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<h2>
Exercises to accompany Predicate Tutorial 3
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5/5/15
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<h3>
Exercise 1 (of 3)
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<p>
Use the <em>Interpretation Tab</em> to show drawing or interpretation.
Consider a) whether the terms ('names') in the following formulas refer
properly to entities in the Universe, and b) whether each formula is
true or whether it is false. Select the formula and click <em>True?</em>
(under the <em>Semantics Menu</em>) to see what the program thinks.
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<blockquote>
a) Fa
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b) Ga
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c) Fb
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d) Gb
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e) Fa≡Gb
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f) Ga∧Fb
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h) Fb⊃Gb
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f) Gc
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g) Ha
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h) Fa ∨ Fc (*see Note*)
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(*Note. There are different ways of handling a case like this. On the
one hand you can say that Fa is true, therefore the disjunct is true; or
you can say that Fc is not entirely proper, therefore it is
inappropriate to ask about the truth value of the disjunct. We follow
the latter here; but there are occasions when it would be better to
follow the former.*)
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<h3>
Exercise 2 (of 3)
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<p>
This exercise is similar to the first, but this time we are interested
in lists of formulas and the 'simultaneous truth' of all the formulas in
a list. A list of formulas is just some formulas separated by commas.
And a list of formulas is true if and only if all the formulas in the
list are true. (So, if at least one formula in a list is false, the
whole list is taken to be false.)
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<p>
Use the <em>Interpretation Tab</em> to show drawing. Consider whether
each list of formulas is true or whether it is false. Select the list of
formulas and click True? to see what the program thinks.
</p>
<blockquote>
a) Fa, Gb
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b) Fa, Fa⊃Gb,~Gb
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c) Ga∧Fa, Ha
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d) Gb∨Ha, Fa
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e) Ra∧Rb,Fa
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<h3>
Exercise 3 (of 3).
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<p>
When we are trying to establish that an argument is invalid, we look for
a counter-example to it. A counter-example is a world or scenario in
which all the premises are true and the conclusion false.
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For example, consider the argument<br>
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If Bert is famous or Arthur is a gambler, then Arthur is famous and
Bert is a gambler
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Bert is a gambler.
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Arthur is famous.
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Therefore
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Bert is famous or Arthur is a gambler.
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which symbolizes to
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<p style="margin-left: 40px">
(Fb∨Ga)⊃(Fa∧Gb)
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<p style="margin-left: 40px">
Gb
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Fa
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∴
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Fb∨Ga
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<p>
ask the program about the truth of each of these formulas individually.
Notice that in the drawn world all the premises are true and the
conclusion false-- the depicted world constitutes a counter-example to
the argument. It proves that the argument is invalid.
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<p>
We can also use a device that we used in propositional logic. Instead of
trying to see if all the premises can be true and the conclusion false
at one and the same time, we can make a list of all the premises and the
negation of the conclusion. Thus
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(Fb∨Ga)⊃(Fa∧Gb), Gb, Fa, ~(Fb∨Ga)
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now ask the program if this entire list of formulas is true.
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It will tell you that they are. In the jargon, the set of formulas is
'satisfiable'.
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<p>
In practical cases you will be given the argument and have to produce
the counter-example drawing yourself.
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<p>
The program can do this for you.
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<p>
Open a new <em>Browser</em> (under the <em>File Menu</em>).
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<p>
Copy the list of formulas
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<p>
(Fb∨Ga)⊃(Fa∧Gb), Gb, Fa, ~(Fb∨Ga)
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and paste it into the new journal. Select it in the new journal and
click <em>Satisfiable?</em>, and look at the drawing.
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[Copying to a new journal is not an essential part of <em>Satisfiable?</em>
, on this occasion we are just trying to preserve this file as is.]
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