1. 7/20/2018
EMR 17 Logical Reasoning
Lecture 7

2. Expressive adequacy: . v –
7/20/2018
Expressive adequacy: . v –
Suppose we have the truth-table for some mystery schema, using (say) the letters p, q, r, s.
Suppose we want to write down a logically equivalent schema, using only the connectives
v –
Can we do this?
Of course.
All we need to do is to produce the disjunctive normal form.

3. Expressive adequacy: . v –
7/20/2018
Expressive adequacy: . v –
One way of putting this result is as follows:
A formal language with just the connectives
v –
is expressively adequate, in the sense that
For any schema A, there is a schema B that
(i) is logically equivalent to A;
(ii) contains only the connectives “.”, “v”, and “–”.

4. Expressive adequacy: . –
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Expressive adequacy: . –
Do we need all three of these connectives?
No: for example, we can make do with just –
This may be obvious. But let’s stop to prove it, anyway.

5. 7/20/2018
A substitution law
Let’s first remind ourselves of one of our substitution laws:
Suppose schema B is a constituent of schema A.
Maybe A has the form “X . B”, or “X v (B ⊃ Y)”, etc.
Suppose schema C is logically equivalent to B.
Suppose schema D is what results from replacing B by C, within A.
Then A and D are logically equivalent.
Why? Because for any row of the truth-table, the truth-value contributed by C will be the same as that contributed by B.

6. Expressive adequacy: . –
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Expressive adequacy: . –
Back to our proof.
Suppose schema A has the form
B v C
Then A must be logically equivalent to
–(–B . –C)

7. Expressive adequacy: . –
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Expressive adequacy: . –
Why? Well, since
(p v q) and –(–p . –q)
are logically equivalent,
(p v q) ≡ –(–p . –q)
must be a tautology. In that case, so is
(B v q) ≡ –(–B . –q)
and by another step of the same sort, so is
(B v C) ≡ –(–B . –C)
Hence, (B v C) and –(–B . –C) are logically equivalent.

8. Expressive adequacy: . –
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Expressive adequacy: . –
Okay. Now suppose schema A has the form
B v C v D
By the result we just established, this is equivalent to
–[ –(B v C) . –D ]
But we know that –(B v C) is equivalent to (–B . –C).
It follows that the disjunction we started with is equivalent to
–( –B . –C . –D)

9. Expressive adequacy: . –
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Expressive adequacy: . –
And so on: any disjunction
B v C v D v E …
is equivalent to a conjunction:
–( –B . –C . –D . –E . …)
(We could have figured that out just by thinking about truth-tables, as well.)
Now, any schema A is equivalent to some schema B in DNF.
So we can use the above trick to convert B into an equivalent schema C that contains only “.” and “–”.
Q.e.d.!

10. Other minimal sets? We’ve shown that we can make do with just . –
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Other minimal sets?
We’ve shown that we can make do with just –
We clearly can’t make do with just “.”. Or with just “–”.
So this is a minimal set of connectives.
Are there other minimal sets, drawing just on our familiar connectives?
. v ⊃ ≡ –
Yes. Which ones?

11. Adequacy of – and v For example, we can make do with just v –
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Adequacy of – and v
For example, we can make do with just
v –
How would you show this?
One way: show that for any schemata A and B,
A . B
must be logically equivalent to
–(–A v –B)

12. Adequacy of – and v So anything we can express using just . –
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Adequacy of – and v
So anything we can express using just –
we can also express using just
v –
But we can express everything using just “.” and “–”. (Well, every truth-table.)
So we can express everything using just “v” and “–”.

13. Adequacy of – and ⊃ In the same way, we can make do with just ⊃ –
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Adequacy of – and ⊃
In the same way, we can make do with just
⊃ –
How would you show this?
One way: show that for any schemata A and B,
A . B
must be logically equivalent to
–(A ⊃ –B)

14. Inventory of minimal sets
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Inventory of minimal sets
So far, we have these minimal sets (drawing just on our familiar connectives): . . v v ⊃ ⊃ ≡ – – – –
Are there any others? (Again, drawing just from these connectives.)
No.

15. Inventory of minimal sets
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Inventory of minimal sets
First, we can show that these are not expressively adequate: . . v v ⊃ ⊃ ≡ ≡ – – – –
How? Well, if they were expressively adequate, we should be able to use them to produce a schema equivalent to –p
Hah! (Try it.)

16. Inventory of minimal sets
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Inventory of minimal sets
So if our minimal set is drawn from these connectives . . v v ⊃ ⊃ ≡ ≡ – – – –
then it must include the connective “–”.
Okay, so maybe we can get by with just “–” and “≡”?
Nope.

17. The poverty of “–” and “≡”
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The poverty of “–” and “≡”
Let’s start by observing that for any schemata A, B, C:
A ≡ (B ≡ C)
is logically equivalent to
(A ≡ B) ≡ C
(Why?)
Upshot: in a schema whose sole connective is the biconditional, it doesn’t matter where the parentheses are.

18. The poverty of “–” and “≡”
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The poverty of “–” and “≡”
Next, for any schemata A and B where B is a tautology
A ≡ B
is logically equivalent to A
(Well, duh.)

19. The poverty of “–” and “≡”
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The poverty of “–” and “≡”
Now look what happens when you put those two facts together.
Suppose schema A has (besides parentheses) only the symbols “p”, “q”, and “≡” in it. For example, A is
{ [ p ≡ (q ≡ p) ] ≡ q }
By what we’ve just shown, we know this must be equivalent to
(p ≡ p) ≡ (q ≡ q)
Which is, um, rather easy to evaluate.

20. The poverty of “–” and “≡”
7/20/2018
The poverty of “–” and “≡”
For a seemingly harder example, consider
( (p ≡ q) ≡ { [ p ≡ (p ≡ q) ] ≡ q } ) ≡ [p ≡ (q ≡ p) ]
Not so difficult, actually. By rearrangements that we know give us something equivalent, this turns into
(p ≡ p ≡ p ≡ p ≡ p) ≡ (q ≡ q ≡ q ≡ q)
And we can easily see that this is just equivalent to p

21. The poverty of “–” and “≡”
7/20/2018
The poverty of “–” and “≡”
What does this mean? Exactly this:
Suppose schema A has (besides parentheses) only the symbols “p”, “q”, and “≡” in it. Then A is equivalent to one of these:
p ≡ q p q
a tautology

22. The poverty of “–” and “≡”
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The poverty of “–” and “≡”
Next, remember that the rule for driving negations into biconditionals also allows us to drive them out:
A ≡ (–B ≡ C)
is equivalent to
A ≡ –(B ≡ C)
is equivalent to
–[ A ≡ (B ≡ C) ]

23. The poverty of “–” and “≡”
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The poverty of “–” and “≡”
Now we can put all this together.
Suppose schema A has (besides parentheses) only the symbols “p”, “q”, “–”, and “≡” in it.
Pull all negations out (cancelling double-negations at will). You’ll end up with a schema B or a schema –B, where B has (besides parentheses) only the symbols “p”, “q”, and “≡” in it.
Then A is equivalent to one of these:
(p ≡ q)
–(p ≡ q) p q –p –q
a tautology
a contradiction

24. The poverty of “–” and “≡”
7/20/2018
The poverty of “–” and “≡”
But in that case, A cannot possibly be equivalent to any of:
p . q
p v q
p ⊃ q
So much for expressive adequacy!

25. 7/20/2018
Really minimal sets
Let’s explore whether we could get by with just one connective.
Not one of these, obviously:
v ⊃ ≡ –
No, we’ll have to make one up.
Let’s call it ↓.
And let’s try working out its truth-table.

26. 7/20/2018
Really minimal sets
We need to be able to ‘define’ negation. So we’d better have
p ↓ p
turn out to be logically equivalent to –p
That gives us this much of the truth table: p q
p ↓ q T T ⊥ T ⊥ ⊥ T ⊥ ⊥ T

27. Really minimal sets How should we fill in the rest?
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Really minimal sets
How should we fill in the rest?
Well, suppose we tried this:
That would make (p ↓ q) logically equivalent to … –p
DISASTER! p q
p ↓ q T T ⊥ T ⊥ ⊥ ⊥ T T ⊥ ⊥ T

28. 7/20/2018
Really minimal sets
No good: we need a truth-table that gives q something to do!
Well, suppose we tried this:
That would make (p ↓ q) logically equivalent to … –q
DISASTER. p q
p ↓ q T T ⊥ T ⊥ T ⊥ T ⊥ ⊥ ⊥ T

29. 7/20/2018
Really minimal sets
No good: we need a truth-table that gives p something to do!
Time to try something intelligent:
That would make (p ↓ q) logically equivalent to …
–p . –q
Now we’re getting somewhere. p q
p ↓ q T T ⊥ T ⊥ ⊥ ⊥ T ⊥ ⊥ ⊥ T

30. is logically equivalent to
7/20/2018
Really minimal sets
Notice that if we negate
–p . –q
we get
p v q
So that means that
(p ↓ q) ↓ (p ↓ q)
is logically equivalent to
p v q
We are done – “↓” is, all by itself, expressively adequate!

31. Really minimal sets Can you find a short expression for (p . q)?
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Really minimal sets
Can you find a short expression for (p . q)?
By the way, in English, (p ↓ q) would read: “neither p, nor q”.
So, in this idiom, (p v q) becomes:
“It’s neither the case that neither p nor q, nor is it the case that neither p nor q.”
Clearly it would improve our logical hygiene if we started speaking this way….

32. Really minimal sets Another example: (p ≡ q) ⊃ [ r v (p . q) ]
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Really minimal sets
Another example:
(p ≡ q) ⊃ [ r v (p . q) ]
becomes:
(((((p↓(q↓q))↓((p↓p)↓q))↓(((p↓(q↓q))↓((p↓p)↓q)))↓(r↓((p↓p)↓(q↓q)))↓(r↓((p↓p)↓(q↓q))))↓
(((((p↓(q↓q))↓((p↓p)↓q))↓(((p↓(q↓q))↓((p↓p)↓q)))↓(r↓((p↓p)↓(q↓q)))↓(r↓((p↓p)↓(q↓q))))

33. 7/20/2018
Really minimal sets
There’s one other way to go. To keep separate track of it, we’ll use a different symbol.
We’ll fill the table in in the only remaining way possible:
That would make (p | q) logically equivalent to …
–p v –q
You should be able to take it from there! p q
p ↓ q
p | q T T ⊥ T ⊥ T ⊥ T T ⊥ ⊥ T