Logic Week 1-2
Fallacy of Affirming the Consequent
(INVALID) If it rained, the ground is wet; the ground is wet; the ground is wet; hence, it rained.
Premise Indictor Expression
- Since - For - Because - On Account Of - Inasmuch as - For the reason that
Conclusion Indictor Expression
- Therefore - Thus - It Follows That - So - Hence - Consequently - As A Result
Logically True
True under all possible situations
Logically Indeterminate
True under some and false under some other possible situations
(Valid/Invalid & Sound/Unsound) All gods are immortal. So all gods have always existed
Invalid/Unsound
Inconsistent
No possible situation under which all members are true
Entailment
Similar to validity but not identical. A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentences false
How is it possible for a valid argument to have a false conclusion?
Some of the premisses of such an argument may be true, but at least one premiss must be false
(T/F) A set containing all the sentences of English is bound to be Inconsistent
TRUE
(T/F) Every set of true sentences is consistent
TRUE - If all the sentences in a set are true then it is possible for all of them to be true so set is consistent
(T/F) If the set {A,B,C} is consistent then {A,B} is also consistent
TRUE - If it is possible {A,B,C} are consistent so it is possible {A,B} are consistent
(T/F) An invalid argument must have a consistent set of premisses.
TRUE - If the argument is invalid then there is a possibility that the premisses are all true and the conclusion is false so that same possibility relates to consistency
(T/F) Every logically true sentence is true
TRUE - Logically true is true in every possible situation. If it is true in actual situation then it must be true.
(T/F) If A is logically false then the set {A,B} is inconsistent
TRUE - Suppose A is logically false so there is not possible situations A is true so there is no possible where A+B are true so the set is inconsistent
(T/F) If an argument is invalid then it conclusion is not logically true
TRUE - Suppose an argument is invalid, then there must be a possibility that the premisses re true and the conclusion is false
(T/F) If {A,B} is consistent and C is logically true then {A,B,C} is consistent
TRUE - Suppose {A,B} is consistent so there is a possible situation in which A+B are both true. Suppose C is logically true in every possible set. Therefore, there is a possible set in which A,B,C, are true
(T/F) If A is logically true and B is logically equivalent to A then B is logically true also
TRUE - Under all possible situations the two sentences have the same truth value to be logically equivalent, A is true under all situations so B is true under all situations
Logically Equivalence
Under all possible situations the two sentences have the same truth value (either both are true or both are false)
(Valid/Invalid & Sound/Unsound) All gods are immortal. So all gods shall always exist
Valid/Sound
Invalid Argument
There is a possible situation under which all the premisses are true and the conclusion is false. Famous Example - Fallacy of Affirming the Consequent
Consistent
There is possible situation under which all members are true together
Holy Grail of Logic
To be able to determine the validity of any arguments
(T/F) From the fact that an argument is not sound you cannot infer that its conclusion is false
True
(T/F) If the premisses are true and the conclusion is false then the argument is definitely invalid
True
(T/F) The conclusion of a sound argument is bound to be true
True
(T/F) There are invalid arguments with a false conclusion
True
(T/F) There are invalid arguments with a true conclusion
True
(T/F) There are valid arguments with a true conclusion
True
Sound Argument
An argument is sound if, and only if, all its premisses are true and its valid
Valid Argument
An argument is valid if, and only if, it is impossible that its premisses are true while its conclusion is false; in other words, if the premisses are true then the conclusion must be true
Predicate Logic
Branch of symbolic deductive logic that predicates and individual terms as the fundamental units of logical analysis
Sentential Logic
Branch of symbolic deductive logic that takes sentences as the fundamental units of logical analysis
What is logic concerned with?
Concerned to understand the validity of arguments
Argument
Consists of one or more statements, called the premisses, and a statement called the conclusion
(T/F) If the set {A,B} is consistent then {A,B,C} is also consistent.
FALSE - Example to Prove: {Obama is president, Kerry is Secretary of State} Consistent {Obama is president, Kerry is Secretary of State, Obama is not president} Inconsistent
(T/F) Any set containing just one sentence is bound to be consistent
FALSE - Example to Prove: {Some ducks are not ducks} Inconsistent
(T/F) If the premisses are consistent, then the argument must be invalid
FALSE - Snow is White. Snow is White. Therefore, Grass is Green (Invalid but Consistent)
(T/F) If {A,B} is inconsistent then A is not logically equivalent to B
FALSE - {Dogs are not Dogs, Cats are not Cats}
Logically False
False under all possible situations
(Valid/Invalid) No nuclear powered submarines are commercial vessels, so no warships are commercial vessels, since all nuclear powered submarines are warships.
Invalid
(Valid/Invalid & Sound/Unsound) We (in this classroom) are all mortal. So there will come a time when none of us will be alive
Invalid & Unsound
Logic Special Cases (5)
1.) An argument who conclusion is logically true. 2.) Every logical true sentence being entailed by every set of sentences, including the empty set, because it is impossible for a logically true sentence to be false and hence impossible for the members of a set, any set, all to be true and that logically true sentence false. 3.) Arguments who premises form logically inconsistent sets 4/5.) Sentences p and q are logically equivalent if and only if it is not possible for one of the sentences to be true while the other sentence is false.
