Truth functional logic (relevant) Flashcards
Three kinds of symbols in TFL
Atomic sentences, connectives, brackets
Formal definition of a sentence of TFL
- Every atomic sentence is a sentence
- If A is a sentence, then ¬A is a sentence
- If A and B are sentences, then (A ˄ B) is a sentence
- If A and B are sentences, then (A v B) is a sentence
- If A and B are sentences, then (A → B) is a sentence
- If A and B are sentences, then (A ↔ B) is a sentence
- Nothing else is a sentence
Every sentence is constructed out of…
An atomic sentence
The connective that was introduced most recently =
Main logical operator (main connective) of the sentence
What is the scope of a negation?
The subsentence for which that connective is the main logical operator
A connective is truth functional if the truth value of the sentence…
Is uniquely determined by the truth value of the constituent sentence
The truth value of a negation is uniquely determined by…
The truth value of the unnegated sentence
The truth value of a conjunction is uniquely determined by…
The truth value of both conjuncts
The truth value of a disjunction is uniquely determined by…
The truth value of both disjuncts
How can we assign truth values?
Valuations- find out if atomic sentences are to be true or false
Valuation
Is any assignment of truth values to particular atomic sentences of TFL
What does a truth table represent?
All possible valuations
What is a gap in a truth table represented by?
-
What can a truth table be used for?
Calculating the truth value of complex sentences, each line represents a variation, can use to determine if a sentence is a tautology
When is a sentence a tautology?
If it is true on every valuation
Necessary truth =
Tautology symbolized
Necessary falsity =
Sentence is a contradiction if it is false on every valuation
When are sentences jointly tautologically consistent?
If there is some valuation which makes them all true
Jointly tautologically inconsistent
There is no valuation which makes them all true
If they tautologically entail…
Then they are valid
We can demonstrate the validity of arguments by?
- Symbolising the arguments in TFL
2. Checking for tautological entailment using truth tables
Can we demonstrate the validity of every argument through TFL and tautological entailment?
No
Why might TFL not demonstrate the validity of an argument?
It is formally valid but our formal language is not strong enough to capture its formal validity
Tautological equivalence
Sentences are tautologically equivalent iff they have the same truth value on every valuation.
