Question 1:First-Order Logic

1.Translate the sentences below from English into first-order logic.For each translated sentence provide an explanation of why your first-order sentence captures its En- glish counterpart.

Use the signature S consisting of the unary predicate symbol Region and the binary predicate symbols Disjoint,Included and Overlap and the constant symbols dorset,fife, scotland,england.

(a)Any two regions are either disjoint,overlapping,or one of them is included in another.                  [0.5  marks]

(b)Every region is included in itself.                                                          [0.5  mark]

(c)If two regions are disjoint,they are not overlapping.                           [0.5  marks]

(d)If two regions are overlapping,none of them is included in another. [0.5  marks]

(e)If one region is included in another,then they are not disjoint.           [0.5  marks]

(f)If two regions are disjoint,then any region included in the first one is disjoint from the second one.              [0.5  marks]

(g)Dorset and England are regions,Dorset is included in England.       [0.5  marks]

(h)Fife and Scotland are regions,Fife is included in Scotland.                [0.5  marks]

(i)Scotland and England are disjoint.                                                    [0.5  marks]

2.Do the sentences from Part 1 above logically entail that Dorset does not overlap with Scotland? Justify your answer by providing a proof using precise semantic argu- ments or by providing a counter-example. [4.5 marks]

3.Does hold? Justify your answer by  providing a proof using semantic arguments or by providing a counter-example. [4 marks]

4.Does hold? Justify your answer by provid- ing a proof using semantic arguments or by providing a counter-example. [4 marks]

5.We want to prove that the following argument is true:

If all quakers are reformists and if there is a protestant that is also a quaker,then

there must be a protestant who is also a reformist.

Define a set of FOL sentences X and a sentence G capturing this argument and show  that  X G using  semantic  arguments.Justify  why  X  and  G  properly  capture the argument.

Hint: You can define the FOL sentences in X and the sentence G using the unary

predicate symbols:Quaker,Reformist and Protestant. [8 marks]