A Summary Sheet About Consequence

Intro to Symbolic Logic (Phil 102)

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.

The Varities of Consequence

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:
(Question to ponder: How would we represent these Consequence relations in an Euler diagram?)

The Connection With Validity

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.)

The Connection With Equivalence

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.)

Two Last Observations:

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.