Logics Flashcards
What is a proposition?
It’s a complete frase that states something; aka a claim.
It is the basic unit of propositional logics.
When analysing texts, these propositions can be substituted by single letters or any other symbol, for simplicity.
What are claims joined in a conjunction called?
Conjuncts.
In basic compound claims, what does OR conote?
Disjunction
In basic compound claims, what do IF and THEN conote?
Conditional
In basic compound claims, what does NOT conote?
Contradictory
What is the only case in which a conjunction is true?
When all claims are individually true.
What is syntax?
The way in which elements are ordered to form a complete claim (phrase).
Give examples of a few words that can serve as a conjunctive.
But
Although
However
Yet
What is semantics?
The meaning (or the study of the meaning) of something specially language. Connotative meaning.
What is a conjunction?
A compound claim formed of two or more claims and in which all claims must be true for the conjunction to fulfil the truth condition.
How many basic compound claims are there?
Four
Which properties makes an argument Valid?
If all the premises are true, than the conclusion CANNOT be false;
i.e.
It is LOGICALLY IMPOSSIBLE for the premises to be true and the conclusion false;
i.e.
The truth of the premises GUARANTEES the truth of the conclusion.
What can validity, in logics, be applied to?
Only entire arguments and not single premises or conclusions.
In basic compound claims, what does AND conote?
Conjunction
What is the truth condition for a disjunction?
At least one of the disjunction must be true.
What are the claims in a disjunction called?
Disjuncts.
When are more than one disjuncts true in a disjunction?
In an Inclusive OR (inclusive disjunction)
For example: “A triangle can be defined as a polygon with three sides OR as a polygon with three vertices”
When can only one disjunct be true?
In an exclusive OR (exclusive disjunction)
When is an inclusive disjunction false?
Only when both disjuncts are false.
When is an inclusive disjunction true?
When either one of the disjuncts is true.
What word gives hint of an exclusive disjunction?
Either
In conditional claims, what is the name of the proposition that comes after the IF and before the THEN?
Antecedent
In conditional claims, what is the name of the proposition that comes after the THEN?
Consequent
What is affirming the consequent?
Assuming the antecedent from the consequent.
What is the assertion in a conditional proposition?
The relationship between the antecedent and the consequent.
What are conditions under which a conditional is false?
When the antecedent is true and the consequent is false.
What is a simple claim?
A claim that has no other claims as a component.
What does a contradictory (not-claim) does to a claim.
It reverses its truth value.
Makes it false. If A is a claim, not-A is the opposite.
What is the difference between a contradictory and a contrary?
In the CONTRARY two claims cannot be true at the same time but both can be false at the same time.
In the CONTRADICTORY they cannot be true at the same time and they cannot be false at the same time.
What is a Contradiction?
It’s a conjunction of the form:
A and not-A
Example: All swans are wholly white and all swans are wholly brown.
When can a contradiction be true?
Never
In logics, what is consistency?
A set of claims is consistent if it is logically possible for all of them to be true at the same time.
What does “logically possible” mean?
Does not entail a contradiction.
n.b. a contradiction is a set of claims that put together form a falsity in all possible worlds.
What can and can’t Logic tell us about inconsistencies?
It can tell us that, on a set of claims, at least one is contradictory; but it doesn’t show us which one it is or how to find it conclusively.
What are the four basic compound claims?
Conjunction (a AND b AND c and etc…)
Disjunction (a OR b OR c OR etc…)
Conditional ( IF a THEN b)
Contradictory (NOT-a)
What effect does a negation of a contradictory ( NOT-(NOT-a)) have?
It restores the original claim.
What does the contradictory of a conjunction produce?
A disjunction.
How is the contradictory of a conjunction constructed?
By changing the AND to an OR and negating each claim with an “Either” in the beginning.
In algebraically terms, describe the negation of a conjunction.
not-(A and B) = (not-A) or (not-B)
What does the negation of a disjunction produce?
A conjunction which is negated.
not-(A or B) = (not-A) and (not-B)
In English language, what is (not-A and not-B) equivalente to?
Neither A nor B
Ps: “nor” is not a disjunction but a CONJUNCTION.
What are the two DeMorgan’s Rules?
not-(A and B) = (not-A) or (not-B)
not-(A or B) = (not-A) and (not-B)
What does the contradictory of a conditional produce?
A conjunction
What is the general rule for the contradiction of a conditional?
not-(if A then B) = (A) and (not-B)
Ps: normally this is phrased as A “but” not-B
B if A is equal to…
If A then B
In (A only if B) which is the antecedent and which is the consequent?
The antecedent come before the “only if” so (A) is the antecedent.v
The standard form is:
If A then B
The roles of the IF and THEN are reversed in the non-standard form.
What is the general relationship between IF and ONLY IF claims?
consequent IF antecedent
antecedent ONLY IF consequent
What is a biconditional?
When the claim in a conditional always have the same truth value.
What is the relationship between a conditional and a buconditional?
A conditional is half of a biconditional.
Conditional = If A then B = B if A
Biconditional = B if, and only if A
What is the rule for (A unless B)
(B) is the antecedent; it is negated. (A) is the consequent.
A unless B = If not-B Then A
Unless = If not
What is the general rule for Contaposition?
If A then B = If not-B then not-A
Explain how are contrapositives created?
Put the consequent first and the antecedent last and negate both claims.
What are the possible forms a disjunction can take?
A or B
A or (not-B)
(not-A) or B
How can a conditional be written as a disjunction?
If A then B = (not-A) or B
Describe the truth table of Conditionals.
A THEN B
t T t
t F f
f T t
f T f
Describe the truth table of a Disjunction.
A OR B
t T t
t T f
f T t
f F f
Describe the truth table of a Conjunction.
A AND B
t T t
t F f
f F t
f F f
What is the general rule for the translation of a conditional into a sentence?
If A then B = A is sufficient for B
i.e. not necessary, but sufficient; there are other ways to achieve B.
What is the general rule for conditionals in relation to necessity?
If A then B = B is necessary for A
i.e. necessity is the inverse of sufficiency for conditionals.
State the rules for necessity and sufficiency for conditionals.
A is sufficient for B = If A then B
A is necessary for B = If B then A = If not-A then not-B
