1

"j,
[j®y] So, y"
is a valid argument form. That is, any argument which results from
replacing j and y
with declarative sentences is a valid argument. You can see that this
is so by looking at the truthtable for [j®y]: whenever it is true and j
is true, y is also true.
(This argument form is often called "modus ponens" – or
"modus ponendo ponens".)
Which of the
following are also valid argument forms? 

(i)

¬y,
[j®y] So, ¬j 

(ii) 
y,
[j®y] So, j 

(iii)

¬j,
[jÚy] So, y 

(iv)

j,
¬[jÙy] So, ¬y. 
2

In
each of the following, say whether the second sentence is an acceptable
translation of the first. (If it isn't, think of situation in which
the sentences would have different truthvalues.) 

(i)

Liverpool
will win the cup just if they beat Tranmere Rovers. [Liverpool will
win the cup « Liverpool will
beat Tranmere Rovers]. 

(ii) 
A
team will win the cup if and only if it wins every round. [A team
will win the cup « it will
win every round]. 

(iii) 
Manchester
United will win the cup only if they beat Valencia. [Manchester United
will win the cup « Manchester
United will beat Valencia]. 

(iv)

Either
Manchester United or Valencia will win the cup, but not both. [Manchester
United will win the cup « ¬Valencia will win the cup]. 

(v) 
John
will not come unless Mary comes. [John will come « Mary comes]. 
3

Let
us abbreviate John will come as "J" and Mary will come as
"M". In the following questions you are given a sentence
made up of "J" and "M" and one or more of the
truthfunctors of our propositional language, and are asked to provide
sentences which are equivalent but use a different selection of truthfunctors.
(Sentences of this sort, which consist of just capital letters and
truthfunctors of the prepositional language are called formulae.
And the process of translating sentences into formulae is called
formalising.)There will be many possible answers – though some
will be simpler and more obvious than others. So, if your answer is
different from the one given, and you are in doubt about whether it
is also right, use a truthtable to check it. 

(i) 
Give
a formula which is equivalent to [J®M] using just Ù
and ¬ (together with "M", "J" and brackets). 

(ii) 
Give
a formula which is equivalent to [J®M]
using just Ú and ¬. 

(iii) 
Give a formula which is equivalent to ¬[J®M] using just Ù
and ¬. 

(iv) 
Give
a formula which is equivalent to ¬[JÙM] using just ¬ and Ú. 

(v) 
Give
a formula which is equivalent to ¬[JÚM] using just ¬ and Ù. 

(vi) 
Give
a formula which is equivalent to [J«M] using just ¬ and Ù. 