What do we mean when we say that a sentence Q is a consequence of a set of sentences {P1, P2, ...Pn}? (Sometimes there will be only one sentence in that set; that's okay.)
What we mean is that whenever all of those sentences {P1, P2, ...Pn} are true, that forces Q to be true as well.
That is, the truth of all of those sentences {P1, P2, ...Pn} together will guarantee that Q is true.
So suppose that you want to know whether a sentence Q is a consequence of {P1, P2, ...Pn}. How can you figure this out?
The way that we will focus on for now is looking for a counterexample.
A counterexample is a case where the purported guarantee is broken; that is, it is a case where all of the sentences {P1, P2, ...Pn} are true, but the supposed consequence Q is false.
If you find a counterexample, you have conclusively shown that the consequence relationship does not hold: Q is definitely not a consequence of those other sentences if there is a case where Q is false while all of those other sentences are true.
What kind of cases do we need to consider when we're looking for a counterexample?
It depends on the kind of consequence we're talking about: Truth-Table, Logical, or Tarksi's World Consquence:
- For Truth-Table Consequence, the cases we need to consider are all of the rows of the truth table for {P1, P2, ...Pn} and Q.
- For Logical Consequence, we need to consider all of the cases we can conceive of, keeping in mind the meanings of the Boolean connectives and the predicates of our language.
- For Tarski's World Consequence, the cases we need to consider are all of the possible Tarski's World World files we could create given the rules of the program as it's designed (e.g., there are only three shapes, three sizes, etc.).
(Question to ponder: How would we represent these Consequence relations in an Euler diagram?)
To say that an argument is valid is just to say that its conclusion is a consequence of its premises. (Thus we can distinguish between different notions of TT-validity, logical validity, and TW-validity.)
Two sentences are equivalent to one another if, and only if, they always have the same truth value; that is, if and only if there is no case where one is true and the other is false.
That means that two sentences are equivalent to one another if, and only if, they are consequences of one another. (Think about why this is true.)
1. Don't slip into thinking that you have found a consequence relationship just because you have found one or more cases where {P1, P2, ...Pn} are all true and Q is also true. Even if that happens, there still might be a counterexample.
2. If you're dealing with Truth Table Consequence and you've
done your table correctly, you can know for sure that you have considered all
of the relevant cases. So if you have done a complete, correct Truth Table and
there is no counterexample row, then you have conclusively demonstrated that
the TT-consequence relationship does hold.
But for Logical and
Tarski's World Consequence, it's harder to know that you have thought of all
of the relevant cases. Just because you haven't thought of a counterexample
doesn't mean that there isn't one: There might be one that you're not thinking
of. So it would be good if we could come up with a general way of
conclusively showing that a consequence relation does hold for logical or TW
consequence.