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What is an argument?
An argument is a set of sentences in which one, the "conclusion," is supported by the others, which are called "premises."
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How do you know an argument is deductively valid?
It is deductively valid if it is not possible for the premises to be true and the conclusion false.
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How do you know an argument is deductively invalid?
It is deductively invalid if it is possible for the premises to be true and the conclusion false.
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How do you know an argument is sound?
It is sound if and only if all the premises are true and it is deductively valid.
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How do you know an argument is unsound?
It is unsound if and only if either one of the premises (at least) is false or it is invalid.
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How do you know if all the sentences in a set are logically consistent with each other?
They are logically consistent if it is possible for all the members of that set to be true.
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How do you know if all the sentences in a set are logically inconsistent with each other?
They are logically inconsistent if it is not possible for all the members of that set to be true.
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Are these sentences (1) conjunct or disjunct, and (2) logically consistent or inconsistent with each other: Mind and Body are distinct substances. Mind and body interact. No distinct substances interact.
(1) They are conjunct, since they overlap. (2) They are logically inconsistent because it is not possible for all of them to be true.
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Are these sentences (1) conjunct or disjunct, and (2) logically consistent or inconsistent with each other: All humans are rational. All angels are rational. No angels are human.
(1) They are disjunct, since they do not overlap. (2) They are logically consistent because it is possible for all of them to be true.
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Are these sentences (1) conjunct or disjunct, and (2) logically consistent or inconsistent with each other: The earth is flat. The moon is made of cheese. Gold is inexpensive. Humans never die.
(1) They are disjunct, since they do not overlap. (2) They are logically consistent because it is possible for all of them to be true.
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How do you know a pair of sentences is logically equivalent?
They are logically equivalent if it is impossible for the sentences to differ in truth value. This means that it is not possible for one to be true while the other is false.
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How do you know a pair of sentences is not logically equivalent?
They are not logically equivalent if it is possible for the sentences to differ in truth value. This means that it is possible for one to be true while the other is false.
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Are the following sentences logically equivalent: All doctors are rich people. All rich people are doctors.
They are not logically equivalent.
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Are the following sentences logically equivalent: No doctors are rich people. No rich people are doctors.
They are logically equivalent.
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Are the following sentences logically equivalent: John is tall. John is not short.
They are not logically equivalent because John could be of average height.
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Are the following sentences logically equivalent: Both the Jets and Giants lose. Neither the Jets nor the Giants win.
They are not logically equivalent because they could have tied.
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How do you know an individual sentence is logically true (a tautology)?
An individual sentence is logically true (a tautology) if it is not possible for the sentence to be false.
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How do you know an individual sentence is logically false (a contradiction)?
An individual sentence is logically false (a contradiction) if it is not possible for the sentence to be true.
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How do you know an individual sentence is logically indeterminate (contingent)?
An individual sentence is logically indeterminate (contingent) if it is possible for the sentence to be true and it also possible for the sentence to be false.
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Is the following sentence a tautology, a contradiction, or contingent: If they win, they win.
It is a tautology, because it is not possible for the sentence to be false.
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Is the following sentence a tautology, a contradiction, or contingent: The mind is the brain and it is not the brain.
It is a contradiction, because it is not possible for the sentence to be true.
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Is the following sentence a tautology, a contradiction, or contingent: Either they win or they don't win.
It is a tautology, because it is not possible for the sentence to be false. Think of it as "A or not A" and you'll see why it cannot be false.
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Is the following sentence a tautology, a contradiction, or contingent: All doctors are rich people but some rich people are not doctors.
It is contingent, because it is possible for the sentence to be true or false.
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How is the truth value of atomic (aka simple or non-compound) sentences determined?
The truth value of atomic sentences are determined by a truth value assignment (aka truth value interpretation or valuation) which assigns a T or an F to each atomic sentence in sentential logic (SL). Note: Truth value assignments may or may not be determined by the actual truth values of the sentences which the atomic sentences stand for.
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How is the truth value of molecular (aka compound) sentences determined?
The truth value of molecular sentences is determined by the truth values of the atomic sentences which occur in it, according to the rules for the truth-functional connectives (and, or, not, if/then, if and only if).
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How is the truth value of the "and" main connective (MC) determined?
- The "and" MC is true if and only if both conjuncts are true.
- The "and" MC is false if and only if one or both conjuncts are false.
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How is the truth value of the "or" main connective (MC) determined?
- The "or" MC is true if and only if one or both disjuncts are true.
- The "or" MC is false if and only if both disjuncts are false.
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How is the truth value of the "not" main connective (MC) determined?
- The "not" MC is true if and only if the original sentence is false.
- The "not" MC is false if and only if the original sentence is true.
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How is the truth value of the "if/then" main connective (MC) determined?
- The "if/then" MC is true if and only if either the antecedent is false or the consequent is true.
- The "if/then" MC is false if and only if the antecedent is true and the consequent is false.
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How is the truth value of the "if and only if" main connective (MC) determined?
- The "if and only if" MC is true if and only if both sentences have the same truth value.
- The "if and only if" MC is false if and only if the sentences differ in truth value.
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What do main logical properties express?
Main logical properties express what truth values (true or false) are possible or not possible with regard to a sentence, a set or pair of sentences, arguments, or to a set of sentences in relation to another sentence.
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How are truth values dependent on form?
Form indicates which truth values are possible or not possible.
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In sentential logic (SL), how is form analyzed?
In SL, form is analyzed in terms of non-compound (aka atomic) sentences and compound ones formed from them using the five connectives (and, or, if/then, if and only if, not).
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What does "truth-functional" mean with regard to a compound sentence with connectives/operators (or, if/then, and, not, if and only if)?
It means that the truth value (either T or F) of the compound sentence is wholly determined by the truth values of its atomic parts.
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What is a truth value assignment (TVA)?
- A TVA is each possible set of truth values (T or F) for the atomic parts of a sentence. If there are "n" atomic parts, then there will be 2 to the nth possible ways of assigning Ts and Fs to those parts...basically, there are 2 to the nth possible TVAs. So each TVA will determine one possible truth value (T or F) of the sentence.
- So all the possible truth values of a sentence, set, or pair of sentences, etc., is going to be identical to its truth values on all TVAs. So, for sentential logic, each of the logical properties can be redefined, replacing talk of what is possible or impossible with talk of its truth values on TVAs.
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What is the difference between full and short truth tables and truth trees?
- Full truth tables display all relevant TVAs.
- Short truth tables display one TVA.
- From a completed truth tree that is open, one can recover all TVAs on which all the members of a set of sentences are true.
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ARGUMENT
An argument is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises.
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DEDUCTIVE VALIDITY
An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false.
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DEDUCTIVE INVALIDITY
An argument is deductively invalid if and only if it is not deductively valid, therefore it is possible for the premises to be true and the conclusion false.
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DEDUCTIVE SOUNDNESS
An argument is deductively sound if and only if it is deductively valid and all its premises are true.
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DEDUCTIVE UNSOUNDNESS
An argument is deductively unsound if and only if it is not deductively sound, therefore either it is deductively unsound or at least one of its premises is false.
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INDUCTIVE STRENGTH
An argument has inductive strength to the extent that the conclusion is probable given the premises.
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LOGICAL CONSISTENCY
A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true.
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LOGICAL INCONSISTENCY
A set of sentences is logically inconsistent if and only if it is not logically consistent, therefore it is not possible for all the members of that set to be true.
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LOGICAL TRUTH
A sentence is logically true if and only if it is not possible for the sentence to be false.
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LOGICAL FALSITY
A sentence is logically false if and only if it is not possible for the sentence to be true.
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LOGICAL INDETERMINACY
A sentence is logically indeterminate if and only if it is neither logically true nor logically false.
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LOGICAL EQUIVALENCE
The members of a pair of sentences are logically equivalent if and only if it is not possible for one of the sentences to be true while the other is false.
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TRUTH-FUNCTIONAL USE OF A CONNECTIVE
A sentential connective is used truth-functionally if and only if it is used to generate a compound sentence from one or more sentences in such a way that the truth-value of the generated compound is wholly determined by the truth-values of those one or more sentences frofm which the compound is generated, no matter what those truth-values may be.
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TRUTH-FUNCTIONAL TRUTH
A sentence P of SL is truth-functionally true if and only if P is true on every truth-value assignment.
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TRUTH-FUNCTIONAL FALSITY
A sentence P of SL is truth-functionally false if and only if P is false on every truth-value assignment.
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TRUTH-FUNCTIONAL INDETERMINACY
A sentence of P of SL is truth-functionally indeterminate if and only if P is neither truth-functionally true nor truth-functionally false.
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TRUTH-FUNCTIONAL EQUIVALENCE
Sentences P and Q of SL are truth-functionally equivalent if and only if there is no truth value assignment on which P and Q have different truth values.
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TRUTH-FUNCTIONAL CONSISTENCY
A set of sentences of SL is truth-functionally consistent if and only if there is at least one truth-value assignment on which all the members of the set are true. A set of sentences of SL is truth-functionally inconsistent if and only if the set is not truth-functionally consistent.
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TRUTH-FUNCTIONAL ENTAILMENT
A set of Γ ("gamma") of sentences of SL truth-functionally entails a sentence P of SL if and only if there is no truth-value assignment on which every member of Γ is true and P is false.
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TRUTH-FUNCTIONAL VALIDITY
An argument of SL is truth-functionally valid if and only if there is no truth-value assignment on which all the premises are true and the conclusion is false. An argument of SL is truth-functionally invalid if and only if it is not truth-functionally valid.
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Are all logically true sentences also logically equivalent?
Yes.
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Are all logically false sentences also logically equivalent?
Yes.
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Give an example of a valid argument with true premises and a true conclusion.
- If dogs are mammals, then dogs are warm-blooded.
- Dogs are mammals.
- Therefore, dogs are warm-blooded.
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Give an example of a valid argument with at least one false premise and a true conclusion.
- If dogs are reptiles, then dogs are warm-blooded.
- Dogs are reptiles.
- Therefore, dogs are warm-blooded.
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Give an example of a valid argument with a false conclusion.
- In 2000, Bush and Gore ran for President.
- The candidate who won the popular vote won the presidential election.
- Gore won the popular vote.
- Therefore, Gore won the presidential election.
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Give an example of an invalid argument all of whose premises are true and whose conclusion is true.
- A dime is worth 10 cents.
- A penny is worth 1 cent.
- Therefore, a quarter is worth 24 cents.
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Give an example of an invalid argument all of whose premises are true and whose conclusion is false.
- A dime is worth 10 cents.
- A penny is worth 1 cent.
- Therefore, a quarter is worth 5 cents.
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Give an example of an invalid argument with at least one false premise and a false conclusion.
- A dime is worth 10 cents.
- A penny is worth more than a dime.
- Therefore, a quarter is worth 5 cents.
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Give an example of a pair of sentences, both of which are logically indeterminate and are logically equivalent.
No one will win. There will be no winner.
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Give an example of a pair of sentences that are not logically equivalent but are both true.
Seattle is the largest city in Washington State. George W. Bush succeeded Bill Clinton as President of the U.S.
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Give an example of a pair of sentences that are logically equivalent, one of which is logically true and one of which is not.
Not possible. The definition of logically equivalence states that a pair of sentences is logically equivalent if and only if it is not possible for one to be true while the other is false.
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Give an example of a pair of sentences that are logically equivalent and both false.
- Honey bees make silk. Silk is made by honey bees.
- Gore succeeded Clinton as President. Clinton was succeeded by Gore as President.
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Give an example of a pair of sentences, at least one of which is logically true, that are logically equivalent.
- A square has four sides. A triangle has three sides.
- Two logically true sentences will satisfy this requirement, because as each must be true, then it is impossible for one to be true while the other is false. Hence, they are logically equivalent.
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Give an example of a pair of sentences that are logically equivalent, one of which is logically false and the other of which is logically true.
Not possible. If one of the sentences is logically false, then by the definition of logically equivalence (which states that two sentences are logically equivalent if it not possible for one to be true while the other is false), they cannot be logically equivalent.
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