Q1. [10 ] Augment the signature cfw_, by and prove the completeness and soundness of the calculus obtained by supplementing the basic rules used so far with the rules:
(1)
X ; X ,
(2)
X, | X, X,
Q2. [10 ] Prove: (Finiteness Theorem for |=) If X |=

Once again, we visit several Islands in the Archipelago of Knights and Knaves along with our Antropologist. In these islands, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave.

Once again, we visit several Islands in the Archipelago of Knights and Knaves along with our Antropologist. In these islands, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave.

Once again, we return to the Island of Knights and Knaves. In this island, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Notation: K = knight. For any inhabitant x, we let

Once again, we return to the Island of Knights and Knaves. In this island, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Notation: K = knight. For any inhabitant x, we let

Once again, we return to the Island of Knights and Knaves. In this island, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [10 ] Once again, we return to the Island of K

Once again, we return to the Island of Knights and Knaves. In this island, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [10 ] Once again, we return to the Island of K

Suppose I put a penny and a gold ducat down on the table and ask you to make a statement if your statement is true, I must give you one of the coins (not saying which); if it is false, I give you neither. Q1. [10 ] What statement can you make so that

Suppose I put a penny and a gold ducat down on the table and ask you to make a statement if your statement is true, I must give you one of the coins (not saying which); if it is false, I give you neither. Q1. [10 ] What statement can you make so that

As we approach end of this class, we all visit the island of knights and knaves. We are now proud to be well-educated and perfectly accurate logicians: that is, anything we prove is really correct we never prove anything false. Recall that on this isl

Once again, we visit the Island of Knights and Knaves along with our Antropologist. In these islands, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Notation: k = knight, k

Once again, we visit the Island of Knights and Knaves along with our Antropologist. In these islands, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Notation: k = knight, k

Q1. [10 ] The axioms of PA in L ar := Lcfw_0, S, +, are as follows:
x Sx = 0 x x + 0 = x x x 0 = 0 xy (Sx = Sy x = y) xy x + Sy = S( x + y) xy x Sy = x y + x x x 0 x ( Sx ) x ( IS)
Prove in PA the associativity, commutativity, and distributivity of

In the Island of Knights and Knaves, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [10 ] An anthropologist, visiting this island, came across three inhabitants, whom h

In the Island of Knights and Knaves, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [10 ] An anthropologist, visiting this island, came across three inhabitants, whom h

Recall, from the last lecture, in the Island of Knights and Knaves, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [5 ] An anthropologist met just two inhabitants, A an

Recall, from the last lecture, in the Island of Knights and Knaves, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [5 ] An anthropologist met just two inhabitants, A an

Once again, we return to the Island of Knights and Knaves along with our Antropologist. In this island, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [15 ] Rumor has i

Once again, we return to the Island of Knights and Knaves along with our Antropologist. In this island, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave. Q1. [15 ] Rumor has i

Once again, we visit several Islands in the Archipelago of Knights and Knaves along with our Antropologist. In these islands, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave.

Once again, we visit several Islands in the Archipelago of Knights and Knaves along with our Antropologist. In these islands, those called knights always tell the truth and knaves always lie. Furthermore, each inhabitant is either a knight or a knave.

As we approach end of this class, we all visit the island of knights and knaves. We are now proud to be well-educated and perfectly accurate logicians: that is, anything we prove is really correct we never prove anything false. Recall that on this isl