Discrete Mathematics
Inference Theory
,Argument(Optional)-Rules of Inference
Suppose we are given a finite set of propositions (called
hypotheses) 𝐻1 , 𝐻2 , … … … . . 𝐻𝑛 , all assumed true.
Also assume that from these premises, we can arrive at a conclusion
“C” through reasoning (argument).
Such a discussion can be written in inferential form as follows
9/4/2024 Indian Institute of Information Technology, Kottayam 2
, Valid and Invalid Arguments
An argument is valid if the conjunction of the hypotheses
𝐻1 , 𝐻2 , … … … . . 𝐻𝑛 logically implies the conclusion C.
That is, 𝐻1 ⋀𝐻2 ⋀ … … … . ⋀𝐻𝑛 ⟶ 𝐶 is a tautology
Otherwise, the argument is invalid.
Note
1. An argument is valid if and only if the conclusion is a logical
consequence of the hypotheses.
2. A set of hypothesis (premises) 𝐻1 , 𝐻2 , … … … . . 𝐻𝑛 is said to be
inconsistent if 𝐻1 ⋀𝐻2 ⋀ … … … . ⋀𝐻𝑛 implies a contradiction.
9/4/2024 Indian Institute of Information Technology, Kottayam 3
Inference Theory
,Argument(Optional)-Rules of Inference
Suppose we are given a finite set of propositions (called
hypotheses) 𝐻1 , 𝐻2 , … … … . . 𝐻𝑛 , all assumed true.
Also assume that from these premises, we can arrive at a conclusion
“C” through reasoning (argument).
Such a discussion can be written in inferential form as follows
9/4/2024 Indian Institute of Information Technology, Kottayam 2
, Valid and Invalid Arguments
An argument is valid if the conjunction of the hypotheses
𝐻1 , 𝐻2 , … … … . . 𝐻𝑛 logically implies the conclusion C.
That is, 𝐻1 ⋀𝐻2 ⋀ … … … . ⋀𝐻𝑛 ⟶ 𝐶 is a tautology
Otherwise, the argument is invalid.
Note
1. An argument is valid if and only if the conclusion is a logical
consequence of the hypotheses.
2. A set of hypothesis (premises) 𝐻1 , 𝐻2 , … … … . . 𝐻𝑛 is said to be
inconsistent if 𝐻1 ⋀𝐻2 ⋀ … … … . ⋀𝐻𝑛 implies a contradiction.
9/4/2024 Indian Institute of Information Technology, Kottayam 3
