2017 T1 Flashcards
Premise
The proposition asserted as evidence (or reasons) for a conclusion.
Conclusion
In any argument, the proposition to which the other propositions in the argument are claimed to give support, or for which they are given as reasons.
Valid argument
An argument is valid if there is no possible world in which the premises are true but the conclusion is false. Meaning the premises logically entail the inevitable conclusion when premises are assumed true.
Preposition
An assertion which holds a truth value, meaning that it can be either true or false but not both. At minimum, a proposition must consist a subject and a verb.
Invalid argument
An argument is invalid if there is at least one possible world in which the premises are true and the conclusion false.
Truth table
A table on which all possible truth values of complex propositions are displayed, as determined by the all of possible combinations of the truth values of their propositional variables, the validity of an argument is solved through the occurrence of a counter-example.
Label tautology, contradiction and contingency on all main columns.
Tautology
A proposition which is true in all possible worlds.
Contradiction
A proposition which is false in all possible worlds.
Contingency
A proposition which is true in at least one possible world and false in at least one possible world (sometimes true and sometimes false)
Dictionary
A list which assigns propositional variables to propositions, this is to keep track of and understand what each variable symbolises.
Propositional Calculus (PC) or Propositional Logic (PL)
The branch of symbolic logic that deals with propositions and the relations between them.
&
Conjunction
Both B and Q
P although Q
P but Q
P however Q
P Q 1 1 1 1 0 0 0 0 1 0 0 0
Symbolise
The use of symbols to represent ideas.
Standard Form
A uniform format through which to present arguments. P1. P2. P3. ------------------ \:.
Matrix
The column to the left of the truth table of a complex proposition, it lists all the possible combinations of the truth values of the propositional variables making up the complex proposition or argument.
≢
Exclusive disjunction
Either P or Q, but not both
Exactly one of the following is true: P or Q
P Q 1 0 1 1 1 0 0 1 1 0 0 0
Logical equivalency
A logical equivalency refers to the relationship between two propositions in which both propositions always hold the same truth values as each other in all possible worlds. For example, ~(P & Q) is logically equivalent to ~P v ~Q. We can demonstrate this by completing a Truth Table for each of these propositions which will produce identical results in the main column for each row. In addition, a logical equivalency can be conveyed through the use of a bi-conditional. When using truth tables, if the result in the main column of a bi-conditional is always 1, then the proposition on the left and right hand side of the tribar are deemed logically equivalent
v
Inclusive disjunction
Either P or Q At least one of the following is true: P or Q P except for Q P or Q or both P unless Q
P Q 1 1 1 1 1 0 0 1 1 0 0 0
~
Negation
It is not the case that
Counter-example
In one specific world, all premises are true while the conclusion in false.
Stated as (example):
P Q R S
1 0 1 1
If there is one counter-example, the entire argument is invalid.
≡
Bi-conditional
P if and only if Q
P is both necessary and sufficient for Q
P is equivalent to Q
P Q 1 1 1 1 0 0 0 0 1 0 1 0
Definitions
1 explain the term
2 example of the term
E.g. A valid argument is…. An example of a valid argument is…
Prove validity of an argument
This argument is valid, as the premises, assuming they are true, logically lead to and guarantee the truth of the conclusion. Premise one uses the term ____is/all etc.___, which leaves no rooms for alternative options; and premise 2 is also certain by saying that ____is/all etc.___, and logically follows premise 1. This means that the conclusion, which is supported by both premises and leaves no alternative answer, must be true, and therefore this makes the argument valid.
Difference between tautology/contradiction/contingency and counter-example
Tautology/contradiction/contingency: the conclusion is true/false/some true some false, in all possible world
Counter example: the conclusion is false when all premises are true in the specific world
Justification for logical equivalence
Identify whether if they are logical equivalent or not
Define “propositions”
Define “logical equivalence” (hold the same truth value: this is evident in the main operator columns in the truth table of the two propositions as an example)
Define all operators in the propositions
Give examples of all
Explain what each proposition mean in English
Apply the explanation to state whether the propositions are logically equivalent.
(If they mean the same thing in English, then they hold the same truth value, thus they are logically equivalent)
Truth table for both propositions
⊃
Conditional
If P then Q P only if Q P is sufficient for Q P implies Q Q is necessary for P Q if P
P Q 1 1 1 1 0 0 0 1 1 0 1 0
Difference between row and column in a truth table
Row: relationship between premises and conclusion
Column: truth value of a proposition
