Editing Hypothetical syllogism (section)

== Applicability ==
The rule of hypothetical syllogism holds in [[classical logic]], [[intuitionistic logic]], most systems of [[relevance logic]], and many other systems of logic. However, it does not hold in all logics, including, for example, [[non-monotonic logic]], [[probabilistic logic]] and [[default logic]]. The reason for this is that these logics describe [[defeasible reasoning]], and conditionals that appear in real-world contexts typically allow for exceptions, default assumptions, [[ceteris paribus]] conditions, or just simple uncertainty.

An example, derived from Ernest W. Adams, <ref>{{Cite book |last=Adams |first=Ernest W. |title=The Logic of Conditionals |publisher=Dordrecht: Reidel |year=1975 |page=22}}</ref>
# If Jones wins the election, Smith will retire after the election.
# If Smith dies before the election, Jones will win the election. 
# If Smith dies before the election, Smith will retire after the election.

Clearly, (3) does not follow from (1) and (2). (1) is true by default, but fails to hold in the exceptional circumstances of Smith dying. In practice, real-world conditionals always tend to involve default assumptions or contexts, and it may be infeasible or even impossible to specify all the exceptional circumstances in which they might fail to be true. For similar reasons, the rule of hypothetical syllogism does not hold for [[counterfactual conditionals]].