Truth Tables and Validity Flashcards
What are all the operators thus far introduced (&, ∨, ¬, and →)
Truth functional operators
Given a sentence A & B, what can we work out?
The truth value just by knowing whether A is true and B is true.
The truth value of any sentence built from truth-functional operators is completely determined by what?
The truth value of the basic propositions in it
What is the way of doing logic with truth function operators called?
Truth Functional Logic (TFL)
What 3 things does the conditional not say concerning the sentence:
‘IF I’m in Paris THEN I have blue hair’?
- It doesn’t say ‘Whenever to go to Paris, I always have blue hair’
- It doesn’t say ‘If I were to go to Paris then I would have blue hair.’;
- It doesn’t say ‘IN any situation in which I’m in Paris, I have blue hair.’
What does the sentence ‘IF I’m in Paris THEN I have blue hair’ actually say?
If it’s true that I’m in Paris then it’s true that I have blue hair
When we think about whether an argument is valid, we think about what?
All possible situations and whether we can imagine a way to make the premises true and the conclusion false.
When we think about whether a Conditional is true in some situation, we think about what?
Whether the antecedent and the conditional are true or false in that particular situation.
What are the four types of conditional?
- Rule;
- Decision;
- Cause;
- General;
- Inference
Concerning the inferential form of the conditional, say you have an argument ‘If the fingerprints match then you’re guilty’ and the fingerprints do not match, is it valid to claim that the person is not guilty?
Invalid
Why is that invalid?
Because the person could be proven guilty for other reasons.
Formalise the argument:
P1 ‘IF the fingerprints match, then you’re guilty’
P2 ‘The fingerprints do not match’
Conc: ‘You’re not guilty’
P1: F => G
P2: ¬ F
Conc: ¬ G
With this argument, what row of the truth table for the conditional proves it incorrect?
Row three
How can we express a stronger conditional?
With the biconditional
What is the biconditional in natural language?
‘If and only if’
What is the logical operator for the biconditional?
<=>
If we were to rewrite the argument ‘The fingerprints match if and only if you’re guilty’, what would happen to the aforementioned issue?
Then if the fingerprints did not match, you would not be guilty.
What are the four possibilities we have to consider with the biconditional?
T T | T
T F | F
F T | F
F F | T
What do we look for to prove an argument’s validity?
A counterexample
Define a counterexample
A situation in which all the premises are true and the conclusion is false
Define a valid argument
In any situation in which all the premises are true and the conclusion is also true
What is meant by ‘situation’ here?
‘A way the world could be’ - it could be a realistic scenario or something completely fantastical
For any situation we consider, what is the sole thing we care about?
Whether the premises and conclusion are true
What is the truth table for the argument
P1: A => B
P2: A
Conc: B
________P1 P2 Conc
A B | A => B | A | B
———————————-
T T | T | T | T
T F | F | T | F
F T | T | F | T
F F | T | F | F
P1 | P2|Conc
How, with a truth table, do we see if it valid?
Check every row in which all the premises are true and if the conclusion is true then the argument is valid
What is the truth table for the argument
P1: A => B
P2: ¬B
Conc: ¬A
_______P1 P2 Conc
A B | A => B | ¬B | ¬A
———————————-
T T | T | F | F
T F | F | T | F
F T | T | F | T
F F | T | T | T
What is the truth table for the argument
P1: A => B
P2: ¬A
Conc: ¬B
_______P1 P2 Conc
A B | A => B | ¬A | ¬B
———————————-
T T | T | F | F
T F | F | F | F
F T | T | T | F
F F | T | T | T
In this example, why is it invalid?
_______P1 P2 Conc
A B | A => B | ¬A | ¬B
———————————-
T T | T | F | F
T F | F | F | F
F T | T | T | F
F F | T | T | T
It is invalid because row 3 and row 4, all premises are yet they don’t both have a true conclusion
What is the form of this complex sentence:
‘It will be sunny on Wednesday and
it will rain on Thursday or it will rain on Friday’
A & (B V C)
A stands for ‘It will be sunny on Wednesday’;
B stands for ‘It will rain on Thursday’;
C stands for ‘It will rain on Friday
The brackets in A & (B V C) are important. Why?
To remove ambiguity. For A & (B V C) is different from (A & B) V C
We can take any two complex sentences and join them together with an operator to make a bigger sentence. Translate:
‘If the train is late or I forget my passport then
I’ll miss my flight and I’ll be sad.’
(A V B) => ( C & D)
A stands for ‘The train is late’;
B stands for ‘I forgot my passport’;
C stands for ‘I’ll miss my flight’; and,
D stands for ‘I’ll be sad’
Concerning this argument what do we want to know?
P1: (A => B) & (B => A)
Conc A <=> B
The truth values of (A => B) & (B => A) and A <=> B
To work out the truth value of (A => B) & (B => A), what do we first have to know?
The truth values of (A => B) and (B => A)
So before we fill in the column for the premise, what do we have to fill in?
The intermediate columns: (A=>B) and (B=>A)
What are intermediate columns for?
Our own convenience
Why do we only need intermediate columns for?
Because we cannot jump straight to figuring out the values of the premises and conclusion
Once the intermediate columns are figured out, what are the values in the intermediate columns?
Not important anymore
