In formal logic, sentences and arguments are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer. This text describes two formal languages which have been of special importance to philosophers: truth-functional sentential logic and quantified predicate logic. The book covers translation, formal semantics, and proof theory for both languages. This can be used as the textbook for a semester long course in logic, for a unit on logic, or for self-directed study. Each chapter contains practice exercises; solutions to selected exercises appear in an appendix.The author is an assistant professor of philosophy at the University at Albany, SUNY.
What is logic?
Logic is the business of evaluating arguments, sorting good ones from bad ones. In everyday language, we sometimes use the word ‘argument’ to refer to bel-ligerent shouting matches. If you and a friend have an argument in this sense, things are not going well between the two of you.
In logic, we are not interested in the teeth-gnashing, hair-pulling kind of ar-gument. A logical argument is structured to give someone a reason to believe some conclusion. Here is one such argument:
(1) It is raining heavily.
(2) If you do not take an umbrella, you will get soaked.
.˙. You should take an umbrella.
The three dots on the third line of the argument mean ‘Therefore’ and they indicate that the ﬁnal sentence is the conclusion of the argument. The other sentences are premises of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion.
This chapter discusses some basic logical notions that apply to arguments in a natural language like English. It is important to begin with a clear understand-ing of what arguments are and of what it means for an argument to be valid. Later we will translate arguments from English into a formal language. We want formal validity, as deﬁned in the formal language, to have at least some of the important features of natural-language validity.
When people mean to give arguments, they typically often use words like ‘there-fore’ and ‘because.’ When analyzing an argument, the ﬁrst thing to do is to separate the premises from the conclusion. Words like these are a clue to what the argument is supposed to be, especially if— in the argument as given— the conclusion comes at the beginning or in the middle of the argument.
premise indicators: since, because, given that conclusion indicators: therefore, hence, thus, then, so
To be perfectly general, we can deﬁne an argument as a series of sentences. The sentences at the beginning of the series are premises. The ﬁnal sentence in the series is the conclusion. If the premises are true and the argument is a good one, then you have a reason to accept the conclusion.
Notice that this deﬁnition is quite general. Consider this example:
There is coﬀee in the coﬀee pot.
There is a dragon playing bassoon on the armoire.
.˙. Salvador Dali was a poker player.
It may seem odd to call this an argument, but that is because it would be a terrible argument. The two premises have nothing at all to do with the conclusion. Nevertheless, given our deﬁnition, it still counts as an argument—albeit a bad one.