Formal Semantics Purpose: formalize correct reasoning

Propositions Statements that are the basic element of reasoning – Defined in different kinds of logic (contexts): e.g., RDF triples; first order predicate logic The set of all propositions ( P ) – P has many specific propositions: p  P – Some propositions (e.g., p3 and p4) may be the logical consequences of other propositions (e.g., p1 and p2) {p1, p2} {p3, p4} Where is the entailment relation, relating sets of propositions to each other A logic (L) is composed of a set of propositions ( P ) with the entailment relation ( ), i.e., L = ( P, )

Model Theoretic Semantics Interpretation, I (think of it as potential ‘realities’ or ‘worlds’) In each type of logic, we use certain mathematical structures as interpretation We want to know if a specific interpretation (I) satisfies a specific proposition p  P (i.e., I p) which reads: ‘I model of p’ I is a model of P (i.e., I P) if it is a model for every p  P Let P and P’ be subsets of set P (i.e., P  P and P’  P ) P’ is entailed by P (i.e., P P’) iff every interpretation (I) satisfying all individual sentences (p) of P (i.e., I P) is also a model of (i.e., satisfies) every sentence p’ from P’ (I P’), for example: igneous_rock rocki.e., if proposition igneous rock entails rock, then If interpretation I satisfies igneous rock (I igneous rock), it must also satisfy (be a model of) the proposition rock (i.e., I rock)

Entailment relation via models logical entailment (I p) reads: I is a model of p, i.e., I satisfies p p 1 : volcanic p 2 : clastic p 3 : volcaniclastic {p 1, p 1 } {p 3 } Models of p 1 (I p 1 ) Propositions Interpretations Models of p 2 (I p 2 ) Models of p 3 (I p 3 ) {p 1, p 2 } {p 3 } iff (I p 1 and I p 2 ), I also satisfies p 3, i.e., I p 3 That is, p 1, p 2 entail p3 Iff the interpretation I satisfies (i.e., is a model of) both p 1 and p 2 as well as p 3.

Simple Interpretations Vocabulary (V) is an arbitrary set of URIs and literals RDF triples relate resources (R) via properties (P) An interpretation (  ) of triples has two sets:  R and  P  R: non-empty set of resources (domain or universe of discourse)  P: the set of properties of  (may overlap  R)  EXT a function that assigns a set of pairs from  R to each property I S a function mapping URIs from V into the union of the sets  R and  P.  L a function from the typed literals from V into the set of  R LV a particular subset of  R called the set of literal values

Interpretation function. 