Tautologous Form: What Does It Mean?

[Bashyboy]

Hello everyone,

I am re-reading the introductory section of Loomis and Sternberg's Advanced Calculus text. The sentence that I have come across which is giving me slight trouble is

"Indeed, any valid principle of reasoning that does not involve quantifiers must be expressed by a tautologous form."

This excerpt is found on page 5, and here is the electronic version of the text http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf

What exactly is being said in this sentence? I do not quite understand it.

 

[gopher_p]

First off, note that this text was originally written in the 1960s, and so some of the language/presentation of the material is a bit dated. Also, and this is becoming a bit of a pet peeve of mine, as is typical of many math texts which aren't wholly devoted to logic but attempt to present logic in a somewhat formal way, this one muddies the waters between syllogism/rhetoric, a topic of philosophy, and symbolic logic/propositional calculus, a topic of mathematics. The distinction is subtle, but it causes much confusion.

The sentence in question here essentially states that the philosopher's valid arguments correspond to the mathematician's tautologies. Or more to the point, if you want to check whether an argument of yours is "correct", translate it into a "truth-functional form" - an appropriate string of "frame variables" and logical connectives - and look at its truth table. If the truth table has all Ts in the right column, then your argument is valid. If not ...

 

[thelema418]

In propositional logic, the reasoning is valid if it is always true, such as the reasoning of modus ponens (direct proof). In propositional logic modus ponens can look like this:

$((P \rightarrow Q) \wedge P) \rightarrow Q$

An example is: "If I like math, then I will get an A on the test. I like math! Therefore, I will get an A on the test."

For the logic with quantifiers... the predicate logic or first order logic, this UC-Davis pdf explains how you show validity:

http://hume.ucdavis.edu/mattey/phi112/validity_ho.pdf