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1.4: Truth Functions

I want to point out one more thing about the way we have defined the connectives '~', '&', and 'v'. Let us start with '~'. What do you have to know in order to determine whether '~A' is true or false? You don't have to know what sentence 'A' actually stands for. You don't have to know whether 'A' is supposed to mean that Adam loves Eve, or that pudding is 10 fattening, or anything like that. To know whether '~A' is true or false, all you have to know is whether 'A' itself is true or false. This is because if you know the truth value of 'A', you can get the truth value of '~A' by just looking it up in the truth table definition of '~'.

   The same thing goes for '&' and 'v'. To know whether 'A&B' is true or false, you don't have to know exactly what sentences 'A' and 'B' are supposed to be. All you need to know is the truth value of 'A' and the truth value of 'B'. This is because, with these truth values, you can look up the truth value of 'A&B' with the truth table definition of '&'. Likewise, with truth values for 'A' and for 'B', you can look up the truth value for 'AvB'.

    Logicians have a special word for these simple facts about '~', '&' and 'v'. We say that these connectives are Truth Functional. In other words (to use '&' as an example), the truth value of the compound sentence 'A&B' is a function of the truth values of the components 'A' and 'B'. In other words, if you put in truth values for 'A' and for 'B' as input, the truth table definition of '&' gives you, as an output, the truth value for the compound 'A&B'. In this way 'A&B' is a function in the same kind of way that 'x + y' is a numerical function. If you put in specific numbers for 'x' and 'y', say, 5 and 7, you get a definite value for 'x + y', namely, 12.

   'A&B' is just like that, except instead of number values 1, 2, 3, . . . which can be assigned to 'x' and to 'y', we have just two truth values, t and f, which can be assigned to 'A' and to 'B'. And instead of addition, we have some other way of combining the assigned values, a way which we gave in the truth table definition of '&'. Suppose, for example, that I give you the truth values t for 'A' and f for 'B'. What, then, is the resulting truth value far 'A&B'? Referring to the truth table definition of 'A&B', you can read off the truth value f for 'A&B'. The truth tables for '~' and for 'v' give other ways of combining truth values of components to get truth values for the compound. That is,'~' and 'v' are different truth functions.

Let's pull together these ideas about truth functions:

A Truth Function is a rule which, when you give it input truth values, gives you a definite output truth value. A Truth Functional Connective is a connective defined by a truth function. A Truth Functional Compound is a compound sentence forked with truth functional connectives. 

Exercise \(\PageIndex{1}\)

1-3.  Try to explain what it would be for a declarative compound sentence in English not to be truth functional. Give an example of a declarative compound sentence in English that is not truth functional. (There are lots of them! You may find this exercise hard. Please try it, but don't get alarmed if you have trouble.)

Contributors

  • Paul Teller (UC Davis). The Primer was published in 1989 by Prentice Hall, since acquired by Pearson Education. Pearson Education has allowed the Primer to go out of print and returned the copyright to Professor Teller who is happy to make it available without charge for instructional and educational use.

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