Inferences Involving Singular Propositions

In Modern Predicate Logic (MPL), logical quantity comes in three 'flavors,' universal, particular, and singular. Thus 'All bloggers are self-absorbed' and 'No bloggers are self-absorbed' are universal; 'Some bloggers are self-absorbed' and 'Some bloggers are not self-absorbed' are particular; 'Bernie is self-absorbed' and 'Bernie is not self-absorbed' are singular. Traditional Formal Logic (TFL), however, does not admit a separate category of singular propositions.

So, just to draw out commenter Edward the Nominalist and Co., how would a defender of TFL account for the validity of the following obviously valid argument:

   1. Mars is red
   2. Mars is a planet
   -----
   3. Some planet is red.

A supporter of MPL could construct a derivation as follows:

   4. Mars is a planet & Mars is red. (From 1, 2 by Conjunction)
   5. There is an x such that: x is a planet & x is red. (From 4 by
        Existential Generalization)
   3. Some planet is red. (From 5 by translation back into ordinary
       language)

No sweat for the MPL boys, but how do you TFL-ers do it? (Of course I am aware that it can be done. The point of this post is mainly  didactic.)

Comments

[Edward the nominalist]

No problem. The scholastics believed that the singular term has the force of the universal, and that that the same conclusion follows as would follow if it were universal. The reason is is that when a singular proposition is true, if affirmative, then the subject (e.g. ‘Socrates’) cannot be truly predicated of anything without the predicate also being truly predicated of it. E.g. if ‘Tully is white’ is true, then whatever ‘Tully’ applies to – let’s say ‘Cicero’, then ‘white’ also applies. You may baulk at ‘Tully’ being ‘predicated’ of anything, but then choose another word. Perhaps ‘applies to’ or ‘refers to’. Ockham explains this here http://www.logicmuseum.com/wiki/Authors/Ockham/Summa_Logicae/Book_III-1/Chapter_8 .

Regarding your ‘Mars’ example. Converting the singular ‘Mars’ to a universal gives a proposition of the form

(1) Every A is B
(2) Every A is C
(3) Some B is C

This is in the third figure, Darapti. It could be proved as follows.

(4) From (2): Some A is C – remember that in TFL, unlike MPC the universal implies the particular
(5) From (4) by conversion: Some C is A
(6) From (1) and (5), Datisi: Some C is B
(7) From (6), conversion: Some B is C - which was to be proved.

The very example you give was also called an ‘expository syllogism’ by the scholastics.

[Bill Vallicella]

Sorry to be pedantic, but aren't you falling into use-mention confusion? 'Tully' applies to Cicero, not to 'Cicero.' After all, 'Tully' is the name of a man, not the name of a word.

Otherwise, well done.

[Account Deleted]

"Converting the singular ‘Mars’ to a universal..."

I have no clue what this means. A universal is by its very nature something that can simultaneously apply to more than one object. 'Mars' is a name of an individual object and this individual object by its very nature cannot be multiply instantiated. Both of Ed's arguments above are based upon an incoherent presupposition.

In addition, the above quotation commits the use/mention fallacy; a universal is a non-linguistic entity, multiply applicable. The word mentioned in the quotation is a linguistic entity; in particular, a name. A linguistic entity such a s a name cannot be "converted" into a universal entity. I wonder the extent to which the above arguments Ed gives presuppose this confusion between use and mention.

[Edward the nominalist]

>> Sorry to be pedantic

Well you are being somewhat pedantic. Using ‘applies to’ in its perfectly ordinary sense, ‘Cicero’ applies to ‘Tully’ in the way I meant.

Peter:
>> 'Mars' is a name of an individual object and this individual object by its very nature cannot be multiply instantiated.

By ‘convert to a universal’ I meant ‘convert to a universally quantified expression that has the equivalent logical force’. By a universally quantified expression I mean a noun phrase of the form ‘every M’.

>>A linguistic entity such a s a name cannot be "converted" into a universal entity.

A linguistic entity such as a name can be converted into a ‘universally quantified expression’, namely a noun phrase of the form ‘every M’.

>> I wonder the extent to which the above arguments Ed gives presuppose this confusion between use and mention.

None whatsoever. The term ‘Mars’ is a term. This term, used in the sense we are using it here, bears a certain relation to a certain planet. Choose the name you like for this relation. Let’s use the scholastic term ‘supposits for’. Thus ‘Mars’ supposits for Mars. Only one thing is the suppositum of that proper name, used in that sense, and the name has that suppositum at whatever time and whatever possible world. That’s why ‘Mars’ is a singular term, and that’s why by its very nature it supposits for just one thing.

By contrast, the term ‘planet’ naturally supposits for more than one thing (nine things, in fact). In scholastic theory, the same relation (supposition) obtains between singular and general terms, and the things they supposit for. In MPC, by contrast, we have two. Reference, for the singular relation, and ‘satisfaction’ for the general one (and in addition, the concept of predication is different also).

Thus, in scholastic logic ‘Mars is a planet’ is true when at least one thing is the suppositum of both ‘Mars’ and ‘planet’. And (by the dici de omni), what a universal proposition signifies is that nothing is the suppositum of the predicate, without also being a suppositum of the predicate. Thus the singular proposition ‘Mars is a planet’ has the force of a universal proposition. ‘Mars is a planet’ signifies that nothing is the suppositum of ‘Mars’, without also being the suppositum of ‘planet’.

In none of what I have just said is there any confusion between use and mention.

[Bill Vallicella]

You are pedantic, not your name [grin]. If I say of Edward that he is pedantic, then I apply 'pedantic' to Edward, not to 'Edward.'

Peter,

The scholastic position that Edward is expounding is actually quite workable and free of obvious mistakes. I refer you to Fred Sommers, *The Logic of Natural Language.*

You find it bizarre because you learned your logic from MPL texts.

[Account Deleted]

Bill,

"The scholastic position that Edward is expounding is actually quite workable and free of obvious mistakes. I refer you to Fred Sommers, *The Logic of Natural Language.*

You find it bizarre because you learned your logic from MPL texts."

No Bill; I do not find it bizarre because of my logic background. I find it bizarre because of the following reason.

In just about one hundred and thirty years or so our understanding of reasoning, logic, mathematics, computer science, and many other fields have advanced to an unparalleled degree. This advance was due primarily to the discoveries of Frege, buttressed by Russell (and others), in logic and Cantor in set theory. The advances these discoveries enabled are several hundred times more profound than whatever the Scholastics achieved in more than two thousand years prior, since Aristotle.

In light of these historical facts, I find it bizarre that anyone would wish to revert back to Scholastic logic (whatever that may be). Would you take seriously a suggestion to abandon post-Newtonian science in favor of Ptolemy’s astronomy or pre-Newtonian physics? What benefit could such a retro-movement offer (with all due respect to Sommers, whom I knew personally and respected)?

[Account Deleted]

Ed,

1)You say: “Let’s use the scholastic term ‘supposits for’.”

Let’s not! There is already a clear term for the relation between the linguistic entity ‘Mars’ and the planet: it is called ‘reference’. If the scholastic terms ‘supposits for’ just means reference, then let’s stick with the later; if it does not, then I have no clue what this scholastic term means and why should we abandon useful usage in favor of obscure terminology.

2) “In scholastic theory, the same relation (supposition) obtains between singular and general terms, and the things they supposit for. In MPC, by contrast, we have two. Reference, for the singular relation, and ‘satisfaction’ for the general one (and in addition, the concept of predication is different also).”

Russell showed how to eliminate singular reference in connection with ‘the’ in favor of quantifiers and identity; Quine proposed to convert a name such as ‘Pegasus’ into a predicate, ‘Pegasizes’, thereby eliminating singular reference.

From a purely logical point of view, satisfaction suffices for all purposes (except with respect to variables). In fact, one can construct a formal language without any names in which all singular reference is carried by quantification and predication. So with respect to this question, I do not see a fundamental difference between modern and scholastic logic in so far as these issues are concerned. (There are of course more fundamental differences between the two logics, differences that render the former so much more valuable and richer compared to the later).

The problem of singular reference as it pertains to names arises when modern logic is used to account for the use of names in normal discourse. The debate between descriptivists (Frege-Russell, etc.,) vs. the new theory of reference (Kripke, Kaplan, half-of-Putnam, etc.,) is precisely about whether the best account of names requires singular reference or such reference can be eliminated in favor of quantifiers and predication (i.e., satisfaction). So scholasticism offers nothing new here. It invites obscurity where some degree of clarity was earned by the advent of modern logic and its conceptual resources.

3) You say: “‘Mars is a planet’ signifies that nothing is the suppositum of ‘Mars’, without also being the suppositum of ‘planet’.”

Let’s then recast the ‘Mars is a planet’ example in a system of modern logic in which names are absent. How would this be done? Simple! Convert the name ‘Mars’ a la Quine into the predicate ‘Marsifies’. Then you get:

(*) ‘Mars is a planet’ is true just in case for every x, if x Marsifies, then x is a planet.

How is scholastic account different than (*)? It is not. Your scholastic account simply uses scholastic verbiage to simulate (*). And I see no advantage to the scholastic account, and its verbiage, that is not already contained in the modern counterpart (*).

So yes, your account above does not commit the use-mention confusion. But then it does not do so simply because it is parasitic on the modern account. When it attempts to depart from the modern account, then I suspect it does so on pain of the use/mention confusion.

4) “And (by the dici de omni), what a universal proposition signifies is that nothing is the suppositum of the predicate, without also being a suppositum of the predicate.”

What does this mean?