Truth-functional Logic Intro PDF

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Truth-functional Logic Intro We’re going to develop a way to study validity. What we’re going to do for the rest of the course is to develop formal languages with which we’ll represent and study information. Calling them formal languages means two related things • They are specifically designed to capture only the logical form of information. • We will define strict rules that capture the logical role of sentences in the language. The first language we will develop is the language of Truth-functional Logic. It studies the way that sentences can be built up out of other sentences Sentential Connectives Ahmad is upset because Kayla is happy An expression which combines with sentences to create more complex sentences is a sentential connective. Different sentential connectives have different kinds of meaning. We will be focusing on truthfunctional sentential connectives. Intuitively, a truth-functional connective is a connective which only makes a claim about the truth-value of the sentences it connects. Truth-functional Logic studies the truth-functional structure of information. Truth-Functional Connectives --A connective is truth-functional if and only if the truth value of a sentence with that connective as its main logical operator is uniquely determined by the truth value(s) of the constituent sentence(s). (forall x. chapter 10) Ahmad is upset and Kayla is happy Suppose you knew whether each of the contained sentences was true (that is, you know whether Ahmad is upset and whether Kayla is happy). Then you would know whether the whole sentence is true. This is because the truth-value of a sentence built with “and” only depends on the truth-value of its parts. A Language for Truth-functional Logic Our language will consist in atomic sentences and complex sentences In our language, atomic sentences will be symbolized using capital letters, e.g., P,Q,R … P : Ahmad is upset Q : Kayla is happy Complex sentences will be built up out of atomic sentences and sentential connectives.

Conjunction ˄ = and Some terminology: When we build a sentence with ˄ , we call the whole sentence a conjunction… … and we call each part a conjunct. This is true even when we build up more and more complex sentences. For example, Truth-values Some more terminology: Logicians – and philosophers more generally – often talk about the truth-value of a sentence. This is just a way of talking about whether a sentence is true or false. So we’ll when we say that a sentence has the truth-value T, we mean that it is true… …and when we say that a sentence has the truth-value F, we mean that it is false. We will define the meaning of connectives in our language by giving a rule which determines how the truth-value of a complex sentence depends on the truth-value of its parts. This will take the form of a characteristic truth-table. This table defines the way in which the truth of a conjunction depends on the truth of the conjuncts. On the whole, the table tells us that conjunction is only true if both conjuncts are true. Otherwise it is false. IMPORTANT NOTE: this applies to any conjunction, even to conjunctions with complex parts – for example, (P ˄ Q) ˄ (R ˄ S). `X’ and ` Y’ aren’t being used as sentences in our language. Rather, they are being used as variables for talking about all sentences in our language. Disjunction  = or Characteristic Truth-table for Disjunction This table tells us that a disjunction is only false if both disjuncts are false. Otherwise it is true. Inclusive/Exclusive Disjunction Sometimes when we say ‘P or Q’ in natural language, we mean to be claiming that one of P or Q is true but not both. This is called an exclusive disjunction. • Example: Given you bought the lunch meal, you can have either the cake or the pie.

Other times when we say ‘P or Q’ , we are claiming that one of P or Q is true, and for all we know both are. This is called an inclusive disjunction. • Example: Given that she is in the class, she is a philosopher major or she is a psychology major. The disjunction we introduced into our language () captures the inclusive meaning of ‘or’. To some extent this is simply a choice we have to make, but there are also good reasons to think that the inclusive meaning if more basic....