Connectives: Meaning of the Connectives

Meaning of connectives:

NOT, AND, OR, IMPLIES, IF AND ONLY IF


Let us define the meaning of the five connectives by showing the relationship between the truth value (i.e. true or false) of composite propositions and those of their component propositions. They are going to be shown using truth table. In the tables P and Q represent arbitrary propositions, and true and false are represented by T and F, respectively.

NOT

P	P
T	F
F	T

This table shows that if P is true, then ( P) is false, and that if P is false, then ( P) is true. 

AND

P	Q	(P  Q)
F	F	F
F	T	F
T	F	F
T	T	T

This table shows that (P  Q) is true if both P and Q are true, and that it is false in any other case.

Similarly for the rest of the tables.

OR

P	Q	(P  Q)
F	F	F
F	T	T
T	F	T
T	T	T

IMPLIES

P	Q	(P  Q)
F	F	T
F	T	T
T	F	F
T	T	T

When P  Q is always true, we express that by P  Q. That is P  Q is used when proposition P always implies proposition Q regardless of the value of the variables in them. See Implications for examples of . 

Also see a note on the truth value of IMPLIES when P is False. 

IF AND ONLY IF

P	Q	( P  Q )
F	F	T
F	T	F
T	F	F
T	T	T

When P  Q is always true, we express that by P  Q. That is  is used when two propositions always take the same value regardless of the value of the variables in them. See Identities for examples of . 

ref: http://www.cs.odu.edu/~toida/nerzic/content/logic/prop_logic/connectives/connectives.html