Basic Notions of Logic Flashcards
What is an argument?
An argument is any set of declarative sentences, one of which is designated as the conclusion of the argument, any others being its premises
What is another term for the words true and false?
Truth-values
What is the Principle of Bivalence?
The assumption that every declarative sentence is either true or false.
Name two reasons why the Principle of Bivalence is controversial.
- “Sherlock lives in 221b Baker Street’ is neither true nor false because Sherlock Holmes doesn’t exist.
- There is ‘no fact of the matter’ to some statements such as whether something takes good or is beautiful.
Define valid argument
An argument is valid if and only if it is not possible for the premises to be true and the conclusion false.
Alternatively, whenever the premises are true, the conclusion must also be true. (everything apart from TT/F)
Define invalid argument
An argument is invalid if and only if it is possible for the premises to be true and the conclusion false. (TT/F)
Name this symbol:
|=
double turnstile
What does Γ |= A mean?
The argument with premises Γ and conclusion A is valid
What does Γ|/= A mean?
The argument with premises Γ and conclusion A is invalid
If an argument is valid and has a false conclusion, one can we deduce from the definition of argument?
That it has at least one false premise.
What are the three kinds of valid arguments?
- True premises and a true conclusion
- At least one false premise and a true conclusion
- At least one false premise and a false conclusion
What is a sound argument?
An argument is sound if and only if it is valid and its premises
are true. (TT/T)
What is an unsound argument?
An argument is unsound if and only if it is either invalid or has at least one false premise.
Why is an argument being sound or unsound significant?
Classing an argument as sound or unsound is not subjective. Proves an argument is convincing objectively.
What is a set of sentences?
Collection of sentences where one is not singled out as a conclusion
What is a consistent set of sentences?
A set of sentences is consistent if and only if it is possible for all the members of that set to be true.
What is an inconsistent set of sentences?
A set of sentences is consistent if and only if it is not possible for all the members of that set to be true.
What is a logically false sentence (or contradiction)? Give an example.
A sentence is logically false or a contradiction if and only it is not possible for the sentence to be true. e.g It is raining and it is not raining.
Would a set consisting only of a single logically false sentence be consistent or inconsistent?
logically inconsistent because members (only one in this case) cannot all be true.
Can a set containing a logically inconsistent sentence be consistent?
No. It will be inconsistent.
If a set is logically inconsistent but contains only one sentence, what type of sentence must it be?
False
What is a logically true sentence (or tautology)? Give an example.
A sentence is logically true or a tautology if and only if it is not possible for the sentence to be false. e.g. It is raining or it is not raining.
What is a logically indeterminate or contingent sentence? Give an example.
A sentence is logically indeterminate or contingent if and only it is neither logically true nor logically false. e.g It is raining.
What are logically equivalent sentences?
The members of a pair of sentences are logically equivalent if and only it is not possible for one of the sentences to be true while the other sentence is false.
Can an argument be logically false?
No, only a sentence can be logically false.
Can a sentence be inconsistent?
No, only sets of sentences can be inconsistent.
Can sentences be valid?
No, only arguments are valid.
When is the only time a sentence could be valid?
But we’ll allow ourselves to call sentences valid if they can be the conclusion of a valid argument that has no premises, i.e. if they are logically true.
Sentences
logical truth, logical falsity and logical indeterminacy
Pairs of sentences
Logical Equivalence
Sets of sentences
Consistency and inconsistency
Arguments
validity, invalidity, soundness and unsoundness
If an argument has a set of inconsistent premises and an arbitrary conclusion while it be valid or invalid?
Always valid because at least one sentence will be false (within the premises) therefore it can never ben the case that the premises are all true when the conclusion is false. The argument will not be sound.
If an argument has a logically true sentence as its conclusion with arbitrary sentences as its premises, will it be valid or invalid?
Valid. It cannot be the case that the premises are true and the conclusion false.
If the conclusion of an argument was identical to one of its premises, would it be valid or invalid?
It cannot be true that the premises are true and the conclusion is false, therefore it is valid.
What is begging the question?
The argument assumes what has already been established. e.g. if the conclusion is identical to one of the premises.
If we have a set consisting of one sentence which is designated as the conclusion, can it still be classed as an argument?
Yes.
If an argument with no premises is valid, what must the conclusion be and why?
Logically true because it can only be the case that it is not possible for the premises of the argument to be true and the conclusion false. If it is not possible for the conclusion to be false, it must be logically true.
What does |=A mean?
A is a tautology, A follows from no premises.
What does “A → B” mean?
If A, then B
A is the antecedent, B is the consequent
What does “A ∧ B” mean?
A and B
Conjuncts
What does “A ∨ B” mean?
A or B
Disjuncts
What does “A ≡ B” mean?
A if and only if B. Biconditional
What does “¬A” mean?
not A, negation
Which principles allow us to produce a valid argument from another argument?
Consequentia mirabilis, conditional proof, proof by cases and reductio ad absurdum
What are logical fallacies?
Logical fallacies are inferences where the truth of the premises does not guarantee the truth of the conclusion.
What are Valid inferences?
Valid inferences are inferences where the truth of the premises guarantees the truth of the conclusion: it is impossible for the premises to be true and the conclusion to be false.
What is a syllogism?
A syllogism is an inference with two premises and a conclusion, where the premises and con- clusions state relations between properties.
Universal affirmative
All A are B
Universal negative
No A are B
Particular affirmative
Some A are B
Particular negative
Some A are not B
