Lecture Notes: Section 2.2

Section 2.2: Truth Tables for Sequents

Definition: A truth value assignment for a sequent $\varphi _{1},\, ...,\, \varphi _{n}\, \vdash \, \psi$ is a truth value assignment for the set $\{\varphi _{1},\, ...,\, \varphi _{n},\, \psi \}$ .

Definition: A sequent $\varphi _{1},\, ...,\, \varphi _{n}\, \vdash \, \psi$ is valid if and only if every truth value assignment for the sequent on which the premises $\varphi _{1},\, ...,\, \varphi _{n}$ are true is also one on which the conclusion $\psi$ is true.

Equivalently:

Definition: A sequent $\varphi _{1},\, ...,\, \varphi _{n}\, \vdash \, \psi$ is valid if and only if there is no truth value assignment for the sequent on which the premises $\varphi _{1},\, ...,\, \varphi _{n}$ are true but the conclusion $\psi$ is false.

Comment: Because truth value assignments correspond to rows of truth tables, we can test for the validity of a sequent by constructing a truth table for it. Hence, we can formulate the definitions above in terms of truth tables:

Alternative Definition: A sequent $\varphi _{1},\, ...,\, \varphi _{n}\, \vdash \, \psi$ is valid if and only if, in the truth table for the sequent, the conclusion $\psi$ is true on every row on which all of the premises $\varphi _{1},\, ...,\, \varphi _{n}$ are true.

Equivalently:

Alternative Definition: A sequent $\varphi _{1},\, ...,\, \varphi _{n}\, \vdash \, \psi$ is valid if and only if, in the truth table for the sequent, there is no row on which the premises $\varphi _{1},\, ...,\, \varphi _{n}$ are true but the conclusion $\psi$ is false.

Comment: We will indicate that we are constructing a truth table for a sequent by putting the turnstile ` $\vdash$ ' between the premises and the conclusion of the sequent.

Example of a valid sequent

Definition: An invalidating assignment for a sequent is a truth value assignment for the sequent on which the premises are truth and the conclusion is false.

Comment: Because a truth table for a sequent represents all of the truth value assignments for it, every invalidating assignment for a sequent will correspond to a unique row of the truth table for the sequent.

Example of an invalid sequent

Comment: Recall the quiz from a few weeks ago with the ``trick'' question (which was not counted against you!) in which you were asked whether the following argument was valid:

All politicians are dishonest.
Phil Gramm is an honest politician.
Therefore, Bill Clinton is a republican.

Though this is a bad argument, it is in fact valid: the badness consists in the fact that the premises are contradictory. But because they are contradictory, you cannot make the premises true. Hence, of course, you can't make the premises true and the conclusion false. Hence, it is valid. We cannot yet represent the above argument formally in a way that reveals the contradictory nature of the premises (that will come in Chapter 3). But consider the following argument instead.

If Phil Gramm is honest ( $\mathrm{H}$ ), he is not a politician ( $\sim \! \mathrm{P}$ ).
If Phil Gramm is wealthy ( $\mathrm{W}$ ), he is a politician ( $\mathrm{P}$ ).
Phil Gramm is honest and wealthy ( $\mathrm{H}\, \&\, \mathrm{W}$ ).
Therefore, Bill Clinton is a republican ( $\mathrm{C}$ ).

It should be clear that the premises of this argument, like the one above, are contradictory. However, this one we can represent in sentential logic. And if you do the truth table for it, you will find that there is no row on which the premises are all truth. Hence, of course, there is no row on which the premises are all true and the conclusion false. So it is indeed a valid argument. The POINT again: validity alone is not enough for a good argument. For that, the argument must be sound, i.e., it must not only be valid but also have true premises!

http://philebus.tamu.edu/~cmenzel/240/LectureNotes/Sec2.2
Last updated 3/31/1998