{"id":5189,"date":"2017-09-30T11:51:14","date_gmt":"2017-09-30T11:51:14","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2017\/09\/30\/the-function-argument-schema-in-the-analysis-of-propositions\/"},"modified":"2017-09-30T11:51:14","modified_gmt":"2017-09-30T11:51:14","slug":"the-function-argument-schema-in-the-analysis-of-propositions","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2017\/09\/30\/the-function-argument-schema-in-the-analysis-of-propositions\/","title":{"rendered":"The Function-Argument Schema in the Analysis of Propositions"},"content":{"rendered":"

The Ostrich of London sends the following to which I add some comments in blue<\/span>.<\/span><\/p>\n

Vallicella: \u2018One of Frege's great innovations was to employ the function-argument schema of mathematics in the analysis of propositions\u2019.  <\/span><\/p>\n

Peter Geach (\u2018History of the Corruptions of Logic\u2019, in Logic Matters<\/em> 1972, 44-61) thinks it actually originated with Aristotle, who suggests (Perihermenias<\/em> 16b6) that a sentence is composed of a noun (\u1f44\u03bd\u03bf\u03bc\u03b1) and a verb (\u1fe5\u1fc6\u03bc\u03b1), and the verb is a sign of something predicated of something else. According to Geach, Aristotle dropped this name-predicate theory of the proposition later in the Analytics, an epic disaster \u2018comparable only to the fall of Adam\u2019, so that logic had to wait more than two thousand years before the \u2018restitution of genuine logic\u2019 ushered in by Frege and Russell. By \u2018genuine logic\u2019 he means modern predicate logic, which splits a simple proposition into two parts, a function expression, roughly corresponding to a verb, and an argument expression, roughly corresponding to a noun. \u2018To Frege we owe it that modern logicians almost universally accept an absolute category-difference between names and predicables; this comes out graphically in the choice of letters from different founts [fonts<\/span>] of type for the schematic letters of variables answering to these two categories\u2019.<\/span><\/p>\n

The Fregean theory of the proposition has never seemed coherent to me. Frege began his studies (Jena and G\u00f6ttinge, 1869\u201374) as a mathematician. Mathematicians naturally think in terms of \u2018functions\u2019 expressing a relation between one number and another. Thus<\/span><\/p>\n

            f(3)  =  9<\/span><\/p>\n

where \u20183\u2019 designates the argument or input to the function, corresponding to Aristotle\u2019s \u1f44\u03bd\u03bf\u03bc\u03b1, \u2018f()\u2019 the function, here y=x2<\/sup>, corresponding to Aristotle\u2019s \u1fe5\u1fc6\u03bc\u03b1, and \u20189\u2019 the value<\/em> of the function. The problem is the last part. There is nothing in the linguistic form<\/em> of the proposition which corresponds to the value in the linguistic form of the mathematical function. It is invisible. Now Frege thinks that every propositional function or \u2018concept\u2019 maps the argument to one of two values, either the True or the False. OK, but this is a mapping which, unlike the mathematical mapping, cannot be expressed in language. We can of course write<\/span><\/p>\n

            ___ is wise(Socrates) = TRUE<\/span><\/p>\n

but then we have to ask whether that equality is true or false<\/em>, i.e. whether the function \u2018is_wise(–) = TRUE\u2019 itself maps Socrates onto the true or the false. The nature of the value (the \u2018truth value\u2019) always eludes us. There is a sort of veil beyond which we cannot reach, as though language were a dark film over the surface of the still water, obscuring our view of the Deep.<\/span><\/p>\n

BV: First a quibble. There is no need for the copula 'is' in the last formula since, for Frege, concepts (which are functions) are 'unsaturated' (ungesaettigt<\/em>) or incomplete.  What exactly this means, of course, is  a separate problem.  The following suffices:<\/span><\/p>\n

\n

___wise(Socrates) = TRUE.<\/span><\/p>\n<\/blockquote>\n

The line segment '___' represents the gappiness or unsaturatedness of the concept expressed by the concept-word (Begriffswort<\/em>).<\/span><\/p>\n

Quibbling aside, the Ostrich makes two correct interrelated points, the first negative, the second positive.<\/span><\/p>\n

The first is that while 'f(3) = 9' displays the value of the function for the argument 3, namely 9, a sentence that expresses a (contingent) proposition does NOT display its truth-value. The truth-value remains invisible. I would add that this is so whether I am staring at a physical sententional inscription or whether I am contemplating a proposition with the eye of the mind.  The truth or falsity of a contingent proposition is external to it.  No doubt, 'Al is fat' is true iff Al is fat.' But this leaves open the question whether Al is<\/strong> fat.  After all the biconditional is true whether or not our man is, in fact, obese.<\/span><\/p>\n

The second point is that there has to be something external to a contingent proposition (such as the one expressed by 'Socrates is wise') that is involved in its being true, but this 'thing,' — for Frege the truth-value — is ineffable.  Its nature eludes us as the Ostrich correctly states.  I used the somewhat vague phrase 'involved in its being true' to cover two possibilities. One is the Fregean idea that declarative sentences have both sense and reference and that the referent (Bedeutung<\/em>) of a whole declarative sentence is a truth-value.  The other idea, which makes a lot more sense to me, is that a sentence such as 'Socrates is wise' has a referent, but the referent is a truth-making fact or state of affairs, the fact of Socrates' being wise<\/em>.<\/span><\/p>\n

Now both of these approaches have their difficulties.  But they have something sound in common, namely, the idea that there has to be something external to the contingent declarative sentence\/proposition involved in its being true rather than false.  There has to be more to a true proposition than its sense.  It has to correspond to reality.  But what does this correspondence really come to? Therein lies a major difficulty.  <\/span><\/p>\n

How will the Ostrich solve it? My impression is that he eliminates the difficulty by eliminating reference to the extralinguistic entirely. <\/span><\/p>\n

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The Ostrich of London sends the following to which I add some comments in blue. Vallicella: \u2018One of Frege's great innovations was to employ the function-argument schema of mathematics in the analysis of propositions\u2019.   Peter Geach (\u2018History of the Corruptions of Logic\u2019, in Logic Matters 1972, 44-61) thinks it actually originated with Aristotle, who suggests (Perihermenias 16b6) … Continue reading “The Function-Argument Schema in the Analysis of Propositions”<\/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":[126,408,108,541],"tags":[],"class_list":["post-5189","post","type-post","status-publish","format-standard","hentry","category-frege","category-language-philosophy-of","category-logica-docens","category-propositions"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/5189","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=5189"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/5189\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=5189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=5189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=5189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}