##### Summary Section 3

In this section we introduced and defined the following terms and concepts:

valid argument form is an argument given in terms of pqr, such that the resulting argument is always valid for any choice of propositions for pqr etc.

Propositions that have the property of being sometimes true and sometimes false are known as contingent propositions.

Propositions that are always true are known as tautologies.

A fundamental proposition whose truth seems to be self-evident and is not proven by use of more fundamental propositions is known as an axiom.

Letters of the alphabet which stand for propositions are known as propositional variables.

When an actual proposition replaces a variable given in an argument form, such substitution is called a replacement instance (of the variable x).

Given any argument, if it is possible to assign statement variables to statements in the argument such that the resulting replacement results in a valid argument form, then the argument is valid.

An invalid argument form is an argument given in terms of pqr, such that the resulting argument may be invalid or may be valid depending on the propositions used to replace the variables pqr, etc.

pseudo-valid argument form is an invalid argument form which is similar in form to valid argument forms. Some texts refer to these forms as invalid deductive arguments.

The term 'inclusive or' means 'p and/or q'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition even when both p and q are true.

The term 'exclusive or' means 'p or q but not both'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition only when exactly one of p or q is true.

We introduced four valid argument forms and compared them to three pseudo-valid argument forms (see table below).

Valid argument forms can be used as 'recipes' to create valid arguments. Also some existing arguments can be shown to have valid forms by isolating certain terms in the argument, looking for a match of corresponding terms in known valid argument forms, and assigning variables to propositions, giving the same variable to the same propositions.

Replacement instances for the variables used in invalid argument forms may or may not produce an invalid argument. To determine which is the case, we use the two step method for determining invalidity.

We noted that in logic the meaning of 'or' is always inclusive, unless otherwise indicated.

Valid argument form

Pseudo-valid argument form

modus ponens/affirming the antecedent

If p then q

p

Therefore q

denying the antecedent

If p then q

not p

Therefore not q

modus tollens/denying the consequent

If p then q

not q

Therefore not p

affirming the consequent

If p then q

q

Therefore p

disjunctive syllogism / process of elimination

p or q

not q

Therefore p

false dilemma

p or q

p

Therefore not q

hypothetical syllogism / chain of reasoning

If p then q

If q then r

Therefore if p then r

(no corresponding pseudo-valid form)