PHIL 210
Week 1 – Monday
Chapter 1 and 2 of Part I and Chapters 4 and 5 of Part II of textbook and Chapter 3 of Part I (introduce
them early)
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No associated practice questions on Carnap
Associated problems in textbook
Deductive logic --> repeated practice is essential
Notion of proof == deductive logic, concerned with:
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Correctly deducing a conclusion from given premises
And thus what is often called a valid inference or argument
Premises of the argument which the arguer thinks will lead to relevant conclusion
Deductive Logic == Formal Logic
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Certain patterns, certain forms of argumentation that do or do not preserve truth
Under what conditions can we know that an argument leads to a true conclusion – that is the
conclusion “truth preserving”
Interested in preserving the truth of the premises
General form of an Argument:
1. One or more premises
2. One conclusion
When judging that the conclusion (what is being argued for) does follow from the premises:
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The conclusion is a logical consequence of the premises
Or that the conclusion follows logically from the premises
Sum up premises and conclusion together by saying:
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The argument (or inference) if logically valid
Interested in the validity of the argument
Example:
Where does Sanji live?
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Sanji lives in Calgary or Edmonton.
Sanji doesn’t live in Edmonton
We can conclude decisively about where Sanjiv lives, guaranteed by the truth of the premises.
Conclusion is guaranteed to be true by a rule of inference and disjunctive syllogism
Argument 1
Sanjiv lives in Calgary or in Edmonton.
Sanji doesn’t live in Edmonton.
Therefore, Sanjiv lives in Calgary.
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Note a few things
o Individual sentences but also some ways of putting them together
o For instance, in the first line, we have put together two sentences (Sanjiv lives in
Calgary, Sanjiv lives in Edmonton) with what we call a connective, the ‘or’
o The first of these appears again in the conclusion which follows the ‘Therefore’
‘Therefore’ marks the conclusion also which indicates the claim that the conclusion (supposedly) follows
from the premises
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Usually write the three dots instead of therefore
The two sentences before the Therefore are called the premises
The individual sentences used are all the kinds that are true or false and sentences of this kind are called
‘assertoric’ because they make assertions
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The sentences must be declarative
Truth-Functional Logic
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Knowing we are working with assertoric/declarative sentences are either true or false, gives
cluses to this logic
Function: given a certain input and certain rule, determine output
If we know the truth-values of the premises, we can determine the truth-value of the
conclusion
Developing the system by looking for the smaller assertoric parts that are repeated
o Example: ‘lives in’ is repeated but is not an assertoric sentence itself == serve as inputs
the system
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Argument 3 would be valid, if we took the ‘or’ as exclusive – what we usually refer to as an
‘either, or ‘situation
But if the ‘or’ is inclusive (always in this course) then invalid
Definitions:
Case: hypothetical; scenario in which each sentence (so premises and conclusion) in an argument is
either true or false
An Argument is valid if there is no case where all its premises are true and the conclusion is false
An argument is invalid if there is at least one case where all its premises are true and the conclusion is
false (i.e. if its not valid)
In this latter circumstance, we call such a case a ‘counterexample’ to an argument
We don’t need to run through all the possible cases, or rule out counter examples
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Sufficient to just now about the form of the argument to know if its valid or invalid
Valid Argument Forms
Disjunctive Syllogism:
Modus Ponens
Modus Tollens
Hypothetical Syllogism
Only Interested in the pattern
Symbolizations key
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Basic language when working in TFL
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Those five connectives are very common
Not clear that if we have used any of the connectives in terms of TFL (for later)
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The premises and conclusions are as before sentences
The sentences we use here are again as before declarative
The conclusion seem inevitable given the two premises
No case where the premises are true and the conclusion is false (def of validity)
No counterexample
Invalid Inference Example:
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Fill in P, Q, a with other examples
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The argument does not preserve the truth of the premises all the way down to the conclusion
The premises are true, but the argument does not guarantee the truth of the conclusion
Specifying with the whale is a counterexample that shows the inference is invalid
Complicated to work with properties, so a lot of work to do to build up to first order logic
The counterexample shows that nothing in the premises does not determine that he is a whale
In this case, if the conclusion is true, what premises would have to be false
The validity/invalidity of an inference has got NOTHING to do with the question of whether the
premises or conclusion are true
When the premises contain a contradiction, the inference is automatically valid
If you play with a valid inference,
Formal Logic:
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Logic is the science of what follows from what; logic investigates what makes the first 2
arguments valid and the third one invalid
Chapter 1:
Arguments can follow different structures:
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2 premises and a conclusion
o Either the butler or the gardener did it.
o The butler didn’t do it.
o The gardener did it.
Single sentence, complete argument
o The butler has an alibi; so they cannot have done it.
Conclusion then premises
o The gardener did it. After all, it was either the butler or the gardener. And the butler
didn’t do it.
Conclusion in the middle
o The gardener did it. Accordingly, it was the gardener, given that it was either the
gardener or the butler.
When approaching an argument we want to know whether or not the conclusion follows from the
premises. First sperate out the conclusion from the premises.
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Conclusion Indicator Words: so, therefore, hence, thus, accordingly, consequently
Premise Indicator Words: since, because, given that
Only interested in sentences that could be declared as true or false
Questions, imperatives (commands, Wake Up!), Exclamations (Ouch!) are usually not sentences in logic
Example:
If the driver did it, the maid didn’t do it. The maid didn’t do it. Therefore the driver did it.
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If the first premise is true and the second premise is true, then then there is only one conclusion
following the premises. There arguments are valid
If the driver did it, the maid didn’t do it. The maid didn’t do it. Therefore the driver did it.
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If the premises are true, the conclusion is not guaranteed that it is also true
Could be a case where the two premises art true but the conclusion is not true. This argument is
invalid
We determine invalidity by determine a counterexample (a scenario where the premises are true but
the conclusion is not)
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If there is a counterexample to an argument, conclusion is not a consequence of the premises
For the conclusion to be a consequence of the premises, the truth of the premises must
guarantee the truth of the conclusion
Valid = conclusion is consequence of premises
Invalid = not valid, has counterexample
Depending on what kinds of cases we consider as potential counterexamples
1. Nomologically Valid: no counterexamples that don’t violate the laws of nature
2. Conceptually Valid: no counterexamples that don’t violate conceptual connections between
words
Formally Valid:
When considering various cases to determine validity of argument, assumes few things;
1. Every case makes every sentence true or not true
a. Any imagined scenario which leaves it undetermined if a sentence in our argument is
true is not considered a counterexample
i. Ex. Scenario where Priya is a dentist but not a ophthalmologist
Sound Arguments
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Validity only rules out the possibility of if the premises are true and the conclusion is not true at
the same time; doesn’t rule out the conclusion by itself
Example:
Oranges are either fruit or musical instruments. Oranges are not fruit. Therefore, oranges are musical
instruments.
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Conclusion is BAD, but it follows the premises
If both premises are true, then the conclusion just has to be true; valid argument
London is in England. Bejing is in China. Therefore Paris is in France.
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Premises and conclusion are true but argument is invalid
If Paris declared independence, it would no longer be true
There is a case where premises are true without the conclusion being true; invalid argument
Validity is not about the actual truth – about whether it is possible for all the premises to be true and
the conclusion at the same time
Sound Argument: argument is sound if it is both valid and all its premises are true
Inductive Arguments
Example:
Every winter so far, it has snowed in Calgary. Therefore, it will snow in Calgary this coming winter.
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Inductive arguments: Argument generalizes based on past cases to a conclusion about all
(future) cases
INVALID argument, even if it has snowed in Calgary every winter, it remains possible it will not
Not even good inductive arguments are not deductively valid
Argument to be valid: it is impossible for all the premises to be true AND the conclusion to be false.
Valid
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It is impossible for the premises to be true
Don’t have enough arguments to say the sentence is false so the argument is true
If the premises is not true, we can’t definitively say that it is false
Example:
o If it rains today, everyone is pink.
o It didn’t rain today. You can’t say for sure that what we said above is false.
Chapter 3
Jointly Possible: sentences are if there is a case where they are all true together
Example Jointly Possible:
Jane’s only brother is shorter than her.
Jane’s only brother is younger than her.
Example Jointly Impossible:
There are at least four giraffes at the wild animal park.
There are not more than two Martians at the wild animal park.
Every giraffe at the wild animal park is a Martian.
Example:
1. It is raining.
2. Either it is raining here or it is not.
3. It is both raining here and not raining here.
Sentence 1 is contingent
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A sentence which is capable of being true and capable of being false
Sentence 2 is necessary truth
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Regardless of circumstance, it is true
Sentence 3 is necessary falsehood
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You don’t need to check to determine if it is true or not, it must be false
John went to the store after he washed the dishes.
John washed the dishes before he went to the store
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Both contingent, might not have gone to store/washed dishes
If either of the sentences is true, then they both are; If either of the sentences are false, then
they both are
Necessarily Equivalent: when two sentences have the same truth value
!!! CHECK OUT G AND H of CHAPTER 3!!!
Chapter 4
Validity in virtue of form
A
If A then C
∴C
A or B
Not A
∴B
Not (A and B)
A
∴ not B
TFL == symbolizing arguments that show that they are valid in virtue of their form
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Sentence letters; basic building blocks of which complex sentences are built
o Used to symbolize certain sentences by providing a symbolization key
Example:
Example:
A: It is raining outside
C: Jenny is miserable
Chapter 5:
Make use of logical connectives in TFL to build complex sentences
Negation
Example:
1. Mary is in Barcelona
2. It is not the case that Mary is in Barcelona
3. Mary is not in Barcelona
Sentence 1 represented by symbolization key ‘B’
Sentence 2 becomes ‘¬B’
Sentence 3 is also ‘¬B’
1. The widget can be replaced
2. The widget is irreplaceable
3. The widget is not irreplaceable
Sentence 1 is represented by R
Sentence 2 can be represented by ‘¬R’
Sentence 3 can be represented by ‘¬¬R’
Conjunction
Example:
1. Adam is athletic
2. Barbara is athletic
3. Adam is athletic, and also Barbara is athletic
Sentence 1 is represented by symbolization key A
Sentence 2 represented by symbolization key B
Sentence 3 is represented by ‘(A ∧ B)’
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1 and 2 are conjucts
If we say something like
Barbara is athletic and energetic
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We can’t say ‘B and energetic’
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Represent E as ‘Barbara is energetic’
Represent as ‘(B ∧ E)’
Although Barbara is energetic, she is not athletic
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Paraphrase to ‘Both Barbara is energetic and it is not the case that Barbara is athletic’
Represent to ‘(E ∧ ¬B)’
TFL does not deal with contrastive such as
Adam is athletic, but Barbara is more athletic than him
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Have to create new symbolization key R to symbolize ‘Barabra is more athletic than Adam’
And then represent as ‘(A ∧ R)’
Brackets are necessary to keep track of things like scope of the negation
Disjunction
Example:
1. Either Fatima will play videogames or she will watch movies.
2. Either Fatima or Omar with play videogames
Symbolization key used:
F : Fatima will play videogames.
O: Omar will play videogames.
M : Fatima will watch movies.
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Sentence 18 represented by ‘(F ∨ M )’
Sentence 19 represented by ‘(F ∨O)’
In TFL, the or is inclusive, it could be both just at least one of them has to be true
1. Either you will not have soup, or you will not have salad.
2. You will have neither soup nor salad.
3. You get either soup or salad, but not both.
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Sentence 1 can be represented with ‘(¬S1 ∨ ¬S2)’
Sentence 2 can be represented with ‘¬(S1 ∨ S2)’
Sentence 3 can be represented with ‘( (S1 ∨ S2) ∧ ¬(S1 ∧ S2))’
o Exclusive or situation
o ‘∧’ can be used for but
Conditional
Example:
1. If Jean is in Paris, then Jean is in France.
2. Jean is in France only if Jean is in Paris.
Symbolization key used:
P : Jean is in Paris.
F : Jean is in France
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Sentence 1 can be represented with ‘(P → F )’
o P is called the antecedent
o F is referred to as consequent
Sentence 2 can be represented with
o Paraphrase as ‘If Jean is in France, then Jean is in Paris’
o Represented with ‘(F → P )’
Conditionals don’t say anything about the causal connection, just that the antecedent is true, then the
consequent is true
Biconditional
Example:
1. Laika is a dog only if she is a mammal
2. Laika is a dog if she is a mammal
3. Laika is a dog if and only if she is a mammal
Symbolization key used:
D: Laika is a dog
M : Laika is a mammal
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Sentence 1 can be represented with ‘(D → M )’
Sentence 2 can be paraphrased as ‘If Laika is a mammal then Laika is a dog’
o Represented with ‘M → D’
Sentence 3 can be represented with ‘(D ↔ M )’
o You can use a biconditional here if it amounts to stating both directions of the
conditional
o Can represent also as ‘(D → M ) ∧ (M → D)’
Unless
Example:
1. Unless you wear a jacket, you will catch a cold.
2. You will catch a cold unless you wear a jacket.
Equivalent sentences; using the symbolization key
J : You will wear a jacket.
D: You will catch a cold.
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Can be represented in a lot of ways
o ‘(¬ J → D)’
o ‘(¬D → J )’ (If you did not catch a cold, you must have worn a jacket
o ‘( J ∨ D)’ (either you wear a jacket or you get a cold)
Week 1 - Tuesday
Chapter 5-6 of Part II and Chapter 9 of PART II, Chapter 7 Part II
Curly letters when we talk about things like the form of an argument
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A, B, C is the place-holder for a sentence (‘object-language’) when operating inside the language
Cursive A,B,C, is the place holder for a place-holder (meta-language) and when you see this front
we are ‘outside’ the language
Symbolization Keys: replacing whole assertoric sentences with single letters; list that pairs sentence
letters with the basic English sentences they represent
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Usually upper case letters and sometimes with subscripts
Fundamental ways of putting sentences together are with the connectives
Cannot break down a ‘basic sentence’ represented by sentence letters by using any one of these
connectives
o Symbolization key should only contain atomic sentences
▪ Ex. Mandy likes skiing or hiking. (bad translation key)
Successful symbolization sometimes requires first paraphrase to ensure basic sentences appear
explicitly
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Watch out for use of pronouns and the tracking of the references
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‘and’ and ‘or’ are ways on joining sentences
Paraphrasing connectives
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English uses many ways of connecting sentences but in symbolization only 5
Not all English connectives can be rendered in any reasonable way by these
Negation
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Conveyed by the word ‘not’
First paraphrase grammatical negation (‘is not’ does not’) using the corresponding basic
sentence prefixed by ‘it is not the case that’
Conjunction
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First paraphrase sentences contained ‘and’, ‘but’, ‘even though’, and ‘although’ using ‘both A
and B’ or just ‘A and B’
Symbolize ‘A and B’ as ‘(A ∧ B)’
Disjunction
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First paraphrase sentences containing ‘or’ or ‘unless’ using ‘either A or B’ or just ‘A or B’
‘(F ∨ M)’
Conditional
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First paraphrase any of these sentences
o ‘If A, B’
o ‘If A, B’
o ‘A only if B’
o ‘B if A’
o ‘B provided A’
Using ‘If A then B’
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Symbolize ‘If A then B’ as ‘(A → B)’
If and only if
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First paraphrase sentences containing ‘if and only if’ or ‘just in case’ using ‘A if and only if B’ can
abbreviate this as ‘A iff B’
Symbolize with ↔
Mixing Connectives
Connectives and Truth-Conditions
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‘True’ and ‘false’ are used to denote what we call truth-values, abbreviate with ‘T’ and ‘F’
Assume that it is just atomic values have truth value and using connectives to connect atomic
sentences generate new truth-values
Propositional logic == truth-functional logic
1. Negation
2. Conjunction
3. Disjunction
4. Conditional
5. Biconditional
Double Negation
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A will always return the truth value back to that of A and so on
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Even number of occurrences of negation symbol applied to A will give the same values as A itself
(A)
Odd number of occurrences with gives the value opposite of A (¬A)
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TFL
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Syntax
o Sentence letters
o Five connectives
o Round brackets
Notion of the sentence of TFL
o Every sentence letter is a sentence (atomic sentence)
o If A is a sentence then ¬A is a sentence
o If A and B is sentences then
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Every sentence is either atomic or built using the five connectives
o There can only be one main connective
TFL Syntax Rules:
(-H/\-S) and -(H\/S) are the same
Either... or... = A ∨ B
Neither... nor = ¬A ∧ ¬B OR ¬(A ∨ B)
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Neither is it the case that Apple is the most popular tech company nor that Microsoft is
the most famous only if Tesla is the fastest-growing car company.
o ((-A /\ -M) -> T)
o (-(A \/ M) -> T)
Inclusive vs Exclusive OR
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Exclusive is (A ∨ B) ∧ ¬(A ∧ B)
If... then... = A → B
… only if.. = 'if A, then B' = A → B
...unless... = 'A unless B' = 'A, if not B’ = ¬B → A
Example:
Calgary is in Canada unless it's not snowing in Paris.
(-S -> -P)
If bachelors are unmarried men, then Miranda is taller than everyone in the room even though she
dances wonderfully.
(B -> (R /\ D))
Either the professor is working in her office or Apple is the most popular tech company unless the TV is
on.
((C \/ A) \/ O) OR (-(C \/ A) -> O)
Chapter 6
Formal definition for a sentence of TFL
In TFL, we decompose sentences into their simpler parts and once we get down to sentence letters, we
know that we are good
Main Logical Operator: dealing with sentence other than sentence letter, some sentential connective
that was introduced last when constructing the sentence
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‘¬¬¬D’, the main logical operator is the first ‘¬’
‘(P ∧ ¬(¬Q ∨R))’, the main logical operator is the ‘∧’
Rules of finding the main logical operator:
The scope of a connective (in a sentence) is the subsentence for which that connective is the main
logical operator for
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Example:
(P ∧ (¬(R ∧ B) ↔ Q))
scope of the negation is just ‘¬(R ∧ B)’
With brackets, we can omit the outermost brackets of the sentence; for example just writing ‘Q ∧ R’
instead of ‘(Q ∧ R)’
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But if we want to embed that sentence into more complicated sentences, we have to put the
brackets back in
We can also use square brackets, no logical difference
Example:
( ( (H → I ) ∨ (I → H )) ∧ ( J ∨ K))
IS SAME AS
[︁ (H → I ) ∨ (I → H ) ]︁ ∧ ( J ∨ K)
Chapter 7 Ambiguity
Sentences can have structural ambiguity and lexical ambiguity
TFL tries to avoid ambiguity by making symbolization keys that do not use ambiguous words or
disambiguate them if a word has different meanings
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e.g., your symbolization key will need two different sentence letters for ‘Rebecca went to the
(money) bank’ and ‘Rebecca went to the (river) bank.’
Scope Ambiguities
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Depend on whether or not a connective is in the scope of another
Example:
‘I like dishes that are not sweet and flavorful’
‘send me a picture of a small and dangerous or stealthy animal’
Example:
‘Avengers: Endgame is not long and boring’
Could be Either:
1. Avengers: Endgame is not: both long and boring.
2. Avengers: Endgame is both: not long and boring.
Use Symbolization Key:
B: Avengers: Endgame is boring.
L: Avengers: Endgame is long.
Sentence 1 can now be ‘¬(L ∧ B)’
Sentence 2 can now be ‘¬L ∧B’
Example:
‘Tai Lung is small and dangerous or stealthy’
Could be Either:
3. Tai Lung is either both small and dangerous or stealthy.
4. Tai Lung is both small and either dangerous or stealthy.
Use Symbolization Key:
D: Tai Lung is dangerous.
S : Tai Lung is small.
T : Tai Lung is stealthy.
Sentence 3 can now be ‘(S ∧ D) ∨ T ’
Sentence 4 can now be ‘S ∧ (D ∨T )’
Wednesday- Week 1
Chapter 10 and 11, 12.4 - 12.5
Exercises from Chapter 11 of Part III but also C, I, J from chapter 12 of Part III
Truth-value assignment for a TFL is assignment of T or F to each basic sentence letter
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Every sentence of TFL receive a truth-value under any given assignment of values to sentence
letters
Nothing else is a sentence other than atomic sentences and complex sentences are with logical
connectives
There is a way of computing the truth-value of the non-atomic sentences
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These are based on the character truth table (the tables are above on this doc)
Computing Truth-Value
(partial true value not focused on this course)
Example (extended (but partial) truth-table – where its already predetermined that H is T and S is F):
Step 1.
Step 2.
Step 3.
* do the main connective last*
Step 4.
Step 5.
Step 6 – Connect the 2 main Connectives
In this case, the negation is the main connective
In truth value, you need 2n lines with n standing for the number of atomic sentences in your target
sentence
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get used to the following procedure: under your first atomic sentence, fill in the first half as T,
and the second half as F. Then under the next atomic sentence, list the first quarter as T, the
second quarter as F, list the third quarter as True, and the fourth quarter as F, so on and so forth
until you hit the last atomic sentence. Your last atomic sentence should have an even number of
alternating T and F.
Example:
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In this example, alternate between every 4, every 3 and every 1
Testing for Validity Using Truth-Tables
Validity in TFL
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Valid if there is no case where all premises true and conclusion is false
Case must make every basic sentence true or false (and not both)
Any assignment make every sentence letter true or false (and not both)
Example --- Disjunctive Syllogism:
Step 1.
Step 2.
Step 3.
Step 4.
Step 5.
Everything checks out here, it is valid
Import Validities to Check:
Example – an Invalid Argument -- Affirming the Disjunct:
Similar Steps to Disjunctive Syllogism (Except Last Step)
The valuation where we find the atomic sentences are both assigned to the value true is a
counterexample of the argument
Import Invalidities to Check:
Affirming the Disjunct and Denying the Conjunct can be logically equivalent, but later on we can see how
we can test for equivalence
Chapter 10
Truth-Functional
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A connective is truth-functional iff the truth value of a sentence with that connective as its main
logical operator is uniquely determined by the truth value(s) of the constituent sentence(s)
o Evert connective in TFL is truth-functional
When treating a TFL sentence as symbolizing an English sentence, we are stipulating that the TFL
sentence is to take the same truth value as that English sentence
Chapter 11
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Valuation is any assignment of truth values to particular sentence letters of TFL
In a truth table, each row of truth table represents a possible valuation; complete one represents all
possible valuations
Example:
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the column underneath the main logical operator tells you the truth value of entire sentence
Complete Truth Table
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Has a line for every possible assignment of True and False
Size of the truth table depends on the number of different sentence letters in the table
Complete truth table with n different sentence letters has 2n lines
Chapter 12.4
Entailment
Testing for Entailment
Example:
‘¬L → ( J ∨ L)’ and ‘¬L’ entail ‘ J ’
Check whether there is any valuation which makes both ‘¬L → ( J ∨ L)’ and ‘¬L’ true whilst making ‘ J ’
false
Entailment is present
ENTAILMENT PRESENT MAKES IT VALID
Thursday- Week 1
Read rest of Chapter 12 of Part III and Chapter 3 Part I (Did we read?)
Should be able to do Practice Problems (4) (5)
Exercises from Chapter 12
Entailment
So if there is entailment, that means the conclusion is valid
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Way of stating validity
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The highlighted is a counterexample where the premises are True and the Conclusion is false
This would make it invalid because a counterexample exists
The premises does not entail the conclusion
Example:
Step 1.
Step 2.
Step 3.
Step 4.
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In the A --> -not C, the --> is the main connective, which means you do the negative first and
then compare the A (T or F) with the negation one
Step 5.
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In every assignment, at least one premise is false or the conclusion is true; which means the
argument is valid and the entailment is correct (no counterexample)
Equivalent Sentences
(truth-functional) equivalence)
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A and B agree in truth value under every valuation
Material Implication (a type of equivalence)
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Equivalent because every line is equivalent in truth value
A counterexample to the equivalence is when the truth values are different
--> could be eliminated, you can replace with each other!! Completely interchangeable
Other Important Equivalences
Other connectives that are not included in the five are in one of these two categories:
1. Can easily find truth-functional equivalents
2. Those which we can’t deal with truth-functionally at all
Exclusive ‘Or’
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You can choose either (the one possibly excludes another, can't have both)
Can already be written by the connectives we already have
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The same as
Neither... nor...
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Can be paraphrased using ‘both ‘[︁it is not the case that A ]︁ and [︁it is not the case that B]︁’
¬A ∧ ¬B OR ¬(A ∨ B)
Unless
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‘(¬ J → D)’ OR ‘( J ∨ D)’ -- simplest way
The Material Conditional
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‘--->’ simply a statement of the way the truth-values of the sentences involved are linked and
nothing more
By itself it does not say anything about why the sentences are linked in that way == became
known as ‘material conditional’; just states the matters of fact about the connection
When A ∧ B is true, A → B is true and B → A is true
A → B doesn’t mean the two have to be logically linked in natural language
Truth Conditions of Conditionals
If X is drinking alcohol (A) , then X is over 18 (B)
-
-
“If A then B” can only be false if
o A is true: we check that the age if X is drinking beer (A true) and we don’t bother if X is
drinking soda (A false)
o B is false: so we check drink if X is underage (B false), we don’t bother if X is over 18 (B
true)
‘If A, then B’ is true if:
o A is false (we don’t bother checking pop)
o Or B is true (we don’t worry about the drink if X is over 18)
o (or both)
Non- Truth Functional Connectives
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Only focuses on the cause and effect relationship
Cannot be represented simply by ‘→’ (implies) cannot fully capture the meaning behind
The situation changes everything and the cause might not always be the cause that led to the
outcome
‘because’ or ‘would have’
No one size fits all translation of words like ‘because’
A →B will be inadequate
Equivalence and Entailment Linked
Tautologies
Example:
Laws of Tautologies
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Law of Excluded Middle
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Law of non-contradiction
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Implications
Dialetheism contradicts Law of Non-Contradiction
A is a contradiction iff it is false on every evaluation (false on every line of truth table)
Example:
‘[︁(C ↔ C ) → C ]︁ ∧ ¬(C → C )’ is a contradiction
-
‘A is entailed by anything’; if A is a tautology, then we always have B is entailed by A, whatever B
is
All F in its truth table would be a contradiction (reserve notion of a tautology)
o If B is a contradiction, then we always have B entailed by A, whatever A is
Example:
‘(H ∧ I ) → H ’ is a tautology
Joint Satisfiability
Example
Finding a counterexample amounts to showing a certain joint satisfiability
Tautologies and Joint Satisfiability
(no contradiction)
Validity (Check Every Assignment) ≠ Satisfiability(finding at least one assignment) ≠ Tautology
Week 2 Monday
Chapter 15 of Part IV and Section 1,3, 4 of Chapter 16
Week 2 Practice Problems (Up to Question 10)
Week 1 Practice Problem (5), 3.4, 3.5, 3.6, 3.7
System of proof: getting logical consequences from a given starting point to prove or deduce
conclusions from premises
1. A proof consists of a finite number of steps
a. First step (first line) and has definite end after finite number of steps
b. If there are n premises to start from, these should be listed as the first n lies/steps
2. The move from each line/step to the next one should take place according to one of a fixed,
finite number of rules
a. Rules of Inference
Because there is only one main connective for each sentence, there is only a small number of rules of
inference
-
Rules of inference can be regarded as mini-proofs (each application of a rule just is a valid
inference)
Deduction/derivation is put together as succession of mini-proofs
Example:
-
-
The first horizontal Line separates the premises (Line 1 and 2)
Once you go below the horizontal line, can only put down a new line if we can justify it by one
of a fixed number of rules, rules which represent as mini proofs
o Example in Line 4, the justification (to the right) consists of 2 things:
▪ Citation of a rule (here ‘--> E’)
▪ Citation of earlier lines which the rule calls on (in this example lines 1 and 3 –
the appropriate number of liens to cite is part of using the appropriate rule)
The second horizontal line represents the assumptions (A on line 3 -- no need to prove that)
Line 4 and 5 are valid inferences from Line 1,2,3
Vertical line at the left tracks the beginning to the end == “main scope line”
Second scope line (In example from lines 3-5) == Subproof
We always end with something, sentence on line 6 (conclusion)
Proof of the sentence ‘Q’ from the assumptions (premises) P1 to Pn
-
Single turnstile
-
When we have a proof of ‘Q’ without calling on any premises, it is represented by
Our immediate aim is to set up a proof-system which respects valid inference or entailment, guarantees
that:
-
When obeyed, proof system is sound
We want to build a sound derivation system
Example Informal Proof:
Formal Proofs – Fitch System of Natural Deduction
Contains:
1.
2.
3.
4.
Numbered lines containing sentences of TFL
A line may be a premise
It it’s not a premise, it must be justified
Justification involves
a. A rule
b. Prior lines (referred to by line number)
Rules of Natural Deduction
-
Rules should be:
o Simple – cite must a few lines as justification
o Obvious --the justification for the rules should be perfectly clear and simple based on
valid inferences
-
-
o Schematic—can be described just by forms of sentences involved
o Few in Number – want to just have a few rules
Rules based on the connectives
Two rules per connectives
o Introduction rule (usually represented as ‘I’)
o Elimination rule (usually represented as ‘E’)
Every line that’s not a premise will be justified
Eliminating ‘/\’
USING P/\Q , you can justify that
With the informal proof example above, for line 4, -S /\ -M can be used and getting –M, you can
conclude that “Marisuz doesn’t enjoy hiking”
Introducing ‘/\’
To justify P/\Q,
Example,
If you had “Sara doesn’t enjoy hiking” and “Marisuz doesn’t enjoy hiking”, conclude that “Neither Sara
nor Marisuz enjoy hiking”
Rules for ‘/\’
‘Simplification’
As long as you have justified in a previous line of P /\ Q (where the conjunction is the main connective)
you can use that line to simplify; only requires one line to justify its use
-
The rule is /\ -E
‘Conjunction’
-
Need two conditions P & Q either given to us or derived
Requires two lines to justify its use
Rule of /\-I
Examples:
Eliminating ‘-->’
To justify using a conditional in the form of ‘P --> Q’
-
To extract information from a conditional, you need the antecedent
Principle behind it is Modus Ponens:
Introducing ‘-->’
To justify ‘P --> Q’ (for example from some premise R),
You need a proof that mirrors
Show that
If we can prove the second, then we can prove the first statement
-
Known as Semantic Deduction Theorem
-
If R, P is entailed by Q then every valuation makes one of R and P false or it makes Q true
If there are no counterexamples to the claim R being entailed by P --> Q that means the entailment
holds
Subproofs
-
-
-
Our aim will be to justify adding the sentence ‘P --> Q’ by giving a proof of the sentence ‘Q’ with
‘P’ as an extra premise
The Fitch Solution: add P as an extra assumption and try to prove Q, but keep track of this P
and what depends on it by further indenting with a vertical line
o Separating the P as a subproof by indenting with another vertical line (with smaller
horizontal line showing that as an assumption)
o Won’t have access to that extra assumption once you close the subproof
Once you finish proving ‘Q’, close the subproof and add ‘P --> Q’ as a new line to the main
vertical line of the proof
o Extra assumption P has been discharged
Nothing inside a subproof is available outside as a justification, since it depends on the extra
assumption made
Rules for ‘--->’
Implication Elimination
Recognize as Modus Ponens
-
Rule is --> -E and it requires two lines to justify its use
As long as you have the conditional and the antecedent of the conditional, you can conclude the
consequence of the conditional
Site the line of conditional and the line of the antecedent
Implication Introduction
-
Rule is --> -I and requires one range of lines to justify its use
Example:
-
Start a subproof where the assumption is the desired antecedent
EVERYTIME YOU WANT TO INTRODUCE A NEW IMPLICATION, YOU NEED TO OPEN A NEW SUB-PROOF
Chapter 16
Formal proof: sequence of sentences, some of which are marked as being initial assumptions (or
premises) and the last line is the conclusion
-
Start the proof by writing the premise
Number each line of the proof
A line is drawn after the premises, everything above the line is an assumption; everything below
the line is either a new assumption or something following the assumption
Conjunction
Perhaps we are working through a proof, and we have obtained ‘R’ on line 8 and ‘L’ on line 15. Then on
any subsequent line we can obtain ‘R ∧ L’ thus
Every line of proof must either be an assumption or must be justified by some rule
Ex. ‘∧I 8, 15’ represents that the line was obtained by conjunction introduction from lines 8 and 15
Conjunction Introduction Rule:
Conjunction Elimination Rule:
YOU CAN ONLY APPLY THESE RULES WHEN THE CONJUNCTION IS THE MAIN OPERATOR
Conditional
Conditional Elimination Rule (modus ponens):
-
In the citation for -->E, always cite the conditional first followed by the antecedent
Example:
P → Q,Q → R ∴ P → R
Tuesday—Week 2
Chapter 16, 2, 5 (Remaining Sections)
Chapter 19 of IV
Week 2 Practice problems 2, 3 4
-
Tomorrow helpful for strategy for complex proofs
Practice Questions at the end of Chapter 16 and Chapter 19
Rules for ‘<->’
Introducing ‘<->’
-
Equivalent to ((P->Q)/\(Q->P))
There can be as many lines as you like between i and j and between k and l
Can come in any order, the second subproof doesn’t have to come immediately after the first
Rule is <->-I and it requires one range to justify its use
Eliminating ‘<->’
-
Can have the antecedent or the consequent, whichever side we have, we can get the other side
Can be as many lines between m and n and n and k
Use sentence P <->Q once with P to get Q and use it again with Q to get P
Rule is <-> -E and it requires two ranges of lines to justify its use
Reiteration
-
Rule only requires one line
You just have to have a sentence
Can introduce the important sub-species of proofs with no premises
Ex. Can show that
Example:
With no premises, you need to always start with the subproof; cannot start without making an
assumption
And then discharge the subproof by using the ->-I rule
EVERYTIME YOU PROVE A SENTENCE WITH NO PREMISES, PROVING THAT THAT SENTENCE IS A
TAUTOLOGY
Example:
Because B is not part of the subproof, and you need B because it’s a conditional, the only way to
introduce B is to open up a subproof and have that subproof start with an assumption that corresponds
to the antecedent of the desired conditional
When P can be proved from no premises, write as
Something is a tautology if it is derivable from anything with no premises
-
Known as the tautological completeness of the fitch system
Completeness
-
Completeness of the Fitch System with Respect to Validity
Whenever an inference is valid, then there exists a Fitch proof of the conclusion from the
premises
Converse to the claim of the soundness of the system
This shows that the Fitch system not just respects valid inference – recall soundness expresses
Subproofs
-
When a rule calls for a subproof cite it as n-m with first and last line numbers of the subproof
After subproof is finished, cite the whole thing and not any individual lines
Subproofs can be nested
Once the surrounding subproof is finished, you can’t cite the subproof again after you close it
Rules for ‘V’
Introduction Rule for ‘V’
-
Given that P entails P \/Q
Q can be anything
Rule only cites one line
No real conditions for using it, just need to have some sentence
Example:
-
The order which your introduce (B \/ A vs. A \/ B) doesn’t matter
Eliminating ‘\/’
-
Justifying using disjunction P\/ Q
If both P and Q separately entail some third sentence R, then we know that R follows from the
disjunction
-
Need two subproofs that show R but in each proof, we are allowed to use one of P, Q as
assumption
Proof by Cases
-
Order of subproofs does not matter
NEED TO CITE 3 THINGS: The line where the disjunction is and the two ranges
Example: A \/ B |- B \/ A
Example: A\/B, A-> B |- B
Contradictions
-
In proofs, we use premises, sentence we’ve proved (new sentences) and extra sentences we’ve
assumed
Sometimes assumptions we have to make are incompatible with the premises
EXAMPLE: to prove the Fitch version of the Disjunctive Syllogism ‘ A\/ B, - B |- A
o Using \/ E
o The assumption of ‘B’ for the second subproof conflicts with the main premises of –B
o We have to use the disjunctive information given so proceed to get A that way
Contradictions – Eliminating ‘¬’ (negation)
-
Negation needs to be the main connective and two contradictory sentences
The order does not matter
The upside down T flags a contradiction
Named ‘-E’ ; cites two contradictory lines
the rule works as long as the upside down T is entered in the same (sub-)proof that P is in; you
can use the rule without reiterating from the column on the left (you can move stuff from the
left column to the right but not the right to the left)
Explosion
-
‘from a contradiction, anything follows’ (Ex Falso Quodlibet)
-
Once you have a contradiction, you can prove anything you like
Whenever we can justify *the upside down T* in a proof, we should be able to justify anything
-
Rule called X, cites one line and this line can only be *the upside down T*
Example: Disjunctive Syllogism
Introduction Rule for ‘¬’
-
An argument is valid iff the premises together with the negation of the conclusion are
jointly unsatisfiable
To justify ¬P, show that P (together with other premises) is unsatisfiable
-
rule cites only one range of lines
Subproof must end with *upside down t* (contradiction) and in the same subproof
(column) as the initial assumption
o only then you are licensed to conclude the initial assumption is unsatisfiable,
hence the contrary must be true
Example:
Indirect Proof Rule
-
Reductio or reductio ad absurdum
Idea is to prove the claim P by beginning a argument/proof with its opposite, ¬P and
showing that this leads to a contradiction (opposite of negation introduction)
Example:
Chapter 16
Reiteration rule allows you to repeat it on a new line
Chapter 19
A is a theorem iff ⊢ A (represented with a single turnstile)
-
There is a proof of A with no undischarged assumptions
Example: showing that ‘¬(A ∧ ¬A)’is a theorem
Example is an instance of Law of Non-Contradiction
Two sentences A and B are provably equivalent iff each can be proved from the other; i.e., both
A ⊢ B and B ⊢ A.
The sentences A1,A2,. . . ,An are provably inconsistent iff a contradiction can be proved from
them, i.e., A1,A2,. . . ,An ⊢ ⊥. If they are not inconsistent, we call them provably consistent.
Week 2 Wednesday
Chapter 17 of Textbook (Good Supplement!)
-
Handout in the Handout Tab
Exercises on Carnap
Exercises on 16, 19 in Textbook
Regular Rule: one that involves no subproofs; either in the same column or in any column to the
right, of all the lines you cite to justify your use of the rule
Escape Rule: involves a subproof; only be used in the context where you have opened a
subproof but the rule is closed to close the subproof; exactly one column to the left of the
subproof(s) that you must cite to justify your use of the rule
Working Backward from a conclusion (goal) means:
-
Find main connective of goal sentence
Match with conclusion of corresponding I rule
Write out (above the goal!) what you’d need to apply that rule
Working forward from a premise, assumption or already justified sentence means:
-
Find main connective of premise, assumption or sentence
Match with top premise of corresponding E rule
Write out what else you need to apply the E rule
Constructing A Proof
1. Write out premises at the top, write conclusion at the bottom
Strategy
1. Look at the conclusion; narrow the last step down to 3 possibilities
a. The I rule associated with the main connective of your conclusion
b. \/E
c. IP (excluding the case where the main connective of conclusion is negation)
2. If an I rule that is a candidate for your last step involves opening a subproof, opening
that subproof should be your first move
3. Look at your premises, if there are no disjunctions in the premises, rule of \/ E
4. Look at your premises, the main connective of each corresponds to the E rule that you
can first use
5. Unpack and evaluate what new E rules you can use
6. Tweak result as you go
7. If you keep hitting dead-ends, you most likely need IP; happens when the argument
depends on hidden assumption or hidden equivalence, such as LEM
Example:
LEM (Law of Excluded Middle)
A \/ ¬A
1. The main connective of the conclusion is \/; so the last step can either be \/I, \/E or IP
2. No premises; no disjunctions in premises so last step cannot be \/ E
3. Assume the last step is \/ I, the rule doesn’t involve a subproof but no premises so a
subproof is necessary
- Trapped within the subproof
4. If IP is the last option, then the first step is to open up subproof with the negation of the
desired conclusion
The structure of this proof is important!! REMEMBER THE
STRUCTURE
Example
(A -> B) \/ (B->A)
1. Main connective is \/ so the last step can either be \/I, \/E or IP
2. No premises, no disjunction so last step cannot be \/E
3. Assume last step is \/I, since the rule doesn’t include a subproof, but no premises so
need to start with subproof
No clear path from A to B
4. Two ways to approach this when you know law of excluded middle exists
De Morgan’s Law
-
One of the instances of De Morgan’s Law is
The main connective is /\ so the options for last step are /\I, \/E, and IP
Premise contains disjunction but main connective is negation so rule of \/E
Could be IP but the main connective is negation; the only way to get information from premise
is to build. Contradiction (what you need to do for IP)
One of the ways:
De Morgan’s Law framed as a Tautology
-
-
Main connective of conclusion is -->, so option for last step is -->I, \/E and IP
No premises, so not \/E
Could be IP, more of a last resort
the main connective of the consequent is a negation, so possibilities for the last step are ¬I or
\/E
The second main connective of the consequent is a disjunction, given the before last
step is ¬I, which means the second step should be A\/B, then the before the last step
may well be \/E
Chapter 17 (Good Supplement!)
Working Backwards from a conjunction
1. If we want to prove A/\B, that means trying to prove it using /\I
2. Prove A first and then prove B
Working Backwards from a conditional
1. Use --> as the end; this requires a subproof starting with A and ending with B
Working backwards from negated sentence
1. Ends with –I, and start with subproof with assumption A
2. Last line of subproof has to be ⊥
Working backwards from a disjunction
1. Work forward, apply the \/I strategy only when working forward
Week 2 – Thursday
Chapter 18 of IV
Simulating the Explosion X
-
You can derive X from the rules we already have
A way to simulate Disjunctive Syllogism is what we need
Example:
Example (Will not get this long proof on assignment):
Week 3 – Monday
Chapter 22.1-3, Chapter 22.4-5
Problem with TFL is that it does not recognize inner structure of these atoms
Names are used to single out objects and use predicates to refer to properties of objects
A particular sentence can be separated into a name and what’s left when removed is the
predicate
-
‘Common Nouns’ -- so pick out collections of objects, unlike proper names which pick
out single object
Every time we substitute a name to fill in the blank, we get a sentence that is either going to be
true of false
Example:
Socrates is older than Plato
-
Two objects are mentioned using proper names
Stated that in some way these objects are related
Same can be done with many relations (two different specific objects), each time we do
this, a sentence which is either a true or false but where the relation on its own _____ is
older than ____ is neither true nor false
-
With the subscripts, the larger of the two numbers indicates how many name places
there are
The numbers also indicate the order in which they are to be considered
-
-
Important because the order of the names make a difference in truth-value to what
we want to say, carries a difference
One place predicates – expresses properties (ex. Property of being a whale, property of being a
wizard)
Two place predicates—expresses ordinary relations (ex. Is older than, younger than, is the
mother of, etc.)
You can turn relations into properties
-
-
-
Ex. ____1 is older than Plato
o The property which some objects have and other’s don’t of ‘being older than
Plato’
There can be relations between more than two objects
o Ex. ‘Betweenness’ is a standard one
▪ Barbara sits between Adam and Claire
- Predicate is ‘between
Can call all these relations ‘predicates’ but can be distinguished by the number of place
(or gaps) they contains called arity -- ‘n-ary’ (replace n with the number of gaps)
o The places which have to be filled in a predicate are called arguments
Symbolization
1. Names: lowercase letters for proper names of English
2. Predicates: uppercase letters with the blanks marked
3. Ex.
-
Putting names in any of the ‘gaps’ will generate atomic sentence
o Atomic sentences are not given, generate atomic sentences by combining names
and predicates
You can make use of connective structure to form sentences such as
Be Careful of:
-
Pronouns (The failing New York Times is fake news)
Combined predicates
Tracking pronouns
Proper Name: to pick out a single specific thing
Common Nouns: ‘hero’ or ‘rock’ which single out collections of things
Only consider names that pick out actual objects in the domain we choose to consider (Ex. Not
Sherlock Holmes in the domain of actual human beings)
Names pick out uniquely and there can be different names for same objects
With predicates, all properties/relations is determinate
-
When we put a name ‘a’ into a predicate ‘B(-)’, what results is always a sentence ‘B(a)’
with a truth value
Example:
Assignment of truth-values to the sentences, atomic sentences (sentence formed with specific
names and predicates) will change
Assigning Truth-Values Depends on:
1. ‘Domain/Universe of Discourse’
a. Ex. The domain of all mammals, domain of all wizards, domain of all natural
numbers – domains can be infinite or finite
2. The selection of objects within that domain to be named by the names given
a. Ex. The name ‘m’ will name Mandy in the domain of all mammals, the name ‘h’
will name Hermione in the domain of all wizards, etc
3. The selection of properties and relations over this domain to be singled out by the
predicates given
3 things combined are called ‘Interpretation’ of a particular FOL
Variables in FOL
-
Object Variable
o Usually using ‘x’, ‘y’, ‘z’ as our object variables
o Can change the earlier formats by using variables instead of gaps with numerical
subscripts
-
Using different variables to stand for the different numerical subscripts
H(g) is a sentence; but H(x) is not a sentence because it is a placeholder (doesn’t specify
anything)
o ‘H(x)’ it is a formula (proto-sentence)
Existential Quantifier
Ex. ‘something’ ‘someone’ ‘there is..’
-
Often go where names and pronouns also go which is why we can mistake that
quantifiers like this functions as names
Quantifiers works differently from names
A quantifier expresses something about the property
Ex. ‘There is a hero’ says something about the property ‘hero’ namely that there is at
least one thing which has that property
Quantifiers provide answers to “How Many” Questions; about the extent of the property: the
number of objects which have it, the 'quantity’
-
‘There is’ Symbol
o Combine it with some formula/proto-sentence (containing a variable) that it
modifies
o Put the variable in question (and which marks the gap) after the symbol
o Is important when there is more than one variable as in ‘A(x,y)’, useful to know
which one of these variables is being picked by the quantifier
▪ Ex. Not making it clear whether we’re going to speak of someone who
admires or someone who is admired
1. In the first case, the same person must be a hero and wear a cape
2. In the second case, one person can be the hero, and yet another can wear a cape
Multiple ‘Ex’ are independent even if they use the same ‘x’
No difference in meaning between
When considering names and predicates, the assignment of truth to those sentence involves
specifying a domain
Existential quantifier as ranging over the domain
The word ‘some’ on its own is a determiner and needs a complement
-
For Example, ‘some hero’ a common noun
o A noun phrase ‘some admirer of Greta’
‘Some F is/has/does G’ OR ‘Some Fs are/have/do Gs’
Week 3—Tuesday
Chapter 22.4-5 of Part V and Chapters 23-25, Chapter 27-28
The Universal Quantifier
-
‘Every’
o With the variable it refers to written after it as in
-
Mnemonic for ‘all’
The quantifier qualifies the predicate that the variable ‘x’ is part of
English has many determiners that express universality such as ‘all’, ‘every’ and ‘any’
o All true in the same case
‘Every F is G’ is true iff everything which is F is G
o The same for ‘All Fs...’ and ‘Any F...’
‘Every F is G’
-
Symbolizing that if it is F then it is G, not that everything has to be F and G
-
This also covers ‘All Fs are Gs’, ‘Every F is G’ ‘Any F is G’
Examples:
‘No Fs are Gs’
Two ways of interpretations
1. There does not exist an F which is also G
2. Whatever F you take, that will not have G
-
There are actually both equivalent
Examples:
‘Only Fs are G’
Example:
Quantifier Equivalences
Other Equivalences:
-
There are going to be systematically two ways of formulating things, one with ‘all’ and
one with ‘exists’ assuming that the negation is put in the right place
The indefinite article of ‘a’
1. If ‘a’ is used to claim existence for example with
a. ‘Greta admires a hero’
2. If ‘a’ is used as a generic indefinite article it is closer to the universal quantifier
a. ‘A hero is someone who inspires’
‘someone’, ‘something’ can require a universal quantifier
-
If these occur in the antecedent, with a pronoun referring back to it in the consequent
For example:
-
‘any’ occurring in the antecedent but without pronouns referring back to them is
existential
Mixed Domains
EXAMPLES LOOK OVER
Identity
-
a 2 place relation which always symbolized with ‘=’
Distinction between what the world is like and what we are able to assert about the
world
One thing can possess two different names but be equal
Everything is identical to itself (Law of Identity)
-
Principle of the Indiscernibility of Identicals
-
Tells you if the two things are identical, then what predicate you can assert as ‘one’
of them must also hold when you refer to another
Another of seeing this is
-
If two things are different, then must be some property which marks the difference
Logical Particle
-
Essential component of FOL, connective, two quantifiers, identity
Identity is logical particle, it is something we adopt in all our FOL
Identity -- ‘Else’
-
In a domain consisting of only people, everyone observes everyone not everyone
observes everyone ELSE, observing itself is included
Can make ‘else’ explicit by using the identity predicate (‘x observes everyone who is not
x’)
Example: There is someone who observes every person other than them
Uniqueness
-
To say that at least one person satisfies a certain thing or there is exactly one person
that satisfies a certain thing and no one else does
‘No one other than x is a hero’
-
All equivalent, choose one that feels the most natural
‘Only’
-
‘Greta is the only hero’, ‘Potter is the only wizard’
o No one other than Greta is also a hero
Only is really ‘exactly one’
If just using the first part it translates to ‘at least one’
‘At Most’
-
There are distinct heroes of a certain account
-
If we say there is at most one heroes, you can deny that there are at least two
-
If x and y are heroes, x and y are the same
o True when there are 0 heroes, true when there is one hero fail when there are
two or more heroes
-
If there are at most two heroes
o Have to deny there are at least three heroes
o Three existential assertions so will involve
(three distinct ones)
o Deny the sentence that there are at least three heroes
OR
‘Exactly’
-
To say there are exactly two heroes is to say both that there are at least two and there
are at most two
OR
-
By increasing the number, the sentences will become very long
‘The’
-
Used to phrases called definite descriptions such as ‘the hero’ or ‘the youngest villain’
‘The A is B’ should be represented in such a way that it is true iff exactly one thing is A
AND that thing is also B
Singular Possessive
-
Example: ‘Autumn wears Greta’s Cape’ which translates to ‘Autumn wears the cape that
belongs to Greta’
OR
‘Both’ ‘Neither’
-
Used as determiners
Example: ‘Neither villain is younger than Greta’
Week 3 – Wednesday
Chapter 29-31 and Chapter 32-33
Term
Bound/Free
Formula
Sentence
FOL
-
Consists of the following
For the predicates, the first number is how many elements it takes, the bottom is the
number of that element in the list
Name + Predicate == Atomic Sentence (no free variables)
Variable + Predicate == Atomic Formula (free variables exist)
If A is a sentences/formulas then B is a sentence/formula, etc
If A is a formula which contains a free occurrence of the variable x (and no other free
occurrences of a variable), then ∃xA is a sentence
If A is a formula which contains free occurrences of the variable x (and no other free
occurrences of a variable) then ∀xA is a sentence
Denote the assignment of an object to a variable ‘x’ by ‘o(x)’ which we can reads as ‘the object
temporarily assigned to ‘x’
-
These assignments are called satisfaction functions ‘o(x) satisfies ‘H’
Truth of Quantified Sentences with Existential Quantifier
Truth of Quantified Sentences with Universal Quantifier
First Order Valid
Example:
To show if it is valid or not, change the predicate and objects to see if it is still valid
Set the domain for counterexample
If the domain has a counterexample, it has a counterexample in the domain that is finite, it’s
good to restrict the domain (ex. Not just all planets, inner planets)
First, second and third premises are true but the conclusion is false, inner planets don’t have
rings, so we provided a counterexample by specifying the interpretation
Produced a counterexample by specifying an interpretation
Extensions--> Example what we have to put on Carnap
Because Carnap isn’t flexible, we need to make our domain only have finitely many numbers
which each number potentially standing in for the name of an object (Usually no more than 5
objects)
-
Associate a with 2, then 2 just is a
Draw Venn Diagrams to know if the
counterexample interpretation is correct
Making “Some As are Bs” True
-
Extension of A and B must have something in common (filled area must contains at least
one object)
A and B can overlap, be equal or be contained
Making “Some As are Bs” False
-
Extension of A and B must have nothing in common
A and B don’t overlap, or one or both is empty
Making “All As are Bs” True
-
Extension fo A must be contained in extension of B
Extensions of A and B are the same
OR Extension of A can be empty
Same situations make...
o “Only Bs are As” True
o “Some As are not Bs” False
Making “All As are Bs” False
-
Extension of A must contains something not in B
Extension of A cannot be empty but B can be empty
Same situations make
o “Only Bs are As” false
o “Some As are not Bs” true
Semantic Notions in FOL
What you can and cannot show in FOL
Week 3 Thursday
Chapter 29-33
Practice Problem 1-7
Dependent On Interpretation
First Order Truth
First Order Truths (First Order Validities) == ‘always true’, ‘true in all cases’
-
All possible cases are substituted for some selection of an interpretation
First Order Equivalence
-
Can't be a situation where one is true and one is false
Example:
Equivalent
-
Example:
Equivalent
Validity of Inference in FOL
-
-
If an inference has a first order truth as its conclusion, it will be valid, regardless of what the
premises are
o Cant find an interpretation and an assignment over it which makes the conclusion false
If an inference where all the premises are jointly unsatisfiable, then that inference is valid
o Cant make all premises true no matter how we choose the interpretation and
assignment
In FOL, there are too many interpretations and impossible to look all possibilities
Counter-interpretations
Interpretations in which certain formulas fail to be true under a carefully chosen assignment while
others hold:
-
In first-order truth, Looking for an interpretation/assignment where a single sentence fails,
showing that the sentence can’t be first-order truth
In first order equivalence, looking for interpretation/assignment where one sentence is true and
the other is not, showing that the sentences can’t be first-order equivalent
In validity, looking for interpretation/assignment in which the premises are true but the
conclusion is false
Producing a Counter-Interpretation
-
-
a simple way is to choose the natural numbers as domain and interpret the predicate ‘A’ by ‘is
even’ and the predicate ‘B’ by ‘is odd’
Example:
First sentence will turn out true with ‘All numbers are even or odd’ while the second is false with
‘All numbers are even or all numbers are odd’
Thus.
Counter Interpretation needs to specify
1. Number of objects in the domain
2. Which objects have whichever property a given predicate represents
3. Which objects stand in whichever relation a given relation symbols represent
All these are answered in a way of lists
1.
2.
3.
4.
List of objects in domain
List of constants noting which object of the domain they pick out
A list of the properties which predicates represent
List of relations which relation-symbols represent
Example:
For an Interpretation you need:
1.
2.
3.
4.
Domain—collections of objects that aren’t empty
Referent for each name (which object it names)
Properties, list of which object has which property, which object has property B, H, F
Relations, which property has relations, pairs of objects standing in the relation, E
Strategy:
Example Continued
Another Example:
Week 4--- Monday
Section1,4 of Chapter 34, parts of Chapter 25 and all of Chapter 37
Extension of the Fitch System for TFL (things such as IP, X, R still apply)
-
The basis of the Fitch proofs is still used, the system of old rules we have for the connectives
To be able to apply the rules, get down to formulas with connective structures – so ‘\/’ or ‘-->’
and this is what the quantifier rules help us do
- Quantifier Rules either
1. Strip off the quantifiers to get the structures
2. Restoring the quantifiers to make sure we have the right form of sentences/formulas we want
to derive
Identity Introduction
-
No conditions to derive it
-
All first order truths
Identity Elimination
-
Represented by ‘=E’
-
Once you have the two conditions, you can replace some or all of the s/t with the other
The place-holder A stands for Formulas/Sentences
Two names of ‘d’ and ‘e’ for example can name the same thing (d=e)
-
Any case where ‘d’ is used is also true stated with ‘e’
-
If something is true when stated using ‘d’ but false when stated with ‘e’, then ‘–d=e’
Example:
-
‘My baby is me’ can be symbolized as ‘b=I'
Equivalent to ‘I am my baby’ so to ‘i=b’
o Referred to as the symmetry of identity
o Can prove it through:
Transitivity of Identity
Universal Elimination (Universal Instantiation)
-
The simplest formula close to ‘AxA(x)’ is ‘A(t)’ where ‘t’ is any term-- ‘A(t)’ can be called instance
of the universal
Can now complete the example from before
-
The ‘b’ goes in for ‘x’ in 1 and the same for 4 using line 2
*LOOOK AT LAST LINE* YOU CAN REPLACE*
Universal Introduction/Generalization
-
-
Have to impose certain restrictions
Two conditions in order to guarantee that the use of the name ‘c’ is arbitrary (nothing specific
be known about the object named) enough
1. The name ‘c’ must not appear in any premise or any assumption of a sub proof not already
ended
a. If ‘c’ does appear in any assumption of a sub proof, that assumption must be discharged
by the time the ‘AI’ rule involving it is applied
2. ‘A(x)’ must be obtained from ‘A(c)’ by replacing all occurrences of ‘c’ by ‘x’ AND this variable
should be new to the formula
a. ‘x’ must not appear at all in the formula ‘A(c)’
b. C must be fully taken out of the formula
This is correct usage:
-
Cannot generalize on a name introduced by universal elimination if that name also appears in
the premises or a sub proof that is not concluded yet
Can generalize on a name introduced by universal elimination if the name does not appear in
the premises and it does not appear in unfinished sub proof
‘All A’s are B’s’
-
Assumption is already discharged, sub proof is over by the time that ‘AI’ is used
Example:
Example:
Week 4 – Wednesday
Remaining Chapters of 34-35
3 is Easy, 4 is something you SHOULD COMPLETE, 5 is something that is hard not necessarily for
assignment
Existential Introduction/Witnessing
-
Can replace one or more occurrences of ‘c’ by the variable ‘x’
You can use ‘AI’ as many times as you want, but ‘EI’ you can’t use as many times as you want
Example:
-
But you can use ‘EI’ as many times as you like on what is ultimately the same statement with
different instantiations
Existential Elimination/ Instantiation
-
The rule
Several Conditions are needed
1. REPLACE ALL INSTANCES OF ‘x’ BY THE SAME ‘c’
2. ‘c’ must NOT occur in ANY PREMISE or ANY DISCHARGED ASSUMPTION OR IN
a. To guarantee it, the name ‘c’ introduced in line ‘i’ is NEW to the proof at that point
3. ‘c’ also must NOT occur in the conclusion of the subproof; ‘B’ must not contain the new name
introduced specifically for using
Example:
Example:
THE ORDER THAT THE QUANTIFIERS APPEAR IN MATTERS AS WELL AS THE ORDER YOU EXECUTE ‘AI’
AND ‘EI’!!
TIPS:
Strategy:
Quantifiers interchanges
Example:
Harder Example:
IP is the candidate for the last step
All of De Morgan’s Law Equivalences are still valid (tautologies)-Example: (Universal Qualifier Version)
Example: (Existential Edition):
Week 4 – Thursday
Example:
Example:
Example: FOL is important for expressing claims involving quantities
Example: Drinker’s Paradox is first-order validity (One of the examples on Carnap)
If someone is drinking, then everyone is drinking