[reposted April 5, 2002]

[1] Notice that these (equivalent) definitions allow three "degenerate" cases of validity: an argument whose premises are contradictory (whatever the conclusion, it can't be false while the premises are true); an argument whose conclusion, although unrelated to its premises, is a tautology (and so can't be false); and an argument whose conclusion is identical with one of its premises. Formally, these are valid arguments, despite their serving no practical function of persuasion.

[2] Although "P and Q" and "P but Q" have the same truth conditions, the latter is standardly used to signal a contrast between "P" and "Q;" because this use of "but" does not serve to assert the existence of the contrast, the truth conditions for the two expressions are the same, and so the two expressions cannot be distinguished in symbolic logic. [See pages 246-7 in Frank Jackson & Philip Pettit, "A problem for expressivism," 58 Analysis 239 (October, 1998); and also Michael Dummett, Frege (Duckworth: 1973), p. 86.]

[3] In virtue of the truth table definitions of the connectives, the truth value of any compound sentence schema is a function of—is uniquely determined by—the truth values of its component sentence letters.

[4] All references to Copi are to Irving M. Copi, Symbolic Logic--5th Edition, Macmillan (N.Y.: 1979).

[5] In case you are curious, the usual, recursive formation rules which define a "well-formed" (i.e., grammatical or proper) sentence schema in sentential logic go like this:

i. Any sentence letter is a well-formed formula;
ii. If "" is a well-formed formula, then "~()" is a well-formed formula;
iii. If "" and "" are both well-formed formulae, then "( ^. )" is a well-formed formula;
iv. If "" and "" are both well-formed formulae, then "( v )" is a well-formed formula;
v. If "" and "" are both well-formed formulae, then "( )" is a well-formed formula;
vi. If "" and "" are both well-formed formulae, then "( )" is a well-formed formula;
vii. Nothing else is a well-formed formula.

[6] In the following example, #1 is the major operator of the entire sentence schema. Within the part controlled by it, the major operator is #2. Within the parts controlled by it, #3 is the major operator; and so on. The scope of an operator is established by the punctuation provided by parentheses, brackets, braces, and so on. Example of ranks of dominance; lower numbers dominate higher:

[7] A pedantic note on terminology: Copi's terminology of "having the statement form" is misleading since a given schema may have many forms, not just one, and because it is too tempting to suppose—quite mistakenly—that if X has the form of Y, Y must in symmetry have the form of X. So I reject Copi's terminology and instead use the notion of "generation," since there is no general temptation to think of the relation of generation as symmetrical; it is of course not symmetrical—X's generating Y does not imply that Y must generate X.

[8] Note on terminology: the word "interpretation" is frequently used ambiguously by logicians to refer both to the assignment under which a particular argument is specified and also to the result of that assignment--the particular argument thus specified, the context determining which of the two is meant. I will mean the first.

[9] See L 8, in Boardman's Supplement, for some examples. Here is a link to a picture of page 8

[10] Consider argument schema # 25, below. Under the interpretation which is specified, it will represent Argument # 2:

Argument Schema # 25	Argument # 2:	Interpretation of Schema # 25 under which it will represent Argument #2:
	If Lawrence University in in Newark, then if Newark is in New Jersey, Lawrence University is in New Jersey. Lawrence University is in Appleton. Therefore, if Newark is in New Jersey, then Lawrence University is in New Jersey.	L = Lawrence University is in Newark. P = Newark is in New Jersey. Q = Lawrence University is in New Jersey. S = Lawrence University is in Appleton.

Argument Schema # 25 can also represent Argument # 3 under the interpretation which is specified:

Argument Schema # 25:	Argument # 3:	Interpretation of Schema # 25 under which it will represent Argument #3:
	If Appleton is in Wisconsin, then if Lawrence University is in Appleton, Lawrence University is in Wisconsin. Appleton is in Wisconsin. Therefore, if Lawrence University is in Appleton, then Lawrence University is in Wisconsin.	L = Appleton is in Wisconsin. P = Lawrence University is in Appleton. Q = Lawrence University is in Wisconsin. S = Appleton is in Wisconsin.

We should notice that Argument # 2 is clearly invalid--since it has true premises and a false conclusion--while Argument # 3 is valid. It is not interesting that Argument Schema # 25 represents a valid argument; however, it is interesting that Argument Schema # 25 represents an invalid argument: for, since Argument Schema # 25 represents at least one invalid argument--Argument # 2--it must be an invalid argument schema. Recall what was said in § 5: "To say that [an argument schema] is valid is to say that there is no interpretation under which it will represent an argument which has true premises together with a false conclusion. And so, just in case there is at least one interpretation of an argument schema under which it represents an argument having true premises and a false conclusion, the argument schema is invalid."

[11] Refer to page L 7, in Boardman's Supplement. Here is a link to a picture of page 7.

[12] Consider Argument Schema # 24, below. Under the interpretation which is specified, it will represent Argument # 1:

Argument Schema # 24:	Argument # 1:	Interpretation under which Schema # 24 represents Argument # 1:
	If Kansas is not near an ocean, then if Lawrence is in Kansas, Lawrence is not near an ocean. Kansas is not near an ocean. Therefore, if Lawrence is in Kansas, Lawrence is not near an ocean.	K = Kansas is near an ocean N = Lawrence is in Kansas O = Lawrence is near an ocean

Notice that the Argument Schema called Modus Ponens generates Argument Schema # 24.

Modus Ponens:	Argument Schema # 24:	Substitutions under which Modus Ponens generates Argument Schema # 24
		Substitute: ~ K / P ; Substitute: (N ~ O) / Q .

As we have learned from § 6, if one argument schema generates a second argument schema, then any argument which is specified under an interpretation of the second must also be specified under some interpretation of the first. So, let us specify the interpretation under which the Modus Ponens argument schema will represent Argument # 1:

Modus Ponens:	Interpretation under which the Modus Ponens argument schema will represent Argument # 1:
	P = It is not the case that Kansas is near an ocean. Q = If Lawrence is in Kansas, then it is not the case that Lawrence is near an ocean.

In the examples above, notice the parallel between what is substituted for "P" and "Q" in order that the Modus Ponens argument schema generate Argument Schema # 24, on the one hand, and the interpretation of "P" and "Q" under which the Modus Ponens argument schema will represent Argument # 1, on the other: you will notice that ^{~ K} / P and ^{( N ~ O )} / Q were used for the substitutions for Modus Ponens and that "P = It is not the case that Kansas is near an ocean. and Q = If Lawrence is in Kansas, then it is not the case that Lawrence is near an ocean. " were used for the interpretation of Modus Ponens to create Argument # 1. Recall the text just prior to this footnote, "In fact, the compound sentences specified in the interpretation of schema A [Modus Ponens in this example] will have, as their component, simpler sentences, those sentences specified in the interpretation of schema B [Argument Schema # 24 in this example]."