1. 7/3/2018
EMR 17 Logical Reasoning
Lecture 11

2. 7/3/2018
Warm-up puzzle
You are visiting the island of knights, knaves, and monks. The usual rules apply.
You encounter five islanders: Alfred, Betty, Clara, David, and Emma.
Alfred says: “Someone tells the truth more often than me.”
Betty says: “We are all the same.”
Clara says: “Either I am a knave, or Alfred is.”
David says: “At least two of us are knaves.”   
What are they?

3. A sample schema, with interpretations
7/3/2018
A sample schema, with interpretations
Consider this schema:
∀x∃y (Pxy . –Pyx)
Can we interpret it in such a way that it comes out true?
Certainly. For example, we could
• take the domain of discourse to be the three dots on this screen,
• take “Pxy” to mean “there is an arrow pointing from x to y”.

4. A sample schema, with interpretations
7/3/2018
A sample schema, with interpretations
∀x∃y (Pxy . –Pyx)
Similarly, if we want it to come out false, we can make use of the same domain of discourse, and the same understanding of “Pxy”, and simply remove an arrow:

5. A sample schema, with interpretations
7/3/2018
A sample schema, with interpretations
∀x∃y (Pxy . –Pyx)
Did we have to be talking about dots and arrows? Of course not.
For example, we might let our domain be Alfred, Betty, and Clara, and understand “Pxy” to mean “x enjoys y’s company”. We could then stipulate that
• Alfred enjoys Betty’s company, but not vice versa;
• Betty enjoys Clara’s company, but not vice versa;
• Clara enjoys Alfred’s company, but not vice versa.
So interpreted, our schema still comes out true!

6. Logically indistinguishable interpretations
7/3/2018
Logically indistinguishable interpretations
∀x∃y (Pxy . –Pyx)
There is something important to notice about the two interpretations we provided, that make our schema true:
They are, in a certain logical sense, exactly the same.
Yes, they talk about different things (dots vs. people).
Yes, they understand the “Pxy” relation differently.
But none of that matters to why they make the schema come out true.

7. Logically indistinguishable interpretations
7/3/2018
Logically indistinguishable interpretations
∀x∃y (Pxy . –Pyx)
All that matters is that each interpretation features
• exactly three objects in its domain, where
• the first bears the P-relation to the second, but not conversely;
• the second bears the P-relation to the third, but not conversely;
• the third bears the P-relation to the first, but not conversely.
It is because of those (very abstract) features that each interpretation satisfies (logician for: “makes true”) our schema.

8. Interpretations using numbers
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Interpretations using numbers
∀x∃y (Pxy . –Pyx)
In fact, if all we are interested in is the purely logical question whether a given schema is satisfiable, we can restrict our attention to interpretations whose domains of discourse contain numbers.
Example:
Logician-speak for “things from the domain that satisfy the condition P”
Domain = {1, 2, 3}
Extension of P =
<1, 2>
Note that order matters, here!
<2, 3>
<3, 1>

9. Working upward It’ s intuitively obvious that this interpretation:
7/3/2018
Working upward
It’ s intuitively obvious that this interpretation:
Domain = {1, 2, 3}
Extension of P =
<1, 2>
<2, 3>
<3, 1>
satisfies this schema:
∀x∃y (Pxy . –Pyx)
But let’s see how to derive this result in an exact manner.

10. 7/3/2018
Working upward
The pairs of numbers <x, y> that satisfy Pxy are obviously just
<1, 2>
<2, 3>
<3, 1>
The pairs of numbers <x, y> that satisfy Pyx (watch the order!!) are
<2, 1>
<3, 2>
<1, 3>
So the pairs <x, y> that satisfy –Pyx are what remains:
<2, 2>
<3, 1>
<1, 1>
<2, 3>
<3, 3>
<1, 2>

11. 7/3/2018
Working upward
So the pairs of numbers <x, y> that satisfy (Pxy . –Pyx) are the pairs that are in both this list:
<1, 2>
<2, 3>
<3, 1>
and in this list:
<2, 2>
<3, 1>
<1, 1>
<2, 3>
<3, 3>
<1, 2>
I.e., these ones:
<1, 2>
<2, 3>
<3, 1>

12. 7/3/2018
Working upward
Now we have to consider which numbers satisfy the (1-place) condition ∃y(Pxy . –Pyx).
Remember what this condition is: It’s the condition that a number x satisfies exactly if there is some number y such that the pair <x, y> satisfies the (2-place) condition (Pxy . –Pyx).
In other words: It’s the condition that a number x satisfies exactly if x is the first member of a pair that satisfies (Pxy . –Pyx).
I.e., if it is the first member of one of these pairs:
<1, 2>
<2, 3>
<3, 1>
I.e., exactly if it is one of these numbers: 1 2 3

13. 7/3/2018
Working upward
So every number x satisfies the condition ∃y(Pxy . –Pyx).
So our schema ∀x∃y(Pxy . –Pyx) is, on this interpretation, true.
There is a general moral:
Once an interpretation specifies
• how many things there are;
• which of them satisfy which basic conditions (Cx, Pxy, etc.);
it thereby determines, in a completely systematic and exact manner, truth-values for all schemas that can be constructed from these basic conditions.

14. Basic vs. derived frames
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Basic vs. derived frames
Suppose we have some complex schema – say, this one:
∃x {Cx . ∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ] }
Notice that many statement-frames appear within it. For example:
(Ty . Pyx)
(Cz . x ≠ z . Pyz)
∃z (Cz . x ≠ z . Pyz)
[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ]
∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ]
{Cx . ∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ] }

15. Basic vs. derived frames
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Basic vs. derived frames
Notice, however, that all of these frames are constructed from these basic frames: Ty
Pyx, Pyz
Cx, Cz
x = z
Constructed how?
• combining with truth-functional connectives;
• negating;
• attaching a quantifier.

16. Constructing a statement
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Constructing a statement
Let’s see how this works, in this case:
x = z
add negation Cz
x ≠ z
Pyz
conjoin Ty
Pyx
(Cz . x ≠ z . Pyz)
conjoin
add existential quantifier
(Ty . Pyx)
∃z (Cz . x ≠ z . Pyz)

17. Constructing a statement
7/3/2018
Constructing a statement
Let’s see how this works, in this case:
(Ty . Pyx)
∃z (Cz . x ≠ z . Pyz)
Combine with ⊃
(Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz)
add universal quantifier Cx
∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ]
conjoin
Cx . ∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ]
add existential quantifier
∃x {Cx . ∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ] }

18. Constructing a statement
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Constructing a statement
Summary:
∃x {Cx . ∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ] }
Cx . ∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ]
∀y[ (Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz) ] Cx
(Ty . Pyx) ⊃ ∃z (Cz . x ≠ z . Pyz)
(Ty . Pyx) Ty
Pyx
∃z (Cz . x ≠ z . Pyz)
(Cz . x ≠ z . Pyz)
x = z
Pyz Cz
x ≠ z

19. Constructing a statement
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Constructing a statement
Here is the take-home point: Every statement has such a “tree”.
What’s more, the tree is unique. Given a statement (or frame), you can – in a completely precise, unambiguous fashion – work backwards to produce its method of construction.
Our general moral, restated: Once an interpretation specifies which ordered sequences of items drawn from its domain satisfy the basic conditions that appear as “starting nodes” in this tree, it thereby determines a truth-value for the schema.

20. 7/3/2018
Adding names
So far, we’ve taken the basic elements from which schemas are constructed to be conditions – Cx, Pxy, etc.
We are also going to allow, as basic elements that can be “slotted” into basic conditions, names (typically lower-case letters a, b, c, etc.).
So we express “b points to everything”, for example, as:
∀yPby
[Warning: In Goldfarb, the function of names is taken over by free variables. This is a much more elegant approach – but also more confusing.]
What does an interpretation do with a name? Easy: assign it an element of the domain as its designation.

21. Interpretations: summary
7/3/2018
Interpretations: summary
Remember that “interpretation”, as we’re using it, has two meanings: thick and thin.
[Recall: a thick interpretation of a truth-functional schema assigns complete English statements to the sentence-letters, whereas a thin interpretation simply assigns truth-values to the sentence-letters.]
To understand key logical notions – equivalence, implication, satisfiability, etc. – thin interpretations are enough.
For quantificational schemas, a thin interpretation need only
• assign a domain;
• assign extensions to the basic predicates (conditions);
• assign designations to the names.

22. Satisfiability defined
7/3/2018
Satisfiability defined
Here it is, as applied to a set S of schemata:
A set S of schemata is satisfiable iff there is some interpretation that makes every schema in S true.
All other logical notions that concern the status of and relations between statements can be defined in terms of satisfiability.

23. Definitions in terms of satisfiability
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Definitions in terms of satisfiability
Contradiction:
Schema A is a contradiction iff it is not satisfiable.
Tautology:
Schema A is a tautology iff the schema –A is not satisfiable.
Equivalence:
Schemata A and B are equivalent iff the schema (A ≡ B) is a tautology.
Implication:
Set S implies schema A iff S, together with the schema –A, is not satisfiable.