Critical thinking is the art of evaluating the judgements and decisions we make by examining the process the leads to them. It improves our rationality, and helps us evaluate arguments.
Arguments aim to persuade, as opposed to explanations, which aim to clarify:
Argument: You shouldn't change currency at the border because the exhange rate is bad
A good argument is not necessarily convincing, nor necessarily saying things that are true. A good argument must simply have acceptable premises and a conclusion that follows from them. Typically, the conclusion is less certain than the premises, whilst in explanations the converse is the case.
Arguments can have premises that aren't explict, called enthymematic premises, e.g.
"Of course I won't vote for Blair. Do you take me for a fool?"
A set of beliefs is consistent if they can all be believed at the same time. Consistency isn't a matter of them persisting through time, i.e. one can be fickle over time but consistent at any one point in time when all your beliefs could possibly be true. Consistent beliefs can even be false. For example, "Anyone with at least one brother or sister is not an only child and Bart is an only child. Bart has no bothers, but he does have two sisters" is an inconsistent belief, whilst "The earth is a planet with one satellite, which revolves around the sun. Planets are spherical. The earth is not flat" is consistent.
A belief is consistent if it could be true, and inconsistent if it is self-contradictory. Again, it needn't actually be true, but only possible. For example, "I invented a new sedative that makes people faster and more excited" is inconsistent, whilst "that tree is made of wood" is consistent.
Sentences express beliefs, but it is not always clear which belief(s) a sentence expresses. Thus it is important to be able to find what beliefs a sentence is expressing before we can establish whether or not it is consistent. Beyond finding enthymematic premises, we face other challanges.
A declarative sentence is one that in English can be put in the form "Is it true that X?". For example, one might ask, "Is it true that cats can swim?" In this case, "cats can swim" is a declarative sentence, and so unambiguously expresses a belief. Spotting declarative sentences is usually easy, but there are more difficult cases, e.g. "Blackmail is wicked" might be said to be declarative, but at the same time could equally be considered imperative.
But even once isolated, declarative sentences can be made difficult to analyse by several problems:
Lexical ambiguity occurs with homononyms, e.g. "she went to
the bank". Without further information on the context it is difficult
to analyse the semantics of this sentence.
Structural ambiguity occurs when the words in a string of
words can be meaningfully grouped together in two or more different
ways, e.g. "fruit flies like a banana". Here there are two possible structural units,
"fruit flies" or "fruit", i.e. the sentence could be describing what
fruit flies like to eat, or how fruit travels through the air.
Vagueness occurs when a claim's meaning is indeterminate, unclear, e.g. "Ted is thin"; what do we mean by "thin"?
Indexicality occurs when sentences that contain indexical terms say different things in different contexts, e.g. "It is raining now" could be referring to this moment in time, or it could have been referring to 12pm on 01/05/2003, etc.
"Pierre est chaud" and "Peter is hot" express the same proposition/belief, despite appearing semantically different. "It is cold today", on the other hand, can express different propositions depending on the context, despite being one single declarative sentence. In other words, a declarative sentence must be understood as distinct from a proposition, and so when analysing sentences, we must look for propositions rather than for declarative sentences, to avoid the problems described above.