Formalizing Aquinas’ Fourth Way

A “less obese” Thomas for a bare-bones formal representation of the 4th way!

I am interested in Aquinas’ Fourth Way, but I find that he lays out the argument so succinctly in the Summa Theologiae that it’s hard to see a valid proof at first blush:

The fourth way is taken from the gradation to be found in things. Among beings there are some more and some less good, true, noble and the like. But “more” and “less” are predicated of different things, according as they resemble in their different ways something which is the maximum, as a thing is said to be hotter according as it more nearly resembles that which is hottest; so that there is something which is truest, something best, something noblest and, consequently, something which is uttermost being; for those things that are greatest in truth are greatest in being, as it is written in Metaph. ii. Now the maximum in any genus is the cause of all in that genus; as fire, which is the maximum heat, is the cause of all hot things. Therefore there must also be something which is to all beings the cause of their being, goodness, and every other perfection; and this we call God (ST I, 2.3).

I would like to eventually work out a stronger version of the argument–stronger, that is, by weaken some of the premises and show that they still lead to the conclusion that God exists. But, my first step is to try and properly depict the essence of the argument in its purest logical form. I think I have to quantify over predicates to capture what I think Aquinas is saying. Also, I’ve used the transcendental of “Truth” as the particular perfection in this formulation of the argument. I chose “Truth” because I didn’t want this to come off as a moral argument by using “Goodness”, and I didn’t want to use “Being” because I fear being slowed down by the “existence is not a real predicate” objection, though I think there are very good responses to that objection. Finally, I confess that I might have tripped up over some of my brackets, so forgive the crudeness of this draft, if crudeness you should find. I happily admit that the errors and misrepresentations are all my own, and not poor Thomas’ fault! So…

Let:
Πxy – x has a greater degree of predicate Π than y
ExΠy –x is the eminent cause of y being Π
Θx – x has godhood
Txy – x has a greater degree of truth than y
ExTy – x is the eminent cause of y being true
(NB: I use x, y, and z as variables and u, v, and w as pseudonyms)

1. (∀x){Θx ≡ (∀y)[(x≠y) → (Txy & ExTy)]} (definition)
2. (∀Π){(∀x)[(∀y)[Πxy → (∃z)[((z≠x) → (Πzx & EzΠx))]]]} (premise)
3. (∃x)(∃y)Txy (premise)
∴(∃x)Θx
Deduction:
4. (∃y)Tuy (3 EI)
5. Tuv (4 EI)
6. (∀x){(∀y)[Txy → (∃z)[(z≠x) →( Tzx & EzTx)]]} (2 UI)
7. (∀y)[Tuy → (∃z)[(z≠u) → (Tzu & EzTu)]] (6 UI)
8. Tuv → (∃z)[(z≠u) → (Tzu & EzTu)] (7 UI)
9. (∃z)((z≠u) → (Tzu & EzTu) (5,8 MP)
10. (w≠u) → (Twu & EwTu) (9 EI)
11. Θw ≡ (∀y)[(w≠y) →(Twy & EwTy)] (1 UI)
12. {Θw → (∀y)[(w≠y) → (Twy & EwTy)]} & { (∀y)[(w≠y) → (Twy & EwTy)] → Θw} (11 Equiv)
13. (∀y)[(w≠y) → (Twy & EwTy)] → Θw (12 Simp)
14. [(w≠u) → (Twu & EwTu)] → Θw (13 UI)
15. Θw (10,14 MP)
16. (∃x)Θx (15 EG)

If you see any problems, let me know in the comments! On a lighter note, here is another version of the Fourth Way that I did with the help of lyrics from the 1980’s classic Higher Love, because if you think about it, there must be higher love…

[Update: My very bright and patient friend, Damon Watson, noticed some problems with the brackets and I have made changes accordingly]

Posted on February 27, 2013, in Arguments for God and tagged , , , , . Bookmark the permalink. Leave a comment.

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