G’day. I’m foregoing the usual news item post for this updated completion to my earlier How to Argue post, in which I wrote on formally setting out arguments, and how we must analyze them, only evaluating them afterward to see whether the truth of the conclusion follows from the premises when an argument is either inductive or deductive in nature.

Here, we’ll evaluate an argument, and I want to use the argument I set out last time, and first going over the argument itself using a truth table to determine it’s validity as given. We’ll also note any insufficiencies of this argument immediately following, and how we can make it better.

Here it is as last time, in one row in formal notation. We are using sentential logic, or basic symbolic logic for simplicity…

P, Q, (P & Q)→R : R→S

And in English, the variables are repeated here from the previous post…

- P – Expressions of belief are claims.

- Q – No claim should get a free pass.

- R – If you express your claims, but don’t like criticism, then you have two choices.

- S – Your choices are to make better claims or to defend your claims.

…where commas are used to separate the premises, a colon to separate the premises from the conclusion, the “&” sign to join premises “P” and “Q” together as sub-premises with “R” in a complex premise, and the “→” sign to designate hypothetical or “If/Then” conditional statements into the 3rd premise, “(P & Q)→R” and the complex conclusion, “R→S.”

Note first that the truth of “R” depends upon the truth of “P” and “Q” together, and that of “S” depends on that of “R.” Already you might suspect that this doesn’t look promising, but let’s work that out, using all four variables and the full argument to see all of the possible worlds that the argument could pertain to, all possible situations, to see if there are any of these in which the premises may be true and the conclusion false.

That would show it to be invalid.

From a filled in truth table, we need find only one row where this is the case, and as we shall see, a truth table’s number of rows, or possible situations, doubles for each variable after the first. This one will have 16 such rows beneath the header showing the guide-columns for the variables and the guide-columns for the argument itself.

Each row on the table shows a possible variation of the truth values of the premises and the conclusion.

For ease of reading the table, I’ll use binary notation, with “1” for true, and “0” for false for the truth values of the variables and their operator symbols.

Well, glancing at the table, we don’t have to look far to see, at row 2 marked in red, that there does exist a possible combination of values in which all of the premises are true and the conclusion false.

P |
Q |
R |
S |
P, |
Q, |
(P |
& |
Q) |
→ |
R, |
R |
→ |
S |

1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |

1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |

1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |

1 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
0 |

1 |
0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |

1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |

1 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |

1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |

0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |

0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |

0 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |

0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |

0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |

0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |

0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |

Sadly, despite any merits at all of it, it’s not *a good argument*, because as good logic students will quickly notice, it’s not complete.

What’s up with this? What’s wrong with it?

First, the original conclusion, “R→S” is better as the premise “R→S,” followed by the premise “R”, and the conclusion “S,” and the argument in full as follows:

P, Q, (P & Q)→R, R→S, R: S

So, let’s look at the truth table with this argument now completed, and see if it’s any better, with the added columns marked in green…

P |
Q |
R |
S |
P, |
Q, |
(P |
& |
Q) |
→ |
R, |
R |
→ |
S, |
R: |
S |

1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |

1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |

1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |

1 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |

1 |
0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |

1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |

1 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |

1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |

0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |

0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |

0 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |

0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |

0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |

0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |

0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |

Much better.

This is a valid argument, though that’s no guarantee of the truth of the conclusion, only of the conclusion’s truth if the premises are likewise true. What’s important to logicians is the truth-preserving qualities of such arguments.

On this now-completed truth table, there exists no possible situation, no possible world in which both the premises’ truth and the conclusions falsehood occur, no small matter even by itself for anybody concerned with truth in whatever form it takes.

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