{"id":7958,"date":"2014-05-13T19:12:58","date_gmt":"2014-05-13T19:12:58","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2014\/05\/13\/more-on-values-and-variables\/"},"modified":"2014-05-13T19:12:58","modified_gmt":"2014-05-13T19:12:58","slug":"more-on-values-and-variables","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2014\/05\/13\/more-on-values-and-variables\/","title":{"rendered":"More on Values and Variables and Logical Form: An Aporetic Hexad"},"content":{"rendered":"

David Brightly comments:<\/span><\/p>\n

\n

. . .  my old copy of Alan Hamilton, Logic for Mathematicians<\/em>, CUP 1978, uses 'statement variables' in his account of the 'statement calculus', as here<\/a>. The justification for 'variable' is surely that statements have values, namely truth and falsehood. The truth value of a compound statement is calculated from the truth values of its component simple statements by composition of the truth functions corresponding to the logical connectives. This is analogous to the evaluation of an arithmetic expression by composition of arithmetic functions applied to the values of arithmetic variables.<\/span><\/p>\n<\/blockquote>\n

I detect a possible conflation of two senses of 'value.'  There is 'value' in the sense of truth value, and there is 'value' in the sense of the value of a variable.<\/span><\/p>\n

If I am not mistaken, talk of truth values in the strict sense of this phrase enters the history of logic first with Gottlob Frege (1848-1925).  Truth and Falsity for him are not properties of propositions, but values of propositional functions.  Thus the propositional function denoted by 'x is wise'  has True for its value with Socrates as argument, and False for its value with Nero as argument.  Please note the ambiguity of 'argument.'  We are now engaging in MathSpeak.  The analogy with mathematics is obvious.  The squaring function has 4 for its value with 2 or -2 as arguments.  Propositional functions map their arguments onto the two truth values.<\/span><\/p>\n

But we also speak in a different sense of the value of a variable.  The bound variables in<\/span><\/p>\n

\n

(x)(x is a man –> x is mortal)<\/span><\/p>\n<\/blockquote>\n

range over real items.  These items are the values of the bound variables but they are not truth values.  Therefore, one should not confuse 'value' in the sense of truth value with 'value' in the sense of value of a variable.  When Quine famously stated that "To be is to be the value of a [bound] variable" he was not referring to truth values.<\/span><\/p>\n

Brightly says that "The justification for 'variable' is surely that statements have values, namely truth and falsehood."  I think that is a mistake that trades on the confusion just exposed.  Agreed, statements have truth values.  But it doesn't follow that that placeholders for statements are variables.<\/span><\/p>\n

I was pleased to see that Hamilton observes the distinction I drew several times between an abbreviation and a placeholder.  He uses 'label' for 'abbreviation,' but no matter.  But I distinguish a placeholder from a variable while Hamilton doesn't.  <\/span><\/p>\n

To appreciate the distinction, first note that with respect to variables we ought to make a three-way distinction among the variable, say 'x,' the value, say Socrates, and the substituend, say 'Socrates.'  Now consider the argument:<\/span><\/p>\n

Tom is tall or Tom is fat<\/span>
Tom is not tall<\/span>
——-<\/span>
Tom is fat<\/span><\/p>\n

This argument has the form of the Disjunctive Syllogism:<\/span><\/p>\n

P v Q<\/span>
~P<\/span>
——-<\/span>
Q.<\/span><\/p>\n

Obviously, 'P' and 'Q' are not abbreviations (labels); if they were then the second display would not display an argument form<\/em>.   It would be an abbreviated argument<\/em>.  But it doesn't follow that 'P' and 'Q' are variables. For if they were variables, then they would have both substituends andf values.  But while they have substituends, e.g., the sentences 'Tom is tall' and 'Tom is fat,' they don't have values.  Why not?  Because we are not quantifying over propositions (or statements if you prefer).   There are no quantifiers in the form diagram.  (This is not to say that one cannot quantify over propositions.)<\/span><\/p>\n

'Tom' is tall' is one of many possible substituends for 'P.'  But 'Tom is tall' is not the value of 'P.'  For we are not quantifying over sentences.  We are not quantifying over propositions either.  So *Tom is tall* is also not a value of 'P.'<\/span><\/p>\n

My thesis is that placeholders in the propositional calculus are arbitrary propositional constants<\/em>.  Since they are constants, they are not variables.   It is a subtle distinction, I'll grant you that, but it seems necessary if we are to think precisely about these matters.  But then one man's necessary distinction is another man's hair-splitting.<\/span><\/p>\n

\n

You also argue that London must wrongly decide that 'if roses are red then roses are red' (RR) is a contingency, because we say it can be seen as having the form 'P–>Q' and in general<\/em> statements of this form are contingencies. Indeed they are. But we don't so decide. We say this is a special case in which P and Q stand for the same simple sentence, 'roses are red', not different ones. P and Q are therefore either both true or both false and either way the truth function for –> returns true. Hence this special case is tautologous. We disagree that the move from RR to 'P–>Q' must be seen as an abstraction. We retain the information that P and Q stand for specific substatements within RR, which may themselves have internal structure. 'Form' is a device for making such structure explicit.<\/span><\/p>\n<\/blockquote>\n

So you are saying that 'P –> Q' has a special case that is tautologous. But that makes no sense to me if RR has both forms.  A sentence (understood to have one definite meaning) is tautologous if its logical form is tautologous, and if RR has the form 'P–> Q' then it it is not tautologous as an instance of that form.  So you seem committed to saying that RR is both tautologous and not tautologous.<\/span><\/p>\n

Isn't that obvious?  If one and same sentence (understood to have one definite meaning) has two logical forms, one tautologous and the other non-tautologous, then one and the same sentence is both tautologous and non-tautologous — which is a contradiction.<\/span><\/p>\n

One solution, as I have suggested several times already, is to say that, while 'P –> P' is a special case of 'P –>Q,' namely the case in which P = Q, the two forms are not both forms of 'If roses are red, then roses are red.'  Only one of them is, the first one.  The second is a form of the first form, not a form of the English sentence.<\/span><\/p>\n

Putting the problem as an aporetic hexad:<\/span><\/p>\n

1. 'P –>P' is a special case of 'P –> Q'<\/span>
2. If a proposition s instantiates form F, and F is a special case of form G, then s instantiates G.<\/span>
3. 'P –> P' is a tautologous form.<\/span>
4. 'P –> Q' is a non-tautologous form.<\/span>
5. No one proposition instantiates both a tautologous and a non-tautologous form.<\/span>
6. 'If roses are red, then roses are red' instantiates the form 'P –> P.'<\/span><\/p>\n

The hexad is inconsistent.  Phoenix and London agree on (1), (3), (4), and (6).  The Phoenician solution is to reject (2).  The Londonian solution is reject (5).<\/span><\/p>\n

But the Phoenicians have an argument for (5):<\/span><\/p>\n

7. The logical form of a proposition is not an accidental feature of it but determines the very identity of the proposition.<\/span>
Ergo<\/span>
8. If s instantiates form F, then necessarily, s instantiates F.<\/span>
ergo<\/span>
5. No one proposition instantiates both a tautologous and a non-tautologous form.
<\/span><\/p>\n

\nRelated articles<\/legend>\n
\n
\"\"<\/a>Placeholders, Variables, and Logical Form<\/a><\/div>\n
\"\"<\/a>Logical Form and the Supposed Asymmetry of Validity and Invalidity: A Defense of Symmetry<\/a><\/div>\n
\"\"<\/a>Logical Form, Instantiation, and Pattern-Matching<\/a><\/div>\n<\/div>\n<\/fieldset>\n","protected":false},"excerpt":{"rendered":"

David Brightly comments: . . .  my old copy of Alan Hamilton, Logic for Mathematicians, CUP 1978, uses 'statement variables' in his account of the 'statement calculus', as here. The justification for 'variable' is surely that statements have values, namely truth and falsehood. The truth value of a compound statement is calculated from the truth … Continue reading “More on Values and Variables and Logical Form: An Aporetic Hexad”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[108],"tags":[],"class_list":["post-7958","post","type-post","status-publish","format-standard","hentry","category-logica-docens"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/7958","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/comments?post=7958"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/7958\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=7958"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=7958"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=7958"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}