Originally Posted by

**romsek** Your last line sums it up nicely, though I'd use the word correct or true rather than consistent. The premise(s) can be consistent and still be incorrect.

"Consistent" is a technical term meaning that all formulas in the set are true under some assignment of truth values to variables. Other ways to say this is that all formulas are true in some row of a truth table, or in some interpretation. The word "correct" is not used in mathematical logic (at least, in this topic), and "true" is relevant to an interpretation: a formula may be true in some interpretation and false in another.

Originally Posted by

**Gayelle** Now with respect to the conclusion being a contradiction, is it not that an argument is “valid if the premises cannot all be true without the conclusion being true as well.”

That's correct.

Originally Posted by

**Gayelle** Therefore when plotted on a truth table there must be no row/case present where all the premises are all found to be true but the conclusion false in that same row for an argument to be deemed valid. However the answer seems to be claiming that since the conclusion is a contradiction a row that consists of anything other than consistent values of “true” would deem the argument valid.

Again, "consistent" is a technical term whose meaning in logic is different from its common English meaning. It can only be applied to a set of formulas. With regard to your remark, if the conclusion is a contradiction, then a single row that has anything other than all "true" for premises would not make this argument valid, but if all rows have this property, then the argument is indeed valid.

Let's think about interpretations (i.e., rows of a truth table) as possible worlds. They give all possible meanings to propositional variables, i.e., basic facts of life. Now, premises are, in general, compound propositions, which may be true or false in any paticular world. If a set of premises is true in at least some world, they are called "consistent" (one could also call such set "possible" or "realizable"). For example, the premises "All inhabitants (of the world) are alive" and "All inhabitants are at least 6 feet tall" are presumable consistent, while "All inhabitants are alive" and "All inhabitants are dead" are inconsistent.

An argument is valid if each world that makes all premises true also makes the conclusion true. Suppose the conclusion is a contradiction, i.e., it is false in all worlds. What would it take for the argument to be valid? If the set of the premises is consistent, the argument cannot be valid. Indeed, in the world that makes all premises true, the conslusion is still false, which contradicts validity. If, on the other hand, the set of premises is inconsistent, then in each world at least some of the premises and the conclusion are false, which is allowed by the definition of a valid argument.

If x and y are propositional variables, p1, p2 and p3 are premises are c is a conclsion with the following truth table

Code:

x y p1 p2 p3 c
--------------
0 0 0 1 0 0
0 1 1 1 0 0
1 0 0 0 1 0
1 1 0 0 0 0

Then the argument that concludes c from p1, p2, p3 is valid because it never happens that all premises are true: the set of premises is inconsistent. The following argument

Code:

x y p1 p2 p3 c
--------------
0 0 0 1 0 0
0 1 1 1 1 0
1 0 0 0 1 0
1 1 0 0 0 0

is not valid because in the world where x = 0 and y = 1, all premises are true, so the conclusion should be true as well.

Originally Posted by

**Gayelle** Sorry just read the rule on posting logic questions. I'm presently attempting Sentential Logic.

You are currently dealing with the semantics of sentential, or proposition, logic. Semantics is about truth/falsity. The forum rule is about various syntactic ways of constructing arguments that do not rely on the concept of truth, but only use the syntactic structure of premises. There are several such formalisms, hence the rule.