We now extend [[L1 (FOL)|L1]] to $L_{2}$ by adding **quantifiers**. We define $L_{2}$ by its symbols, syntax (grammar) & semantics: # Symbols > [!NOTE] Symbols of $L_{2}$ >1. Individual constants: $a,b,c,a_{1},b_{1},\dots$ >2. Variables: $x,y,z,x_{1},y_{1},\dots$ >3. Predicates: $F(x),R(x,y),\text{Smiling}(x)$ >4. Sentence letters: $A,B,C,P,Q,R,P_{1},P_{2},\dots$ >5. Connectives: >6. Quantifiers: >7. Brackets # Syntax > [!NOTE] Definition (Atomic well formed formulas (wff)) > 1. Any predicate of arity $n$ combined with $n$ occurrence of constants or variables is an atomic wff. > 2. All sentence letter are atomic wffs. > [!NOTE] Definition (wff of $L_{2}$) > Contents # Semantics The semantics for $L_{2}$ requires an interpretation: - a non-empty domain of discourse (a [[Sets|set]] of objects); - an assignment of objects from the domain to the constants (names); - a pairing of $n$-place predicates with $n$-place properties or relations; - an assignment of truth values to the sentence letters. An **$L_{2}$ structure** is an abstract situation, similar to the notion of a *possible situation* we have for $L_{1p}$ (i.e., assignments of truth values to the sentence letters. They too are often called structures – $L_{1p}$ structures): - a non-empty domain of discourse (a non-empty set of objects) - an assignment of objects from the domain to the constants, - an assignment of a set of ordered $n$-tuples of objects from the domain to each $n$- place predicate. - an assignment of truth values to the sentence letters. > [!Example] > | $\forall x(\neg x=a\to\neg R(x,x))$ > |----- > |$\neg\exists xR(x,x)$ > >The following structure shows that the above argument is not [[Logically Valid Argument|logically valid]]. > > Domain: $\{ 1 \}$ > $a: \quad 1$ > $R(x,y): \{ <1,1> \}$ # Deductive System See [[Fitch System for L2]]. # Applications - [[Logical Equivalences in L2]]. - [[Enumeration using L2]]. - [[Definite descriptions]]. - [[Truth Functionally Complete Set of Connectives]]. - [[Zermelo Frankel set theory (ZFC)]].