Is rational logic a proper tool of philosophy? Why? How?

[wmw]

I’ll first own up to my ulterior motive to stick to the keyboard versions of the symbols, but I’ll let it go.
So your thinking this is enough of a “cheat sheet”: (I’ll use a “. . .” here and there, to spread things out better)

---

~ (not) . . . & (and) . . . | (inclusive or) - only “false | false” is false, other 3 combinations are true
. . .
antecedent ⊃ consequent (conditional, if-then, or implication) - false only when the antecedent is true and the consequent is false (if a.rain then c.wet, yet still true if sprinkler then wet also)
. . .
≡ (Biconditionals, if-and-only-if statement) - true when both atomic sentences connected are true, or both false (if water is ice then water is solid, no liquid ice or solid steam or solid liquid)
. . .
∃ (existential quantifier) - states that something exist. (∃xPx (exists an x that is a Peach that is x))
. . .
∀ (everything, universal qualifier) -∀xMx ‘for all Xs, X is made of matter.’

---

Though I don’t understand why a peach needs to be assigned to x if ‘P’ is already defined. This seems to have come from the Department of Redundancy Department why not just ∃P (an existing Peach)? Is x somehow “understood” as every possibility in the domain, as you said, “for all Xs” in the domain of furniture?

 

[Rhubarb]

I’ll first own up to my ulterior motive to stick to the keyboard versions of the symbols, but I’ll let it go.
So your thinking this is enough of a “cheat sheet”: (I’ll use a “. . .” here and there, to spread things out better)

---

~ (not) . . . & (and) . . . | (inclusive or) - only “false | false” is false, other 3 combinations are true
. . .
antecedent ⊃ consequent (conditional, if-then, or implication) - false only when the antecedent is true and the consequent is false (if a.rain then c.wet, yet still true if sprinkler then wet also)
. . .
≡ (Biconditionals, if-and-only-if statement) - true when both atomic sentences connected are true, or both false (if water is ice then water is solid, no liquid ice or solid steam or solid liquid)
. . .
∃ (existential quantifier) - states that something exist. (∃xPx (exists an x that is a Peach that is x))
. . .
∀ (everything, universal qualifier) -∀xMx ‘for all Xs, X is made of matter.’

---

Though I don’t understand why a peach needs to be assigned to x if ‘P’ is already defined. This seems to have come from the Department of Redundancy Department why not just ∃P (an existing Peach)? Is x somehow “understood” as every possibility in the domain, as you said, “for all Xs” in the domain of furniture?

The biconditional uses the ‘if and only if’ form. So… “Socrates is mortal if and only if Socrates is a human” can be “M≡H”. M is true - Socrates is mortal. H is true - Socrates is a human. So, the compound sentence is true. “Santa Claus is real if and only if I live on Mars” is also true, because S≡R is the symbolization and both S and R are false. “Socrates is mortal if and only if I live on Mars” is false, the first part about Socrates is true, but the part about me is false.

For the part about peaches, you need the full syntax. The quantifiers are used in first-order logic so they have more nuanced rules than the sentential logic I used to describe the biconditional above. For first-order logic, we use predicates and variables. In the sentence “that is a peach”, the “… is a peach” part is the predicate. So, Px is “X is a peach.” So, when we want to talk about quantifying these sentences, we need to specify what variables we’re talking about. So, ∃xPx translates to “Some X exists, and that X is a peach” or “there is something, that something is a peach.” It’s important because we can get to sentences like this: ((∃x Px) & (∀y Py⊃x=y)) This translates to “There exists some x and that x is the Pope AND for all y’s, if y is the Pope, then x is identical to y” which is better translates into English as this: There is one, and only one, Pope. (At any given time, of course. This is the form Russell gives in On Denoting, if it looks familiar, and also used by Quine in his metaphysics) The x’s and y’s let us differentiate between different things in the same sentence or argument. It makes sure we’re applying predicates to the right variable that we want to quantify over.

 

[Rhubarb]

In sentential logic, whole sentences are symbolized into a variable. “Santa Claus exists” is just S.

In first-order logic, we’re getting into the ‘inter-sentential relationships’ of the sentence. So, we see that the above sentence contains an existential claim - something exists. X exists. Well, we then have to say what that X is. X exists, and that X is Santa Claus. That’s why the more simple sentence logic symbolize sentences like… A&B is simple and easy to read. But, to say that in first-order logic we are more precise and the symbolized sentences look more complex.

 

[wmw]

You must have seen an error in my biconditional example. Are you too polite to say so directly? It would be best to tell me what is wrong with my “phases of water” example.
I seem to read through Santa Clause and living on Mars as both false so the biconditional is true, but am adrift as to whether my cheat sheet (between the lines) is correct. Or is it that it was only the True side (T, T) not the (T, F) and (F, F) so mine is an incomplete example and you had to find a new example to make a complete one.

Trying to get a explanation through my thick cranium just may be too much when it comes to questions of why P has to be set to a variable like x before you can say it exists. I’ll handwave that as just “something we do” for now.

On x’s. Most of what your saying is not sticking. I’m trying a theory of what you may be saying. But basic structure is still alluding me. What is paired with what first? Let me develop the question so you might have a means to quickly display an answer and in the process I’ll display my guess as to the order of decoding.

On the ordering of where the x’s appear. let me insert parenthesis brackets & braces to help you see an order of how I’m trying to read this read. Taking your example:
∃xPx . . . exists an x that is a Peach that is x
{(∃x)[Px]}. . . {(existing is an x) that is [a Peach that is x]}
So, this is a compound of three full statements (1), [2] & {3}.

It’s really rather bold to stack on more questions on a guess that may be wrong, but I’ll go bold.
Then is it a rule ∃ is never in the predicate-object side and is only on the subject side as in (1) & {3}? or just (1)? then is this the same “subject” rule also applicable for ?

 

[Rhubarb]

You must have seen an error in my biconditional example. Are you too polite to say so directly? It would be best to tell me what is wrong with my “phases of water” example.
I seem to read through Santa Clause and living on Mars as both false so the biconditional is true, but am adrift as to whether my cheat sheet (between the lines) is correct. Or is it that it was only the True side (T, T) not the (T, F) and (F, F) so mine is an incomplete example and you had to find a new example to make a complete one.

Trying to get a explanation through my thick cranium just may be too much when it comes to questions of why P has to be set to a variable like x before you can say it exists. I’ll handwave that as just “something we do” for now.

On x’s. Most of what your saying is not sticking. I’m trying a theory of what you may be saying. But basic structure is still alluding me. What is paired with what first? Let me develop the question so you might have a means to quickly display an answer and in the process I’ll display my guess as to the order of decoding.

On the ordering of where the x’s appear. let me insert parenthesis brackets & braces to help you see an order of how I’m trying to read this read. Taking your example:
∃xPx . . . exists an x that is a Peach that is x
{(∃x)[Px]}. . . {(existing is an x) that is [a Peach that is x]}
So, this is a compound of three full statements (1), [2] & {3}.

It’s really rather bold to stack on more questions on a guess that may be wrong, but I’ll go bold.
Then is it a rule ∃ is never in the predicate-object side and is only on the subject side as in (1) & {3}? or just (1)? then is this the same “subject” rule also applicable for ?

“If water is ice then water is solid” is a conditional, not a biconditonal. The biconditional would be “water is solid if and only if water is ice.” I didn’t understand your example, I’m afraid. But yes. A biconditional is true when the atomic sentences are TT or FF, but not TF or FT. Another trick about biconditionals that I find helpful is that a biconditional is the same as two conditionals but with flopped antecedents. (A≡B) is logically equivalent to ((A⊃B)&(B⊃A)).

As for Px - remember that is the more complicated first order logic - but I’ll try to explain anyway. In a sentence of first order logic, sentences need a subject and a predicate just like in our sentences of English. So for the peach example, P itself just symbolizes the predicate “is a peach.” The X gives that predicate subject. Px, then, symbolizes “X is a peach.” When we put the quantifier out in front, we tie the quantifier to the variable we’re discussing: ∃xPx So that way we know that this thing we’re saying exists is what we’re applying the predicate “is a peach” to. The existential and universal quantifiers are neither subjects nor predicates. They are outside that. They tell us how we’re talking about subject-predicate combination. A lot of the way we write first order logic is based on convention - and some parts of the convention are skipped over by practices logicians. There’s even an ‘order of operations’ of sorts that most hold to so they can avoid writing so many parenthesis and brackets. Maybe this will help. If I add in all the brackets I should have, the sentence would look like this: ((∃x)Px) The first bit says “There exists some thing” and the second bit says “that thing is a peach.” The order is important too. (∃x)(∃y)(Py&Px) is wrong - the x’s should come first.

It should also be noted that you can regiment the sentence about peaches in sentential logic much easier - “A peach exists” would just be P in sentence logic. We aren’t worried about predicates in sentence logic. The sentence becomes the level of evaluation rather than looking within the sentence like they do in first order logic. Generally, you learn sentence logic first - because all those rules apply to first order logic. You just make things more complicated by lengthening the sentence. P becomes ∃xPx - and they both mean the same thing in their respective logical languages. (provided we assign things properly. We we do logic, we assign a key to our symbols so everyone knows what we’re talking about. P could be “a peach exists” or “Peter was the first Pope” or “I like pie”, so long as in our sentence and arguments we use the same variable for the same information consistently. Also, different information can’t have the same variable. For instance, the sentence “Peaches exist, and Peter was the first Pope” couldn’t be symbolized as “P&P”, but maybe “P&F”.

 

[Rhubarb]

This would be so much easier with a chalkboard. xD A lot of this is much clearer with a visual diagram.

 

[wmw]

It all seemed fine. I just had to get used to a predicate then subject order.

Then you put down this, “(∃x)(∃y)(Py&Px) is wrong - the x’s should come first”.

And I just go, "why, why, and what? just the x’s and not the y’s? you must have meant the variables (x’s&y’s).

yP = “Peach is y” and xP = “Peach is x” but the ∃x stays in this order? (=“x exists”).

 

[Rhubarb]

It all seemed fine. I just had to get used to a predicate then subject order.

Then you put down this, “(∃x)(∃y)(Py&Px) is wrong - the x’s should come first”.

And I just go, "why, why, and what? just the x’s and not the y’s? you must have meant the variables (x’s&y’s).

yP = “Peach is y” and xP = “Peach is x” but the ∃x stays in this order? (=“x exists”).

Oh, I’m sorry. Yeah, I meant that the variables in the sentence proper needs to align with the variables in the quantifiers. So, if you say (∃x∃z∃y) at the start, then the variables in the sentence should show x first, then z, then y.

And yeah, for some reason they want the predicate before the object - it relates to set theory, as I recall. Objects are in the extension of the set of all things that have the predicate. So, you write Px instead of xP. It’s just convention.

 

[wmw]

Then is this our cheat sheet?

---

~ (not) . . . & (and) . . . | (inclusive or) - only “false | false” is false, other 3 combinations are true
. . .
antecedent ⊃ consequent (conditional, if-then, or implication) - false only when the antecedent is true and the consequent is false (if a.rain then c.wet, yet still true if sprinkler then wet also)
. . .
≡ (Biconditionals, if-and-only-if statement) - true when both atomic sentences connected are true, or both false (Socrates is mortal if and only if Socrates is a human" can be “M≡H”. M is true - Socrates is mortal. H is true - Socrates is a human. So, the compound sentence is true. “Santa Claus is real if and only if I live on Mars” is also true, because S≡R is the symbolization and both S and R are false. “Socrates is mortal if and only if I live on Mars” is false, the first part about Socrates is true, but the part about me is false)
. . .
∃ (existential quantifier) - states that something exist. (∃xPx (exists an x that is a Peach that is x))
. . .
∀ (everything, universal qualifier) -∀xMx ‘for all Xs, X is made of matter.’

---

Then using these constructors lets see how to represent a rather complex but manageable sentence. Yet, at the same time, a something of a complete argument. First let’s see what it might look like in symbols then reconvert it back to see if this “game of telephone” works.

Let me present the starting point:

Among the known living things man is the only creature that can make rational choices; thus, employ free will.

In order not to demand you do all the work I’ll make an attempt:

C=“known living things” = “creatures”
H=“man” (or Humanity)
R=“rational choices”
F=“free will”
variables - s, t

∃sCs∃tHt
∀(s&~t)~R(s&~t)
(Ht&Rt)⊃Ft

I’ll be betting heavily against this being even close, but one or two things may look like the real thing. Yet, now the interlinear re-translation back to English:

There exists s, s are creatures. There exists t, t are Human creatures.

Everything that is a (creature and not Human) is a (creature and not Human) that is not Rational.

if a t is a Human and t is Rational then t has Free Will.

 

[Rhubarb]

Then is this our cheat sheet?

---

~ (not) . . . & (and) . . . | (inclusive or) - only “false | false” is false, other 3 combinations are true
. . .
antecedent ⊃ consequent (conditional, if-then, or implication) - false only when the antecedent is true and the consequent is false (if a.rain then c.wet, yet still true if sprinkler then wet also)
. . .
≡ (Biconditionals, if-and-only-if statement) - true when both atomic sentences connected are true, or both false (Socrates is mortal if and only if Socrates is a human" can be “M≡H”. M is true - Socrates is mortal. H is true - Socrates is a human. So, the compound sentence is true. “Santa Claus is real if and only if I live on Mars” is also true, because S≡R is the symbolization and both S and R are false. “Socrates is mortal if and only if I live on Mars” is false, the first part about Socrates is true, but the part about me is false)
. . .
∃ (existential quantifier) - states that something exist. (∃xPx (exists an x that is a Peach that is x))
. . .
∀ (everything, universal qualifier) -∀xMx ‘for all Xs, X is made of matter.’

---

I’ll lay out the connectives in a sort of table - I’ll show you the combinations of truth values for atomic sentences and what truth value for the compound sentence they yield. I’ll use negation as my example. (These are the basic “truth tables” for the connectives. A google search for truth table will tell you more)

Negation:

~T|F This says that a sentence that is true, when negated, is false
~F|T and when a false sentence is negated, it returns true.

Conjunction:

T&T| F
T&F| F
F&T| F
F&F| F

Disjunction:

TvT| T
TvF| T
FvT| T
FvF| F

Conditional:

T⊃T| F
T⊃F| F
F⊃T| F
F⊃F| F

Biconditional:

T≡T| T
T≡F| F
F≡T| F
F≡F| T

The truth and falsity of your quantifiers depend on your domain - there is no cheat sheet.
For instance, if your domain is “all whole numbers”, the sentence “There exists some X, and that X is negative” is false. If the domain was “all numbers” then it would be true.

 

[Rhubarb]

Then using these constructors lets see how to represent a rather complex but manageable sentence. Yet, at the same time, a something of a complete argument. First let’s see what it might look like in symbols then reconvert it back to see if this “game of telephone” works.

Let me present the starting point:

Among the known living things man is the only creature that can make rational choices; thus, employ free will.

In order not to demand you do all the work I’ll make an attempt:

C=“known living things” = “creatures”
H=“man” (or Humanity)
R=“rational choices”
F=“free will”
variables - s, t

∃sCs∃tHt
∀(s&~t)~R(s&~t)
(Ht&Rt)⊃Ft

I’ll be betting heavily against this being even close, but one or two things may look like the real thing. Yet, now the interlinear re-translation back to English:

There exists s, s are creatures. There exists t, t are Human creatures.

Everything that is a (creature and not Human) is a (creature and not Human) that is not Rational.

if a t is a Human and t is Rational then t has Free Will.

Firstly, as a matter of convention, variables in first-order logic are x-z. A-w are reserved for names. For example, if we define r=Rhubarb and P=is a philosopher, the sentence “Pr” translates to “Rhubarb is a philosopher.” Now I’ll try translating the sentence you put forth.

(1) Among the known living things man is the only creature that can make rational choices; thus, employ free will.

In simple sentence logic:

A= Among the known living things, man is the only creature that can make rational choices.
F=Man employs free will.

The above will be our key. Now, as written we can do this a lot of ways. We can just say A&F. Or… as an (invalid) argument:

(1) A​

(c)F

This doesn’t work so well because the sentence is more complex than sentence logic really can handle. For sentence logic we want to break apart compound sentences at their connectives so we can identify which sentences are atomic. Then we can use sentential logic to analyze them. We can’t really do that with the sentence you provided. Now, if I were to do it in first-order logic I’d do something like this.

(1) Among the known living things man is the only creature that can make rational choices; thus, employ free will.

∀x((Mx&Rx)⊃Fx)

Domain: Known living things. (we can say this is identical to ‘creatures.’ If we don’t want to say that known living things are identical to creatures, we would need to include a predicate ‘is a creature’ to be as specific as possible.)
Mx=x is a man
Rx=x makes rational choices
Fx=x employs free will.

This would translate to “For all living things (because of our domain), if it is a man and makes rational choices, then it employs free will” It needs to be in the form of a universal quantifier because you are making a general claim about all things that are men and makes rational choices. I think this simple sentence is the most elegant way you can express this information. The translation of logic to a natural language, or vice versa isn’t precise. The connectives don’t exactly capture the way we use the words we assign them. A philosopher named Paul Grice noticed this - and wrote a pretty smart way to deal with it.

As some side-notes, the above would be falsified if there are things that are human and also can make rational choices, but cannot employ free will. (Falsified by counter-example) Also, the above sentence, even if true, does not actually attest to the existence of anything. It could be true even if no humans exist, rational choices don’t exist, for if both Humans and rational choices AND the employing of free will doesn’t exist.

 

[Rhubarb]

For your purposes you’d probably want another sentence that says “For all x’s, if x is not a man and capable of making rational choices, then it does not employ free will” which is written the same way, but with negations in front of the conjunction and the antecedent of the conditional.

∀x(~(Mx&Rx)⊃~Fx)

 

[grannymh]

Then is this our cheat sheet?

---

~ (not) . . . & (and) . . . | (inclusive or) - only “false | false” is false, other 3 combinations are true
. . .
antecedent ⊃ consequent (conditional, if-then, or implication) - false only when the antecedent is true and the consequent is false (if a.rain then c.wet, yet still true if sprinkler then wet also)
. . .
≡ (Biconditionals, if-and-only-if statement) - true when both atomic sentences connected are true, or both false (Socrates is mortal if and only if Socrates is a human" can be “M≡H”. M is true - Socrates is mortal. H is true - Socrates is a human. So, the compound sentence is true. “Santa Claus is real if and only if I live on Mars” is also true, because S≡R is the symbolization and both S and R are false. “Socrates is mortal if and only if I live on Mars” is false, the first part about Socrates is true, but the part about me is false)
. . .
∃ (existential quantifier) - states that something exist. (∃xPx (exists an x that is a Peach that is x))
. . .
∀ (everything, universal qualifier) -∀xMx ‘for all Xs, X is made of matter.’

---

Then using these constructors lets see how to represent a rather complex but manageable sentence. Yet, at the same time, a something of a complete argument. First let’s see what it might look like in symbols then reconvert it back to see if this “game of telephone” works.

Let me present the starting point:

Among the known living things man is the only creature that can make rational choices; thus, employ free will.

In order not to demand you do all the work I’ll make an attempt:

C=“known living things” = “creatures”
H=“man” (or Humanity)
R=“rational choices”
F=“free will”
variables - s, t

∃sCs∃tHt
∀(s&~t)~R(s&~t)
(Ht&Rt)⊃Ft

I’ll be betting heavily against this being even close, but one or two things may look like the real thing. Yet, now the interlinear re-translation back to English:

There exists s, s are creatures. There exists t, t are Human creatures.

Everything that is a (creature and not Human) is a (creature and not Human) that is not Rational.

if a t is a Human and t is Rational then t has Free Will.

When I first landed on CAF, I considered Free Will as more important than intellective reasoning when it comes to the “scientific” concept that humans are different in degrees, not in kind, from other species. That idea was dead in the water. Currently, I consider Free Will as the base for a number of Catholic doctrines regarding the origin of human nature.

At the moment, I will continue learning what wmw and Rhubarb are proposing and how that can be logically understood by a granny.

 

[grannymh]

When I first landed on CAF, I considered Free Will as more important than intellective reasoning when it comes to the “scientific” concept that humans are different in degrees, not in kind, from other species. That idea was dead in the water. Currently, I consider Free Will as the base for a number of Catholic doctrines regarding the origin of human nature.

At the moment, I will continue learning what wmw and Rhubarb are proposing and how that can be logically understood by a granny.

One additional note: Human Free Will is more than choosing this or that or nothing, fight, flight or freeze. Human will can choose the super-natural.

 

[Rhubarb]

One additional note: Human Free Will is more than choosing this or that or nothing, fight, flight or freeze. Human will can choose the super-natural.

A description, and theory, of free will is elusive. There are many theories about how it works. If it works at all.

 

[grannymh]

A description, and theory, of free will is elusive. There are many theories about how it works. If it works at all.

I agree just from reading posts on CAF.

I have this granny type theory which works with grannykids. I simply say that this is my game and you play by my rules. One of the best rules is when a grannykid wants a treat, we go get one. Another rule is if a door is unlocked, we can look inside. If there is water we can put our hands in it.

My point is that it is important to choose a theory, description, guidelines, whatever when the game of logic begins. I respect the worldviews of others. So I ask if they would use a “willing suspension of disbelief.” Since I learn a lot from others, I reserve the right to change the “rules.” We need everyone’s creative minds.

What is your choice for a theory of free will?

 

[Aloysium]

A description, and theory, of free will is elusive. There are many theories about how it works. If it works at all.

And, this goes to show that all logic, all the various arguments are only as good as the capacity of their assumptions to approximate the truth.
Truth can be revealed or realized or intuited. I suppose it can be arrived at logically, if one is in possession of an equally valid truth.
That free will exists is obvious, being reflected in our finite ability to self-create.
That said, if someone assumes all creation is founded on randomness and determination, that is what they will come up with at the end of their long analysis.

 

[Rhubarb]

What is your choice for a theory of free will?

I find what they call a ‘soft deterministic’ theory most plausible. Soft determinism admits that the world around us follows a deterministic model - the state of affairs at T2 depends upon the state of affairs at T1+laws of nature, in the scientific sense. Basically, the world is a giant Rube Goldberg machine. A bird can peck a branch and drops an acorn which hits a bicyclist which distracts him and makes him fall down. Everything that happens depends on what came before plus physics and biology and chemistry and etc. However, soft deterministic theories also allow for free will. It says that just because the world is deterministic, that doesn’t mean that there isn’t room for free will. We all seem to make choices based on our reflections and rational thought.

I find these theories attractive because they allow for some choices we make to NOT be free. Not everything we do is due to our free will. We can’t will our heart to stop beating in the same manner I can will my eyelids to close. We can be coerced, which is typically considered to be an infringement of free will. We need to eat, breath and drink or we’ll die - we can’t really will ourselves to stop, or to live off sunlight like a plant. So, clearly our actions aren’t always ‘free.’

There are many soft deterministic theories - I don’t have one particular I endorse fully. But it seems like these sorts are the best bet to capture what happens in regard to our will. It is also is generally acceptable to Christians, who’s theology depends on the freedom of the will.

 

[Rhubarb]

And, this goes to show that all logic, all the various arguments are only as good as the capacity of their assumptions to approximate the truth.
Truth can be revealed or realized or intuited. I suppose it can be arrived at logically, if one is in possession of an equally valid truth.
That free will exists is obvious, being reflected in our finite ability to self-create.
That said, if someone assumes all creation is founded on randomness and determination, that is what they will come up with at the end of their long analysis.

Fair enough. There are plenty of valid arguments that aren’t sound. Justifying the truth of our claims plays a big part. When we’re doing logic, we can’t really find THE truth. We find ‘truth’ given stipulated rules: Given A, B, C and D, we know that E MUST be true.

Though it is contentious to say it’s obvious free will exists. Lots of things have a finite ability to self-create (If I understand what you mean by self-create) and aren’t considered to have free will in the same sense that we humans do. And there are some really good arguments for the non-existence of free will. Furthermore, it seems most plausible to me that all creation is founded on randomness and determination - yet I do think that humans (at least) have free will.

 

[Aloysium]

. . . Though it is contentious to say it’s obvious free will exists. Lots of things have a finite ability to self-create (If I understand what you mean by self-create) and aren’t considered to have free will in the same sense that we humans do. And there are some really good arguments for the non-existence of free will. Furthermore, it seems most plausible to me that all creation is founded on randomness and determination - yet I do think that humans (at least) have free will.

What possibly has led you to believe things are random?