Download Workshop5. Logic in Language

Elements of Computer Science
Workshop5. Logic in Language
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Information
Guided Enquiry
Details for the Portfolio
Extension Material
Purpose
(a) To learn how logic is used in language.
(b) To explore the similarities between digital logic and logic in languages.
(c) To use Boolean Algebra
Portfolio
This session addresses ILO1
There is a lot of work here, too much for one session. Any of the activities from 3 onwards are suitable for
inclusion in your portfolio. I suggest after attempting activities 1 and 2, you choose one or more
“Consistency” tests and one or more “Validation” tests. But try to have some fun solving some interesting
problems.
Activities
1
Logic Connectives : The AND, OR connectives and the NOT
Let us take the two atomic sentences
P = <Holmes was on the case>
Q = <Watson was on the case>
and consider the compound sentence
Either Holmes was on the case, or both Holmes and Watson were on the case.
(a) Work from this compound sentence to fill in the final column in the truth table below which gives the
truth of the compound sentence for the various values of P and Q
Q
0
1
0
1
P
0
0
1
1
mini-terms
(b) Now write down the min-terms (using Boolean algebra) for both rows in the table which have a
1 in the final column.
(c) The Boolean expression for the compound sentence is the conjunction of your mini-terms
(𝑃. 𝑄̅ + 𝑃. 𝑄)
Complete the truth table below for both mini-terms and for the conjunction. You may omit the Os
to make it more readable. Check it agrees with your answer to (a)
P
0
0
1
1
Q
0
1
0
1
𝑃. 𝑄̅
𝑃. 𝑄
(𝑃. 𝑄̅ + 𝑃. 𝑄)
(d) For the rows where the conjunction of the mini-terms is 1, write down the meaning of the miniterm in simple English using the atomic sentences P and Q.
(e) Construct a simulated electronic circuit to represent the problem and its solution. Remember to:
(i) create each mini-term using and AND gate
(ii) create the conjunction of the mini-terms using a single OR gate
(f) Express the solution to the problem as code. Start with each mini-term, then form the
conjunction. Test out your code on the Arduino
(g) Use Boolean Algebra to simplify your expression in (c). You should find it becomes P.
2.
A Simple Logic Problem
Consider the sentence P = <Cows can fly>.
(a) Write down, in English, the meaning of the sentence 𝑷. 𝑷
(b) Now work out the truth table for it, by thinking
P
𝑷. 𝑷
0
1
(c) Find a rule from Boolean algebra which will prove the result.
3
The Conditional IF. First Investigation.
The Philonian Conditional; “If P then Q” can be represented by the truth functor
“(Not P) or Q”,
“If P then Q” is represented by P  Q and is equivalent to 𝑷 + 𝑸 in logic
Let’s consider the two atomic sentences
P = <The switch is pressed>
Q = < The Light is on>
(a) Now consider the compound sentence
If the switch is pressed the light is on
Write down the Boolean expression for this sentence.
(b) Now consider the compound sentence
If the light is on the switch is pressed
Write down the Boolean expression for this sentence
(c) In the truth table below write your Boolean expression from (a) at the top of the third column
and fill in the rows where the expression is true.
(d) Repeat this for your second expression using the fourth column
(e) Now look at each row, and in the fifth column write a 1 where both terms are true together, ie
where columns 3 and 4 both contain 1s.
(f) Now in the sixth column write the mini-terms corresponding to these rows
P
0
0
1
1
Q
0
1
0
1
(g) Finally in the seventh column write down in simple English the state of the switch and light
where the compound expression is true. Do your sentences agree with the compound expression.
(h) Now let’s apply some Boolean algebra. Take your two terms and AND them together and you
should obtain
(𝑃̅ + 𝑄). (𝑄̅ + 𝑃)
Expand this using the distributive rule and you should obtain two non-0 terms. Check that these
agree with your mini-terms in the above table.
(i) Build two simulated digital circuits, one for the expression given in (h) and the second for your
simplified expression in (h). Check that both give identical results.
(j) Write code for both expressions and test that they too have identical behaviour when run on the
Arduino.
4.
The Conditional IF. Second Investigation
Repeat 3. for the compound sentence
If the switch is pressed the light is on AND If the light is on the switch is not pressed.
5.
Consistency – Holmes, Watson and Lestrade.
Let’s use the following atomic sentences
P = <Holmes solved the crime>
Q = <Lestrade took the criminal>
R = <The criminal escaped>
Consider the following text
"If Holmes. solved the crime, then Lestrade took the criminal. Of course, if the criminal escaped,
then Lestrade did not take him. On this particular day, Holmes solved the crime, but the criminal
escaped"
(a) Convert each of the sentences in the above text into Boolean terms involving combinations of
P,Q,R
(b) Add your terms at the top of each column in the truth table below
P
0
0
0
0
1
1
1
1
Q
0
0
1
1
0
0
1
1
R
0
1
0
1
0
1
0
1
(c) You should find that there is only one row where all terms are true together. This identifies the
consistent set of atomic sentences. Write the corresponding mini-term down and the associated
sentence in the row. Does the sentence make sense?
Now you should attempt either (d) or (e)
(d) Construct a Boolean expression by AND-ing your three terms. Simplify this using Boolean
algebra. You should end up with a single term
(e) Construct a simulated digital circuit using your three terms and verify that this produces the
consistent set of sentences you have found in the truth table.
6.
Consistency – Watson and Holmes in Confusion.
Test the consistency of this set of sentences using the method above.
a) Holmes and Watson were on the case.
b) If Holmes was on the case, then Watson was not.
(a) First find the atomic sentences P and Q;
P
=
Q
=
(b) Now transcribe the set of sentences a) and b) in terms of P and Q using Boolean Algebra
a)
b)
(c) Now AND these sentences using Boolean algebra
(d) Now complete the truth table as above and look for any consistent solutions. Did you expect this result?
Look at the starting sentences and think out the answer.
P
0
0
1
1
7.
Q
0
1
0
1
Consistency – Smiley’s People
Test the consistency of this set of sentences;
"Smiley is an English spy. Smiley is not both a Russian and an English spy. If Smiley is a Cad, then
he is a Russian spy".
(a) Find atomic sentences for this text
P
=
Q
=
R
=
(b) Convert each of the sentences in the above text into Boolean terms involving combinations of
P,Q,R
(c) Add your terms at the top of each column in the truth table below
P
0
0
0
0
1
1
1
1
Q
0
0
1
1
0
0
1
1
R
0
1
0
1
0
1
0
1
(d) You should find that there is only one row where all terms are true together. This identifies the
consistent set of atomic sentences. Write the corresponding mini-term down and the associated
sentence in the row. Does the sentence make sense?
Now you should attempt either (d) or (e)
(e) Construct a Boolean expression by AND-ing your three terms. Simplify this using Boolean
algebra. You should end up with a single term
(f) Construct a simulated digital circuit using your three terms and verify that this produces the
consistent set of sentences you have found in the truth table.
8.
Consistency – Too Many Detectives (VERY OPTIONAL)
Test the consistency of these sentences;
"If Holmes. solved the crime, then Lestrade took the criminal. Of course, if the criminal escaped,
then Lestrade did not take him. On this particular day, Holmes solved the crime, but the criminal
escaped"
P
Q
R
=
=
=
(a) Now transcribe each of the three sentences in to a boolean expression.
(b) Now write down the boolean expression for all three sentences together
(c) Build a circuit using the boolean expression and find the truth table.
(d) Identify any consistent solution.
The Method of testing Validity of Arguments
An argument will have several premises and one conclusion. Testing the validity of an argument is done by
constructing a Boolean expression for each premise. To this is added the Boolean expression for the inverse
of the conclusion, ie the conclusion is taken to be false. All expressions are then tested for consistency. If a
consistent solution (a true solution) is found then this is an example of where the conclusion is false. Hence
the argument is proved to be invalid. If there are no consistent solutions then the conclusion cannot be false
for the premises, in other words the argument is true. This approach is called reductio ad absurdum.
9.
Argument Validity – God, Matter and Homer Simpson.
Take the following atomic sentences
P = < Matter always existed>
Q = < God Exists >
R = < Homer Simpson created the Universe >
and consider the text
Matter always existed
If God exists then Homer Simpson created the Universe
If Homer Simpson created the Universe then Matter did not always exist
Therefore there is no God
(a) Transcribe the three premises of the argument into individual Boolean expressions
(b) Now transcribe the conclusion and negate this (Reductio ad Absurdum)
(c) Add your four expressions to the headings of columns in the truth table below
P
0
0
0
0
1
1
1
1
Q
0
0
1
1
0
0
1
1
R
0
1
0
1
0
1
0
1
(d) For each expression insert a 1 into any row where the expression is true.
(e) Look for any row where all expressions are true. If you find one then this is a counter example
and the argument is invalidated.
(f) Optional. Write down an expression consisting of the products of your terms (ANDed). Now
simplify this and show that it evaluates to 0. Start with the distributive rule.
(g) Optional. Either construct a digital circuit which combines (ANDs) all of your individual
expressions and check that this agrees with your truth table or write the combination of your
expressions as code and check this agrees with your truth table using Arduino.
10.
Argument Validity – From Monty Python
Take the three atomic sentences
P = < she’s made of wood>
Q = < she weighs the same as a duck >
R = < she’s a witch >
and the following argument whose validity you must investigate
If she’s made of wood she’s a witch
If she weighs the same as a duck she’s made of wood
She weighs the same as a duck
Therefore she’s a witch
(a) Transcribe the premises of the argument into Boolean expressions
(b) Now transcribe the conclusion and negate this (Reductio ad Absurdum)
(c) Add your four expressions to the headings of columns in the truth table below
P
0
0
0
0
1
1
1
1
Q
0
0
1
1
0
0
1
1
R
0
1
0
1
0
1
0
1
(d) For each expression insert a 1 into any row where the expression is true.
(e) Look for any row where all expressions are true. If you find one then this is a counter example
and the argument is invalidated.
(f) Optional. Write down an expression consisting of the products of your terms (ANDed). Now
simplify this and show that it evaluates to 0. Start with the distributive rule.
(g) Optional. Either construct a digital circuit which combines (ANDs) all of your individual
expressions and check that this agrees with your truth table or write the combination of your
expressions as code and check this agrees with your truth table using Arduino.
11.
Argument Validity – Fallacy of Denying the Antecedent
Take the atomic sentences
P = <There is a fault>
Q = <It will blow up>
and working as above prove that the following argument is invalid
If there is a fault, it will blow up. There is no fault therefore it will not blow up.
12.
Argument Validity – Fallacy of Affirming the Consequent
Take the atomic sentences
P = <There is a fault>
Q = <It will blow up>
and working as above prove that the following argument is invalid
If there is a fault, it will blow up. It will blow up, therefore there is a fault
13.
Argument Validity – Fallacy of Conversion
Take the atomic sentences
P = <There is a fault>
Q = <It will blow up>
and working as above prove that the following argument is invalid
If there is a fault then it will blow up, therefore if it blows up, then there is a fault!
14.
Argument Validity – Modus Tollens
Take the atomic sentences
P = <There is a fault>
Q = <It will blow up>
and working as above test the validity of the following argument
If there is a fault it will blow up. It will not blow up, therefore there is no fault
15.
Consequentia Mirabilis
(a) Here a proof which is true and a little challenging to understand. Prove the validity of the following
argument using either the truth-table approach or Boolean algebra. Try to understand why the proof is so
strange.
If there is proof, then there is proof
If there is no proof, then there is proof
Therefore there is proof
(b) Now let’s drop the first line. Again prove the validity of the argument and try to understand
If there is no proof, then there is proof
Therefore there is proof
16.
Aristotelian Syllogism
Prove that the following argument is correct. It is presented in a general form then an example.
If P then Q and
If Q then R
Therefore P then R
All men are mortals
Socrates is a man
Therefore Socrates is mortal