Logical “And”

Learning Target & Do Now

LT: By the end of today, I will know the definition of the truth functional connective “and”.

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DN: Could we translate each of the following into our current logical language? Why or why not?

“A number x is divisible by 2 and 3.” “Tom will go to Berlin and Paris.”

“Steve is from Texas but not from Dallas.”

Conjunctions

• Each of the sentences in the DN could have been stated with atomics
• Just like conditionals, the relationship between terms is lost
• Today’s new connective fulfills the same role as the English “and”
• Say the following translation key is used:

P = Tom will go to Berlin Q = Tom will go to Paris P and Q

R = x is evenly divisible by 2 S = x is evenly divisible by 3 R and S

T = Steve is from Texas U = Steve is from Dallas T and ~U

Conjunctions, cont.

T = Steve is from Texas U = Steve is from Dallas T and ~U

• Original English sentence:“Steve is from Texas but not from Dallas.”
• The meaning has changed slightly - ‘but’ is superfluous
• Can be represented with other words/combinations
• Associated with surprise - not something a logical language needs to represent
• Using just ‘and’, original meaning appears to be preserved.

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Symbolic logic syntax: T ^ ~U

P ^ Q

• This is also called a conjunction
• The two sentences are called conjuncts
• Definition is defined by a truth table :)

Alternative Phrasings

Some alternative phrasings are explicit:

• “But” (Steve is from Texas but not from Dallas)
• “Even though” (even though they lost the battle, they won the war)

Others, implicit. These usually state multiple properties/conditions:

• Tom, who won the race, also won the championship.
• The star Phosphorus, that we see in the morning, is the Evening Star.
• Dogs are vertebrate terrestrial mammals.

Can we turn each of these into logical conjunctions?

English Traps with “And”

• In some English sentences, “and” is not used as a logical conjunction
• E.g. “Rochester is between Buffalo and Albany”
• The problem here: ‘between’ is a predicate that requires two or more values
• Our logical and only combines sentences, not predicate parts
• Some context dependence, but this case is rare

Conjunction Inference Rules

1. P ^ Q P ^ Q� _____ _____� P Q�In other words, if the conjunction is true then either conjunct is individually true. This is called simplification
2. This works in the opposite direction as well. Called adjunction� P, Q� ______� P ^ Q

Practice

Tom and Steve will go to London. If Steve goes to London, then he will ride the Eye. Tom will ride the Eye too, provided he goes to London. So, both Steve and Tom will ride the Eye.

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Make a translation key. What is our argument? Can we prove it is valid?

Solution

Tom will go to London = P Tom will ride the Eye = Q�Steve will go to London = R Steve will ride the Eye = S

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1. P ^ R Premise 4. P Simplification�2. R → S Premise 5. R Simplification�3. P → Q Premise 6. S Modus Ponens, 2, 5� _________ 7. Q Modus Ponens, 3, 4�S ^ Q | 8. S ^ Q Adjunction, 6, 7

You Do

Prove the following:

(Q → ~S)�(P → (Q ^ R))�T → ~R�P�_____________

(~S ^ ~T)

Solution

1. (Q → ~S) Premise�2. (P → (Q ^ R)) Premise�3. T → ~R Premise�4. P Premise�____________________________________

5. Q ^ R Modus Ponens, 2, 4�6. Q Simplification, 5�7. ~S Modus Ponens, 1, 6�8. R Simplification, 5

9. ~~R Double Negation, 8�10. ~T Modus Tollens, 3, 9�11. ~S ^ ~T Adjunction, 7, 10

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(This is only one of many possible solutions. You may have a different one!)

Homework

DeLancey 5.6 #1, 2

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Today’s animal: pink fairy armadillo

Complex Sentences

A lesson, “and” then some!

Learning Target & Do Now

LT: By the end of today, I will have some experience working with sentences containing several connectives.

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DN: Consider the following three sentences. How does the truth value(s) of each differ?

~(P → Q) (~P → Q) (~P → ~Q)

Parenthese Placement

• Parentheses will define how sentences are read and order of operations
• Reading left to right, the last ‘(‘ matches with the first ‘)’
• Each pair of parentheses will represent a distinct sentence
• Like math parens, should be solved on its own before applying elsewhere

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How should we determine the truth of such sentences?

Truth Tables… But Bigger

• Each of the sentences can be its own column in the truth table
• Allows us to break down the problem

Sample: ~(P → Q)

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You Do

Create the truth tables for (~P → Q) and (~P → ~Q).

I recommend 4 and 5 columns, respectively

(~P → Q)

(~P → Q)

(~P → ~Q)

(~P → ~Q)

Validity of Complex Sentences

• Simplifying complex sentences and checking for validity can be done with one truth table
• Can still identify premises and break down the problem
• Alternately, proofs!

Homework

• DeLancey 5.6 #4 a-l

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Today’s animal: Secretarybird

Conditional Derivations

Learning Target & Do Now

LT: By the end of class, I will see how conditional derivation can occur.

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DN: Determine if the following argument is valid (simplification of a quote from Thomas Hobbes’ Leviathan):

If we want to be safe, then we should have a state that can protect us.�If we should have a state that can protect us, then we should give up some freedoms.

Therefore, if we want to be safe, we should give up some freedoms.

DN Semantics and Syntax

Semantically, the argument is valid (table from textbook).

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Our conclusion is an implication - we have not yet seen how to prove these syntactically.

Conditional Derivation

• For convenience, inference rules can be thought of:
• As a way to show a kind of sentence, or
• Make use of a kind of sentence
• The structure of the conditional truth table is not very helpful
• Can be true when the antecedent is true or false�
• Instead, structure of the proof itself is helpful - why?

The Relation of Proofs to Conditionals

At its core, a proof is a conditional statement

If all of the premises are true, then the conclusion should be true (roughly)

• Imagine there is some sentence W
• Using W, we are able to prove X
• This means that if W is true, X is true (W → X)

Therefore, a proof is a conditional!

Conditional Derivation

• Technically, application of the deduction theorem
• The deduction theorem will be proven later
• For now, think of conditional derivation as a proof method
• Additional traits of conditional derivation:
• A special assumption is made, the assumption for conditional derivation
• What needs to be shown is not the conclusion, but the consequent of the conclusion
• The conditional derivation is a subproof that aids a direct proof
• Example on the next slide

Conditional Derivation Example

• Steps 3, 4, and 5 are the conditional subproof
• The inference rule used in the direct proof is conditional derivation

You Do

Prove the following:

(P → Q)�(R → S)�____________

((P ^ R) → (Q ^ S))

Hint: assume the conjunction P^R

Solution ((P ^ R) → (Q ^ S))

1. P → Q premise 6. R Simplification, 2

2. R → S premise 7. S Modus Ponens, 2, 6

__________________________ 8. (Q ^ S) Adjunction, 5, 7

3. P ^ R Asmp. for conditional deriv 9. ((P ^ R) → (Q ^ S))

4. P Simplification, 1 Conditional Deriv, 3-8

5. Q Modus Ponens, 1, 4

Homework

DeLancey 6.5 #1

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Today’s animal: Siphonophore

Tautologies and Theorems

Learning Target & Do Now

LT: By the end of today, I will know what tautologies and theorems are.

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DN: Think-pair-share: what is conditional derivation?� With your partner, prove P → P.

A Tautology and a Theorem

Let us build the truth table for the following:

((P → Q) → (~Q → ~P))

A Tautology and a Theorem

The expression is always true! This is a tautology.

((P → Q) → (~Q → ~P))

A Tautology and a Theorem

Prove the following is valid:

((P → Q) → (~Q → ~P))

Remember: Assume the antecedent, prove the consequent

A Tautology and a Theorem

Prove the following is valid:

((P → Q) → (~Q → ~P))

1. P → Q Assumption for cond. deriv�2. ~Q Assumption for cond. deriv�3. ~P Modus Tollens, 1, 2�4. (~Q → ~P) Conditional derivation, 2 - 3�5. ((P → Q) → (~Q → ~P)) Conditional Derivation, 1-4

A Theorem

• The proof on the last slide had no premises!
• A statement that can be proven without premises is called a theorem
• Reveal truths that follow from logic alone
• All tautologies happen to be theorems and vice versa!

Homework

DeLancey 6.5 #2, 3a-c

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Today’s animal: Aye-aye

Free Day

Homework

• DeLancey 6.5 # 5
• Read Hillis chapter 2

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Today’s animal: Kiwi bird