# What does this table regarding the 'connectives' '&' and '~' mean?

I'm a modern philosophy student and I'm reading content on the truth-value of sentences in logical structure, but I had trouble underatanding how connectives contribute to the truth-values of say sentences represented as 'p' and 'q'. I know that if q is True, then it supposes that p is also true, since p refers to q. Next it introduced the idea of connectives as truth-value functions to the sentences 'p' and 'q', here was suppose to be photos attached to this, but the option is not available to me as of this moment, so let me try to explain the table as best I can: say there are three seperate columns labled 'p' '&' and 'q' and under each them, there are a set of values of either T(truth) or F(false). For every single set of T or F values there exists a row to set the apart from the other set of T and F values: For the first there is TTT (p=T; &=T, p=T); the next row says FFT (p=F; &=F, p=T); the third row says TFF (p=T; &=F, p=F); and the last says FFF (p=F; &=F, p=F); the book later explains this:

Consider the connective & (and). We could assign to this the following function: p & q has the value true, if p has value true and q has value true. Otherwise it has the value false. Sometimes this is represented in a truth-table as follows: The middle column represents the value that must be assigned to p & q, when p and q each has the value given in the columns beneath the symbols ‘p’ and ‘q’."

What do the other set of values of T and F mean?

• Would it be possible to give author, title, and page of the book you are referring to? Jan 19, 2017 at 0:34
• I can't tell if you're asking "How do I read a truth table?" or "Why is `true and false` equal to `false`?" or both, or something else entirely.
– user6559
Jan 19, 2017 at 1:04
• "p refers to q" -- What does that mean? In truth tables, p and q are independent of each other. Jan 19, 2017 at 2:21
• I am not clear about what the book might have meant by saying that the connective (&) is itself true or false. The connective does not have content as P and Q do. Jan 19, 2017 at 19:35
• "the other set of values of T and F" - what "other set of values"?
– user20153
Jan 19, 2017 at 21:14

Truth tables define the "meaning" of the propositional connectives.

Consider the very simple table for negation: it define the "meaning" (or "usage") of the negation sign specifying that the truth-value of the negated sentence is reversed:

if p is True, then ¬p is False, and vice-versa.

Similarly for conjunction, disjunction, and the others conncetives.

For example, the truth table for conjunction ("and") specify that p ∧ q is True only when both p and q are True; in all other three cases (the other three rows of the t-t) the formula is False.

There is nothing more than this : the verbal instructions for reading the truth table that, in turn, is a tabular device to compute the truth-value of a "complex" formula starting from its elementary components.

• good answer glad you quoted "meaning" - truth tables do not actually have anything to do with meaning, afak, they're just a calculus.
– user20153
Jan 19, 2017 at 21:19