Abstract
If the words in Spinoza’s Ethics are considered as symbols, then certain words in the definitions of the Ethics can be replaced with symbols from set theory and we can reexamine Spinoza’s first definitions within a logical–mathematical frame. The authors believe that, some aspects of Spinoza’s work can be explained and illustrated through mathematics. A semantic relation between the definitions of the philosopher and set theory is presented. It is explained each chosen symbol. The ontological argument is developed through modal logic. And finally, we present some conclusions drawn from this work.
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Notes
The authors use the nomenclature of the Ethics to indicate the respective propositions.
The axiom does not explicitly appear in the Ethics, however, is implicitly used in the demonstration.
This axiom uses the connection between modalities.
Russell analyzes definite descriptions similarly to indefinite descriptions, except that the individual is now uniquely specified. Take as an example of a definite description the sentence "the current Emperor of Kentucky is gray". Russell analyzes this phrase into the following component parts (with 'x' and 'y' representing variables):
- 1.
there is an x such that x is an emperor of Kentucky
- 2.
for every x and every y, if both x and y are emperors of Kentucky, then y is x (i.e., there is at most one emperor of Kentucky).
- 3.
anything that is an emperor of Kentucky is gray.
Thus, a definite description (of the general form 'the F is G') becomes the following existentially quantified phrase in classic symbolic logic (where 'x' and 'y' are variables and 'F' and 'G' are predicates—in the example above, F would be "is an emperor of Kentucky", and G would be "is gray"):
∃x[(Fx ˄ ∀y(Fy → x = y)) ˄ Gx]. Informally, this reads as follows: something exists with the property F, there is only one such thing, and this unique thing also has the property G.
This analysis, according to Russell, solves the two problems noted above as related to definite descriptions:
- 1.
"The morning star rises in the morning" no longer needs to be thought of as having the subject-predicate form. It is instead analyzed as "there is one unique thing such that it is the morning star and it rises in the morning". Thus, strictly speaking, the two expressions "the morning star…" and "the evening star…" are not synonymous, so it makes sense that they cannot be substituted (the analyzed description of the evening star is "there is one unique thing such that it is the evening star and it rises in the evening"). This solves Gottlob-Frege’s problem of informative identities.
- 2.
Since the phrase "the current Emperor of Kentucky is gray" is not a referring expression, according to Russell's theory it need not refer to a mysterious non-existent entity. Russell says that if there are no entities X with property F, the proposition "X has property G" is false for all values of X. Russell says that all propositions in which the Emperor of Kentucky has a primary occurrence are false. The denials of such propositions are true, but in these cases the Emperor of Kentucky has a secondary occurrence (the truth value of the proposition is not a function of the truth of the existence of the Emperor of Kentucky).
- 1.
For Spinoza, God or infinite substance is the source of all that exists; there are things in the world because they are manifestations of such substance. And, of course, if there are specific things that exist, then there must be a fortiori, that from which they emerge.
From the Imaginatio to Amor Dei intellectualis, understood as looking from the minimum sensitivity to look at from the point of view of the substance.
Corresponds to the twenty-ninth letter of the edition and is called, in fact, Letter on infinity.
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Usó-Doménech, J.L., Nescolarde-Selva, J.A. & Gash, H. Ontological Argument and Infinity in Spinoza’s Thought. Found Sci 25, 385–400 (2020). https://doi.org/10.1007/s10699-019-09627-2
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DOI: https://doi.org/10.1007/s10699-019-09627-2