7/20/2018 EMR 17 Logical Reasoning Lecture 7

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.

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 “–”.

Expressive adequacy: . – 7/20/2018 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.

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.

Expressive adequacy: . – 7/20/2018 Expressive adequacy: . – Back to our proof. Suppose schema A has the form B v C Then A must be logically equivalent to –(–B . –C)

Expressive adequacy: . – 7/20/2018 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.

Expressive adequacy: . – 7/20/2018 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)

Expressive adequacy: . – 7/20/2018 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.!

Other minimal sets? We’ve shown that we can make do with just . – 7/20/2018 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?

Adequacy of – and v For example, we can make do with just v – 7/20/2018 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)

Adequacy of – and v So anything we can express using just . – 7/20/2018 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 “–”.

Adequacy of – and ⊃ In the same way, we can make do with just ⊃ – 7/20/2018 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)

Inventory of minimal sets 7/20/2018 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.

Inventory of minimal sets 7/20/2018 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.)

Inventory of minimal sets 7/20/2018 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.

The poverty of “–” and “≡” 7/20/2018 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.

The poverty of “–” and “≡” 7/20/2018 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.)

The poverty of “–” and “≡” 7/20/2018 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.

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

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

The poverty of “–” and “≡” 7/20/2018 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) ]

The poverty of “–” and “≡” 7/20/2018 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

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!

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.

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

Really minimal sets How should we fill in the rest? 7/20/2018 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

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

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

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!

Really minimal sets Can you find a short expression for (p . q)? 7/20/2018 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….

Really minimal sets Another example: (p ≡ q) ⊃ [ r v (p . q) ] 7/20/2018 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))))

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