1. 1 CA 208 Logic Ex3 Define logical entailment  in terms of material implication  Define logical consequence |= (here the semantic consequence relation between a set of premises and a conclusion) in terms of logical entailment  and then in terms of material implication 

2. 2 CA 208 Logic Ex3 PQR (P  Q)(Q  R)(P  Q)  (Q  R)(P  R)((P  Q)  (Q  R))  (P  R) 111 110 101 100 011 010 001 000 Show that {P  Q, Q  R} |= P  R, i.e. ((P  Q)  (Q  R))  (P  R)

3. 3 CA 208 Logic Ex3 Complete the following definition of the syntax of propositional logic with negation, conjunction, disjunction, material implication and the bi-conditional: Let Π be a (coutably infinite...) set of propositional variables Π = {A, B, C,...} (this is the lexicon, the basic building blocks..) If Φ  Π, then Φ is a............... If Φ is a formula, then...... is a formula If Φ and Ψ are formulas, then........ is a formula If Φ and Ψ are formulas, then......... is a formula Nothing else is a formula.

4. 4 CA 208 Logic Ex3 Complete the following definition of the semantics (the meaning M..) of propositional logic (Tarski-style): Let V be a valuation, i.e. an assignment of truth values to each propositional variable in Π: (formally) V:Π  {0,1} (V is a total function from Π to {0,1}) If Φ  Π, then M(Φ) =........ M(  Φ) = 1 iff............ M(Φ  Ψ) = 1 iff.............................................. M(Φ  Ψ) = 1 iff.............................................. M(Φ  Ψ) = 1 iff.............................................. M(Φ  Ψ) = 1 iff..............................................