+### 1.2 The ancients
+Aristotle invented syllogistic or categorical logic. The building block of the logic is categorical propositions, which
+is a subject-predicate sentence appearing in one of four forms: every $S$ is $P$, no $S$
+is $P$, some $S$ is $P$, and some $S$ is not $P$.
+
+
+<!-- Since Aristotle viewed language as expressing ideas that signified entities and the properties they instantiate in 
+the "world", these *categorical* propositions are "evaluated" in some world $w$,
+so this is the model theory and semantical part of the logic. -->
+A syllogism has the syntax $(p, q) \to r$ where $p, q, r$ are all categorical propositions.
+$p$ is the major premise, $q$ is the minor premise, and $r$ is the conclusion. 
+
+To show the validity of a syllogism, we need to be able to derive $r$ from $(p, q)$, using the following rules:
+
+$$
+\begin{align*}
+\text{Axiom 1:} & & (\text{every }X\text{ is }Y \land \text{every }Y\text{ is }Z) & \to \text{every }X\text{ is }Z \\
+\text{Axiom 2:} & & (\text{every }X\text{ is }Y \land \text{no }Y\text{ is }Z) & \to \text{no }X\text{ is }Z \\
+\end{align*}
+$$
+
+and
+
+$$
+\begin{align*}
+
+\text{Rule 1:} & \text{ From } & (p \land q) \to r         & \text{ infer } & (\lnot r \land q) \to \lnot p \\
+\text{Rule 2:} & \text{ From } & (p \land q) \to r         & \text{ infer } & (q \land p) \to r \\
+\text{Rule 3:} & \text{ From } & \text{no }X\text{ is }Y    & \text{ infer } & \text{no }Y\text{ is }X \\
+\text{Rule 4:} & \text{ From } & (p \land q) \to \text{ no }X\text{ is }Y & \text{ infer } & (p \land q) \to \text{ some }X\text{ is not }Y
+\end{align*}
+$$
 
+The axioms are valid syllogisms. The inferenced rules derive one syllogism from another. A prototypical example of a derivation follows:
+
+1. All men are mortal
+2. Socrates is a man
+3. Socrates is mortal
+
+First, let's convert these to categorical propositions:
+1. Every man (Y) is mortal (Z)
+2. Every thing identical to Socrates (X) is men (Y)
+3. Every thing identical to Socrates (X) is mortal (Z)
+
+The categorical expressions are less natural, but this is what it takes to lift singular terms into categorical proposition form.
+After conversion, our starting form almost matches Axiom 1 already. We have 
+$$(\text{every }Y\text{ is }Z \land \text{every }X\text{ is }Y) \to \text{every }X\text{ is }Z$$
+, applying Rule 2, we get
+
+$$(\text{every }X\text{ is }Y \land \text{every }Y\text{ is }Z) \to \text{every }X\text{ is }Z$$
+
+and hence our syllogism is valid by Axiom 2.
+
+
+The syllogistic logic is intimately related with natural language — categorical propositions 
+deal with subjects and predicates by default. This logic is less abstract than first-order logic
+but aligns with our natural reasoning a bit more.
+
+**More interestingly, we may use set theory to provide the semantics of syllogistic logic.
+Then, due to the fact that syllogisms always come in 3 categorical propositions, we can use Venn diagrams to prove things (see [this](https://phil240.tamu.edu/LectureNotes/6.3.pdf))!**
+Previously, we proved validity (derivability in proof theory) using axioms and inference rules, all of which were syntactic.
+However, with set theory and Venn diagrams, the semantic validity is visual!
+
+Usually, we don't get such nice visual semantics, but with syllogisms coming in precisely 3 terms, we can always use Venn diagrams.
+In general, proving is done with inference rules (syntactic), as they are mechanized and efficient. The underlying, semantic model of things
+contains the spirit of the logic. They give meaning to our logic and guide us in designing syntactic rules. Such semantics could be set-theorectical (as for
+syllogistic logic), category theorectic (used in type theory), and operational and denotational (used in programming language).
 
-An axiom system is 
 
 
 ## Resources
-Handbook of satisfiability
+Handbook of satisfiability
+
+[Syllogism and Venn Diagrams](https://phil240.tamu.edu/LectureNotes/6.3.pdf)