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Exercise 2.7
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 truth-table 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 truth-values.) | |
| (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 truth-functors of our propositional language, and are asked to provide sentences which are equivalent but use a different selection of truth-functors. (Sentences of this sort, which consist of just capital letters and truth-functors 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 truth-table 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. | |
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