Consider this argument:
No dogs are skunks. Some dogs are pets. ∴ Some pets are not skunks.
This is valid, but it cannot be proven by sentential logic alone. We need to symbolize the content of the premises. The first premise is a universal statement, which we've already learned about, but it is different than the ones seen in the past two lessons. This one is negative. Instead of stating that one category is a subcategory of another, it states that two categories are mutually exclusive. The next premise is an existential premise. It asserts the existence of something, though it does not name the subject who exists. Unlike the first premise, it asserts that two categories intersect. This is the opposite of two categories being mutually exclusive. A statement in the form of the first would contradict a statement in the form of the second if they used the same terms. The conclusion is also an existential statement. Unlike the previous existential statement, it is negative, claiming that members of one category lie outside of another category. Using the same terms, it would contradict a statement of the form "All pets are skunks," the sort of universal statement we already encountered in the past two lessons.
To symbolize these existential statements, we will need a new symbol:
∃ means "there exists"
It can be written in HTML as ∃
With this symbol in hand, we can symbolize our argument.
(∀x)(Dx ⊃ ~Sx) (∃x)(Dx & Px) ∴ (∃x)(Px & ~Sx)
Here is a proof of it.
1. (∀x)(Dx ⊃ ~Sx) 2. (∃x)(Dx & Px) ∴ (∃x)(Px & ~Sx) 3. Da & Pa // 2 Existential Instantiation 4. Da ⊃ ~Sa // 1 Universal Instantiation 5. Da // 3 Simplification 6. ~Sa // 4,5 Modus Ponens 7. Pa // 3 Simplification 8. Pa & ~Sa // 6,7 Conjunction 9. (∃x)(Px & ~Sx) // 8 Existential Generalization
This proof makes use of two new rules. In line 3, Existential Instantiation lets us go from an existential statement to a particular statement. In line 9, Existential Generalization lets us go from a particular statement to an existential statement. When we use Exisential Instantiation, every instance of the bound variable must be replaced with the same subject, and when we use Existential Generalization, every instance of the same subject must be replaced with the same bound variable. The bound variable is the x you see with the ∃ symbol. It doesn't have to be an x, but in this example, it is.
Notice that Existential Instantiation was done before Universal Instantiation. This is because of a restriction on Existential Instantiation. Whenever it is used, the bound variable must be replaced with a new name that has not previously appeared in any premise or in the conclusion. This is because an existential statement doesn't tell us which individuals it asserts the existence of, and if we use the name of a known individual, there is always a chance that the use of Existential Instantiation to that individual would be mistaken. Whenever we use Existential Instantiation, we must instantiate to an arbitrary name that merely represents one of the unknown individuals the existential statement asserts the existence of. Because of this restriction, we could not instantiate to the same name as we had already used in a previous Universal Instantiation. If we are to use the same name for both, we must do Existential Instantiation first. To better illustrate the dangers of using Existential Instantiation without this restriction, here is an example of a very bad argument that does so.
Holly is a cat. Dogs exist. ∴ Something is both a dog and a cat. 1. Ch 2. (∃x)(Dx) ∴ (∃x)(Dx & Cx) 3. Dh // 2 Existential Instantiation? 4. Dh & Ch // 1,3 Conjunction 5. (∃x)(Dx & Cx) // 4 Existential Generalization
In this argument, the Existential Instantiation at line 3 is wrong. Since Holly is a known individual, we could be mistaken in inferring from line 2 that she is a dog. Since line 1 tells us that she is a cat, line 3 is obviously mistaken. But even if we used categories that are not exclusive, such as cat and pet, this would still be invalid. And, obviously, it doesn't follow from dogs exist that just anything is a dog. You're not a dog, or you wouldn't be reading this.
There is no restriction on Existential Generalization.
This table recaps the four rules we learned in this and the past two lessons:
(∀μ)(Φμ) ∴ Φν
Φν ∴ (∀μ)(Φμ)
The name ν must identify an arbitrary subject, which may be done by introducing it with Universal Instatiation or with an assumption, and it may not be used in the scope of an assumption on a subject within that scope.
(∃μ)(Φμ) ∴ Φν
The name ν must be a new name that has not appeared in any prior premise and has not appeared in the conclusion.
Φν ∴ (∃μ)(Φμ)
Does God Exist?
[…] would be. For further details on the existential quantifier, I’ll refer you to my post Introducing Existential Instantiation and Generalization. So, it is not a quality of a thing imagined that it exists or not. Rather, there is simply the […]
I would like to hear your opinion on G_D being The Programmer. A person’s dna generally being the same was the base class then man and woman inherited person dna and their own customizations of their dna to make their uniquely prepared for the reproductive process such that when the dna generated sperm and dna generated egg of two objects from the same base class meet then a soul is inserted into their being such is the moment of programmatic “instantiation” the “spark of life” of a new person whether man or woman and obviously with deformities there seems to be a random chance factor of low possibility of deformity of one being born with both woman and male genitalia at birth as are other random change built into the dna characteristics indicating possible disease or malady being linked to common dna properties among mother and daughter and father and son like testicular or breast cancer, obesity, baldness or hair thinning, diabetes, obesity, heart conditions, asthma, skin or ear nose and throat allergies, skin acne, etcetera all being pre-programmed random events that G_D does not control per se but allowed to exist in G_D’s PROGRAMMED REAL FOR US VIRTUAL FOR G_D REALITY WE ALL LIVE IN just as the virtual game environment seems real to the players but behind the scenes technically is much more real and machine like just as the iron in our human body’s blood stream like a magnet in an electrical generator spins and likely just as two electronic wireless devices communicate their are likely remote communications both uploads and downloads when each, human body, sleeps. This possibly could be truly controlled through literal “STRINGS” in the human heart as these vibrations could easily be used to emulate frequencies and if readable by technology we don’t have could the transmitter and possibly even the receiver also if we only understood more about what is occurring beyond what we can currently see and measure despite our best advances there are certain spiritual realms and advances that are beyond our understanding but are clearly there in real life as we all worldwide wherever I have gone and I rose from E-1 to become a naval officer so I have traveled the world more than most but less than ya’ know, wealthy folks, hmmm but I AM GOOD an honest and I realize the more I come to know the less and less I really understand and that it is very important to look at the basics of every technology to understand the beauty of G_D’s simplicity making it possible for us to come to learn, discover and understand how to use G_D’s magnificent universe to best help all of G_D’s children.
Just some thoughts as a software engineer I have as a seeker of TRUTH and lover of G_D like I love and protect a precious infant and women.
ENTERTAIN NO DOUBT.
WE ARE MANY.
WE ARE GOOD.
WE ARE CQMING.
G_D IS WITH US AND GOOD IS COMING.
Love to hear thoughts specifically on G_D and INSTANTIATION of us as new human “objects” in an “OBJECT ORIENTED WORLD” G_D programmed and the relation of INSTANTIATION being the SPARK OF LIFE process of reproducing and making a new “man” or new “woman” object allocating new memory for the new object in the “universal computer of time and space G_D programmed in G_D’s allocated memory space”.
“Make a WAY”
Evolution is an algorithmic process that doesn’t require a programmer, and our apparent design is haphazard enough that it doesn’t seem to be the work of an intelligent creator. See my previous posts The Algorithm of Natural Selection and Flaws in Paley’s Teleological Argument.