Greetings.

In the immediately previous post, the “Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics” of September 24, 2020, I said that in the present post I would “spell out how one can prove, demonstrate, that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents.” And that is what I will do, in four steps.

The Four Steps. The first step to be taken is that of proving that any existent is one existent.

I will slip three notes in here. The first is that such propositions as “any existent is one existent” stands in need of more precise reformulations, first in the “nearly standard English” and then in the abbreviations of “mathematical logic” that I made use of in the previous post. I’ll broaden the scope of that observation to include, in principle, all of the propositions playing a role in the coming proofs, although I will provide the reformulation in the “nearly standard English” of only a few of them, chiefly those serving as premises and conclusions.

The second note is that some may see in the proposition, that “any existent is one existent,” a statement of the classical scholastic metaphysical or ontological doctrine that “unity is a ‘transcendental’ property of being,” i.e., “all beings are units.”

The third note, more pertinent to the present post than the second note, is that the conclusion of the first step’s proof will be hereinafter identified as the Principle of Universal Unity, i.e., less grandly, the P.U.U., and will serve as a premise in the proofs that will constitute the second and third of the four steps.

The second step to be taken is that of proving that one plus one are two, while the third is that of proving that two plus one are three. The fourth step is that of spelling out how to go about proving that two plus two are four.

In the first three steps, then I do not simply spell out, how the three conclusions can be proven, I prove them. In the fourth step, I forego the full proof; it is rather lengthy. (Were I to offer a proof, directed to the attention of the scholars of the philosophy of mathematics of Immanuel Kant, that seven plus five are twelve….).

I. The Proof That Any Existent Is One Existent.

The Conclusion. First, I’ll state the conclusion of the proof here on offer, that any existent is one existent, in both its nearly standard English and its “mathematical logic” reformulations.

In the nearly standard English: For any existent x, x is one existent.

In the rendering of “mathematical logic”: (x)1x

The Premises. The proof has three premises.

The First Premise. The Principle of the Symmetry of Identity. The principle of the symmetry of identity (hereinafter, P.S.I.) reads:

In the nearly standard English: For any existent x and any existent y, if x is identical with y, then y is identical with x.

In the rendering of “mathematical logic”: (x)(y)(Ixy –> Iyx)

The Second Premise. The Principle of the Transitivity of Identity. The principle of the transitivity of identity (hereinafter, P.T.I.) reads:

In the nearly standard English: For any existent x, any existent y, and any existent z, if x is identical with y and y is identical with z, then x is identical with z.

In the rendering of “mathematical logic”: (x)(y)(z)((Ixy & Iyz) –> Iyx)

The Third Premise. The Definition of What It Is to Be One Existent.

In the nearly standard English: For any existent x, x is one existent if and only if it is not the case that there is an existent y and there is an existent z such that y is not identical with z and such that both y is identical with x and z is identical with x.

In the rendering of “mathematical logic”: (x)(1x iff ~(Ey)(Ez)(~Iyz & (Iyx & Izx)))

(As I I noted in the immediately previous post, not knowing how to express, using the means available to users of this blog’s platform, WordPress, the “if and only if” relation in the double arrow notation standard in mathematical logic, I have to make do with with “iff.” And, while I’m at it, not knowing, again, how to express the backwards “E,” meaning “There exists…,” using the resources offered by tWordPress, I have had to make do with the standard “E.”)

It would help no one if I were to explicitly spell out every step in any of the four proofs using our nearly standard English. So, I will express the several steps of the first three proofs in the renderings of “mathematical logic.” All of the steps taken in any of the proofs are quite elementary and can be understood by anyone who has studied elementary “mathematical logic” or “symbolic logic.”

The Proof That Any Existent Is One Existent.

1. (x)(y)(Ixy –> Iyx) [Pr., P.S.I.]
2. (x)(y)(z)((Ixy & Iyz) –> Ixz) [Pr., P.T.I.]
3. (x)(1x iff ~(Ey)(Ez)((Iyx & Izx) & ~Iyz) [Pr., Def.]
4. 1a iff ~(Ey)(Ez)((Iya & Iza) & ~Iyz) [3, U.I.]
5. (1a –> ~(Ey)(Ez)((Iya & Iza) & ~Iyz)) & (~(Ey)(Ez)((Iya & Iza) & ~Iyz)) –> 1a) [4, L.E.]
6. (~(Ey)(Ez)(Iya & Iza) & ~Iyz)) –> 1a) [5, Comm.]
&
(1a –> ~(Ey)(Ez)((Iya & Iza) & ~Iyz)))
7. ~(Ey)(Ez)((Iya & Iza) & ~Iyz)) –> 1a [6, Simp.]

Now here, in Step 8, one assumes the contradictory of the anticipated conclusion and then derives a contradiction from the conjoining of that assumption to the premise set. That proves that that contradictory of the anticipated conclusion is inconsistent with the premise set and thus that the anticipated conclusion necessarily follows from the premise set.

8. ~1a [Ass., C.P.]
9. ~~(Ey)(Ez)((Iya & Iza) & ~Iyz)) [7, 8, M.T.]
10. (Ey)(Ez)((Iya & Iza) & ~Iyz)) [9, D.N.]
11. (Iba & Ica) & ~Ibc [10, E.I.]
12. Iba & Ica [11, Simp.]
13. Iba [12, Simp.]
14. Ica [13, Comm., Simp.]
15. Ica –> Iac [1, U.I.]
16. Iac [15, 14, M.P.] {Correction, 11/14/2020, “P” for “T”}
17. Iba & Iac [13, 16, Conj.]
18. (Iba & Iac) –> Ibc [2, U.I.]
19. Ibc [18, 17, M.P.]
20. ~Ibc [11, Comm., Simp.]
21. Ibc & ~Ibc [19, 20, Conj.]

22. ~1a –> (Ibc & ~Ibc) [8-21, C.P.]
23. ~(Ibc & ~Ibc) [Taut.]
24. ~~1a [22, 23, M.T.]
25. 1a [24, D.N.]
26. (x)1x [25, U.G.]

The Proof That One Existent Plus One Existent Are Two Existents.

1. (x)1x [Pr., P.U.U.]
2. (x)(y)(2xy iff ~Ixy) [Pr., Def.] {Correction, 11/14/2020, Insertion of “iff”}
3. 2ab iff ~Iab [2, U.I.]
4. (2ab –> ~Iab) & (~Iab –> 2ab) [3, L.E.]

“And,” “Plus,” and the Distinction Condition. I interrupt this proof to note that in Step 5, the first of two applications of the conditional proof technique, there are two propositions conjoined by the second “&” (i.e., of course, “and”). The first of the two conjuncts, the “(1a & 1b),” is itself a conjunction of two propositions, “1a” and “1b.” Were that the entire story, then the common way of stating the arithmetical truth, “One and one are two,” would be acceptable. It is not, however, acceptable, for one and one not need be two. If, for example, a and b are the same person, under different names, then one and one are one. And so there is the need for the second conjunct of the whole conjunction, “~Iab,” a recognition that a is not identical with b. It is because of this, as I have taken to call it, distinction condition, that I want to insist on the formulation, “one plus one are two,” rather than “one and one are two.”

5. (1a & 1b) & ~Iab [Ass., C.P.]
6. ~Iab –> 2ab [4, Comm., Simp.]
7. ~Iab [5, Comm., Simp.]
8. 2ab [6, 7, M.P.]

9. ((1a & 1b) & ~Iab) –> 2ab [5-8, C.P.]

10. 2ab [Ass., C.P.]
11. 2ab –> ~Iab [4, Simp.]
12. ~Iab [11, 10, M.P.]
13. 1a [1, U.I.]
14. 1b [1, U.I.]
15. 1a & 1b [13, 14, Conj.]
16. (1a & 1b) & ~Iab [15, 12, Conj.]

17. 2ab –> ((1a & 1b) & ~Iab) [10-16, C.P.]
18. (((1a & 1b) & ~Iab) –> 2ab) & (2ab –> ((1a & 1b) & ~Iab)) [9, 17, Conj.]
19. ((1a & 1b) & ~Iab) iff 2ab [18, L.E.]
20. (x)(y)(((1x & 1y) & ~Ixy) iff 2xy) [19, U.G.]

The Proof That Two Existents Plus One Existent Are Three Existents.

1. (x)1x [Pr., P.U.U.]
2. (x)(y)(2xy iff ~Ixy) [Pr., Def.]
3. (x)(y)(z)(3xyz iff (~Ixy & ~Ixz & ~Iyz)) [Pr., Def.]
4. 2ab iff ~Iab [2, U.I.]
5. (2ab –> ~Iab) & (~Iab –> 2ab) [4, L.E.]
6. 3abc iff (~Iab & ~Iac & ~Ibc) [3, U.I.]
7. (3abc –> (~Iab & ~Iac & ~Ibc)) & ((~Iab & ~Iac & ~Ibc) –> 3abc) [6, L.E.]

Let’s notice that in Step 8, there are two conjunctions of conjunctions. The second such conjunction of conjunctions, “~Iac & ~Ibc,” is the expression of the distinction condition for the addition at hand, that of two existents plus one existent. The distinction condition, one can surmise, is a function of the addenda.

8. (2ab & 1c) & (~Iac & ~Ibc) [Ass., C.P.]
9. 2ab & 1c [8, Simp.]
10. 2ab [9, Simp.]
11. 2ab –> ~Iab [5, Simp.]
12. ~Iab [11, 10, M.P.]
13. ~Iac & ~Ibc [8, Comm., Simp.]
14. ~Iac [13, Simp.]
15. ~Iab & ~Iac [12, 14, Conj.]
16. ~Ibc [13, Comm., Simp.]
17. ~Iab & ~Iac & ~Ibc [15, 16, Conj.]
18. (~Iab & ~Iac & ~Ibc) –> 3abc [7, Comm., Simp.]
19. 3abc [18, 17, M.P.]

20. ((2ab & 1c) & (~Iac & ~Ibc)) –> 3abc [8-19, C.P.]

21. 3abc [Ass., C.P.]
22. 3abc –> (~Iab & ~Iac & ~Ibc) [7, Simp.]
23. ~Iab & ~Iac & ~Ibc [22, 21, M.P.]
24. ~Iab –> 2ab [5, Comm., Simp.]
25. ~Iab [23, Simp.]
26. 2ab [24, 25, M.P.]
27. 1c [1, U.I.]
28. 2ab & 1c [26, 27, Conj.]
29. ~Iac & ~Ibc [23, Comm., Simp.]
30. (2ab & 1c) & (~Iac & ~Ibc) [28, 29, Conj.]

31. 3abc –> ((2ab & 1c) & (~Iac & ~Ibc)) [21-30, C.P.]
32. (((2ab & 1c) & (~Iac & ~Ibc)) –> 3abc) [20, 31, Conj.]
&
(3abc –> ((2ab & 1c) & (~Iac & ~Ibc)))
33. (((2ab & 1c) & (~Iac & ~Ibc)) iff 3abc [32, L.E.]
34. (x)(y)(z)(((2xy & 1z) & (~Ixz & ~Iyz)) iff 3xyz) [33, U.G.]

The Proof That 2 Plus 2 Are 4.

It would help no one if I were to explicitly spell out every step in the fourth proof even using the renderings of mathematical or symbolic logic. It will not help those who are not adept in that logic. And it will not help those who are adept in that logic, for they will not need the explicit spelling out, as each of the (very many) steps is utterly elementary. I’ll merely suggest to them that the use of conditional proofs, as was done in the previous proof, provides an easy way to the conclusion.

I’ll just get things started.

1. (x)(y)(2xy iff ~Ixy) [Pr., Def.]
2. (x)(y)(z)(z)(4xyzw iff (((~Ixy & ~Ixz & ~Ixw) & (~Iyz & ~Iyw)) & ~Izw)) [Pr., Def.]
3. 2ab iff ~Iab [1, U.I.
4. 2cd iff ~Icd [1, U.I.
5. 4abcd iff (((~Iab & ~Iac & ~Iad) & (~Ibc & ~Ibd)) & ~Icd) [2, U.I.
6. (2ab iff ~Iab) & (~Iab  2ab) [3, L.E.
7. (2cd iff ~Icd) & (~Icd  2cd) [4, L.E.
8. (4abcd –> (((~Iab & ~Iac & ~Iad) & (~Ibc & ~Ibd)) & ~Icd)) [5, L.E.]
&
((((~Iab & ~Iac & ~Iad) & (~Ibc & ~Ibd)) & ~Icd) –> 4abcd)

Note the distinction condition operative in the second conjunction of the proof’s next step.

9. (2ab & 2cd) & (~Iac &~ Iad & ~Ibc & ~Ibd)) [Ass., C.P.]
.

.
.
45. (x)(y)(z)(w)((((2xy & 2zw) & (~Ixz & ~Ixw & ~Iyz & ~Iyw))) iff 4xyzw) [44, U.I.]

Summing Up.

In this post I have, as promised in the previous post, spelled out how one can prove, demonstrate, that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents; in fact, I have provided complete proofs of the conclusions of the first two.

But I also stated, in the previous post, that in the present post I [would] “affirm that the philosophical realism of the ontology and arithmetic at hand is an apodicticism, according to which they are, not just theories of extra-mental and extra-linguistic existents, but demonstrative sciences thereof. With that done, I believe[d,] I [would] have two realistic feet in the door of the philosophy of mathematics.” My thinking now, however, is that I should do more than so affirm before allowing myself to believe that I have two realistic feet in the door of the philosophy of mathematics. So it is my intention to say, in the next post, or perhaps in the post after next, a little bit more about the philosophical understanding that the realistic philosophy of mathematics that I have in mind brings with it.

Until next time.

Richard