The fallacy of equivocation is committed when a term is used in two or more different senses within a single argument.
For an argument to work, words must have the same meaning each time they appear in its premises or conclusion. Arguments that switch between different meanings of words equivocate, and so don’t work. This is because the change in meaning introduces a change in subject. If the words in the premises and the conclusion mean different things, then the premises and the conclusion are about different things, and so the former cannot support the latter.
(1) The church would like to encourage theism.
(2) Theism is a medical condition resulting from the excessive consumption of tea.
(3) The church ought to distribute tea more freely.
This argument is obviously fallacious because it equivocates on the word theism. The first premise of the argument is only true if theism is understood as belief in a particular kind of god; the second premise of the argument is only true if theism is understood in a medical sense.
(1) Christianity teaches that faith is necessary for salvation.
(2) Faith is irrational, it is belief in the absence of or contrary to evidence.
(3) Christianity teaches that irrationality is rewarded.
This argument, which is a reasonably familiar one, switches between two different meanings of “faith”. The kind of faith that Christianity holds is necessary for salvation is belief in God, and an appropriate response to that belief. It does not matter where the belief and the response come from; someone who accepts the gospel based on evidence (e.g. Doubting Thomas) still gets to heaven, according to Christianity.
For the kind of faith for which (1) is true, (2) is therefore false. Similarly, for the kind of faith for which (2) is true, (1) is false. There is no one understanding of faith according to which both of the argument’s premises are true, and the argument therefore fails to establish its conclusion.
Another argument relating to Christianity that crops up from time to time goes like this:
(1) Jesus is the Word of God.
(2) The Bible is the Word of God.
(3) Jesus is the Bible.
This is usually used to to support some further conclusion about the authority of the Bible or something similar, but there’s no need to go any further to see that there’s a problem here: the phrase “Word of God” means very different things in the two premises, so this argument rests on an equivocation.