## Franklin's Notes

### Syntax and Semantics

Syntax refers to the formalities governing how symbols may be combined in a formal language, whereas semantics refers to the meanings attributed to combinations of symbols. At its heart, Model Theory studies the relationship between syntax and semantics in general.

The following posts explore specific definitions of syntax and semantics in different contexts:

The symbol $\vdash$ always refers to a syntactical idea such as "tautological", i.e. "the truth value of this sentence is always true regardless of the truth values assigned to its symbols", whereas $\vDash$ always refers to a semantical idea such as "valid", i.e. "this sentence holds in every model".

For reference, here's a table of equivalent syntactical and semantical notions (in both sentential and first-order logic):

|Syntax|Semantics|
|-------|---------|
|$\vdash\varphi$ means $\varphi$ is a tautology, or a theorem|$\vDash\varphi$ means $\varphi$ is valid|
|$\psi\vdash\varphi$ means $\varphi$ is deducible from $\psi$|$\psi\vDash\varphi$ means that $\varphi$ holds in any model in which $\psi$ holds, or $\varphi$ is a consequence of $\psi$|
|$\Sigma\vdash\varphi$ means $\varphi$ is deducible from $\Sigma$|$\Sigma\vDash\varphi$ means that $\varphi$ holds in any model in which all sentences in $\Sigma$ hold, or $\varphi$ is a consequence of $\Sigma$|
|$\varphi$ (or $\Sigma$) is consisent |$\varphi$ (or $\Sigma$) is satisfiable |
|$\varphi$ is equivalent to a positive sentence|$\varphi$ is increasing, and not valid or refutable|
|$\varphi$ is equivalent to a conditional sentence|$\varphi$ is preserved under intersections|