Consider the following argument...

• "If you have a current password, then you can log on to the network"
• "You have a current password"
  
  Therefore:
  
  
• "You can log on to the network"

This has the form:

p

→

q

p

∴

q

This form of argument is calls

Modus Ponens

(latin for "mode that affirms")

Note that an argument can be valid, even if one of the premises is false. For example, the argument above doesn't say whether you do or don't have a current password.

Maybe you do, and maybe you don't . But either way, the argument is still valid.

Consider this argument:

• You can't log into the network
• If you have a current password, then you can log into the network
  
  Therefore
  
  
• You don't have a current password.

(ii) Chain rule

(Chain rule) Conditional probabilities can be used to decompose conjunctions. For any propositions α1,...,αn:

P(α1∧α2∧...∧αn)

=

P(α1)�

P(α2|α1)�

P(α3|α1 ∧α2)�

...

P(αn|α1∧���∧αn-1)

=

∏i=1n P(αi|α1∧���∧αi-1),

where the right-hand side is assumed to be zero if any of the products are zero (even if some of them are undefined).

Note that any theorem about unconditional probabilities is a theorem about conditional probabilities if you add the same evidence to each probability. This is because the conditional probability measure is another probability measure.