Idealism and materialism are not proper categories, for most of the theories attempt to combine them. In order to have categories that the different theories don't try to combine, I will categorize them according to the truth values they assign to the functions in T1(2).
Since no universal statement, as such, is required to must maintain the same truth value forever, there are 2^4-1, or 15, ways of regarding it. I will now consider all of these, eliminating uninteresting ones, and grouping some into broader categories, beginning with those that maintain the same truth value for all time. Since there is only one way for it to be false, and three ways for it to be true, there are (2^1 + 2^3) - 2, or 8, ways that maintain the same truth value forever. Four of these always maintain the same truth value in the same way. These are as follows:
Mentalism: (/\x) (Mx & ~Px)
Panpsychism: (/\x) (Mx & Px)
Physicalism: (/\x) (~Mx & Px)
Neutral Monism: (/\x) (~Mx & ~Px)
Each of these theories claims that all the stuff of the universe is one way and no other. Mentalism claims that the universe is made of non-physical mind, Panpsychism of physical mind, Physicalism of non-mental matter, and Neutral Monism of neutral stuff. The other ten theories claim that the universe is made of more than one kind of stuff. The other four that always maintain the value of true for T1(2) claim different combinations of mind, matter, and neutral stuff. Since Neutral Monism's neutral stuff is supposed to replace mind and matter, the combinations with it may be dismissed as superfluous. The remaining combination, mind and matter, is known as Mind/Body Dualism.
We now have seven combinations left to examine. All of these include physical mind. Since any theory that has both physical mind and neutral stuff is superfluous, we can dismiss all four remaining combinations with neutral stuff. We have two combinations left with pure mind. Both of these are variants on Emanationism, the world's most ancient solution to the mind/body problem. According to Emanationism, mind is more fundamental than matter, and the physical world is an emanation from mind. Our remaining combination, physical mind and non-mental matter, is a weak version of Supervenience. According to Supervenience, all minds are physical, but there may also be physical things that are not minds. Weak Supervenience is the opposite of Emanationism, for it regards the physical as primary, and it considers mind to be an outgrowth from the physical. Since Supervenience is the claim that all minds are physical, Panpsychism may be subsumed under it.
Thus, we have six categories of solutions to the mind/body problem. These categories are Substance Dualism (D), Mentalism (M), Physicalism (P), Neutral Monism (N), Emanationism (E), and Supervenience (S). They may be distinguished from each other as follows:
D - (/\x) (Mx -> ~Px) & [(\/x) (Px) & (\/x) (Mx)]
M - (/\x) (Mx & ~Px)
P - (/\x) (~Mx & Px)
N - (/\x) ~(Mx v Px)
E - (\/x) (Mx & ~Px) & (\/x) (Mx & Px)
S - (/\x) (Mx -> Px) & (\/x) (Mx)
By examining these propositions, you may note that Dualism splits the universe in two. It divides it into a physical half and a mental half. Mentalism and Physicalism retain this split and choose one half while rejecting the other. Neutral Monism agrees that mind and matter are conceptually irreconcilable. So it rejects both halves and claims that the same neutral stuff makes up both mental events and physical events. Mental and physical events are separate for Neutral Monism, and they differ according to how they structure the neutral stuff. Finally, Supervenience and Emanationism both deny that the universe can be split in two. They differ on which is more fundamental, mind or matter. Let's now review how the theories in these various categories answer the mind/body problem as posed by the two tetrads.
In terms of the two tetrads, these theories differ as follows. First, they part ways over whether an agent is a body or a mind. ID and OD claim that agents are minds, but PD and ED claim that agents are bodies. Since the claim that bodies are not agents is the denial of T2(3), ID and OD escape from T2 by contradicting T2(3) with (/\x) (Bx -> ~Ax). This is the contradiction of T2(3), because T2(3), (\/x) (Bx & Ax), is an I statement, and (/\x) (Bx -> ~Ax) is the corresponding E statement.
Since PD and ED affirm T2(3), each must deny another proposition of T2. Since PD can be true only if there is a pre-established harmony between the mental world and the physical world, both must be determined from the start. This means that determinism must be true if PD is true. Thus, PD escapes T2 by denying T2(2) with (\/x) (Ax) & (/\x) (~Sx). These together give us (\/x) (Ax & ~Sx). This contradicts T2(2), because T2(2), (/\x) (Ax -> Sx) is an A statement, and it is the corresponding O statement. ED escapes T2 in the same way, for it also implies determinism. This is because it considers the will to be mental, and the mental to be nothing more than an epiphenomenon. Therefore, it does not regard the will as having any effect on the body. So, it denies free will and affirms determinism.
The different forms of dualism differ also according to how they answer T1. ID denies T1(4) by asserting (\/x) (\/y) [(Px & ~Py) & (Ixy & Iyx)], that material and immaterial substances can interact. This contradicts T1(4) because T1(4), (/\x) (/\y) [(Px & ~Py) -> ~(Ixy & Iyx)], is an E statement, and (\/x) (\/y) [(Px & ~Py) & (Ixy & Iyx)] is the corresponding I statement. OD rejects the same proposition in the same way that ID does. Both claim that non-physical minds interact with physical bodies, and differ merely in the details of the interaction. OD claims that it is through the medium of God, and ID claims that it is direct.
Since PD denies that the mind and body have any influence on each other, it denies T1(3) by asserting (/\x) (/\x) [(Bx & My) -> (~Ixy & ~Iyx)]. This implies its contradiction as follows:
1. (/\xy) [(Bx & My) -> (~Ixy & ~Iyx)]
2. (/\y) [(Ba & My) & (~Iay & Iya)] U.I. 1
3. (Ba & Mb) -> (~Iab & ~Iba) U.I. 2
4. Ba & Mb assumption
5. ~Iab & ~Iba M.P. 3,4
6. ~Iab Simp. 5
7. ~Iab v ~Iba Add. 6
8. ~(Iab & Iba) De.M. 7
9. (Ba & Mb) -> ~(Iab & Iba) C.P. 4-8
10. (/\y) [(Ba & My) -> ~(Iab & Iba)] U.G. 9
11. (/\xy) [(Bx & My) -> ~(Iab & Iba)] U.G. 10
12. ~{(\/xy) [(Bx & My) & (Iab & Iba)]} Q.N. 11
Since ED claims that mind has no affect on the body, it denies the same proposition with (/\x) (/\y) [(Bx & My) -> ~Iyx]. A similar proof can be constructed to show that this statement also contradicts T1(3).
All theories in M deny T1(1). M claims to deny the existence of matter, for it holds that nothing exists on its own except perceivers. So it denies (\/x) (Px). It does, however, hold that bodies exist. Therefore, it asserts (\/x) (Bx & ~Px), which is the contradiction of T1(1). In claiming that there are non-physical bodies, however, it does not mean to claim that our minds are housed in non-physical bodies. What it means is that our bodies exist, but as nothing more than our perceptions of them, not as physical things that exist separately from minds, and that our minds are not housed in them at all.
All theories in M also deny T2(3), that anything is both an agent and a body. For M, the body is nothing more than the perceptions we have of it, and not the same thing as the mental agent that perceives it. Although M does not consider the body to be physical, M still claims that the mind and body interact. This is because the mind reacts to its perceptions of the body, and the perceptions of the body change when we will the body to do something.
OM and PM disagree on the truth value of T2(1), which claims that bodies are subject to natural law. For OM, God has created natural law, and our sensations, including our sensations of our bodies, conform to God's natural law. For PM, the world is nothing more than what all the minds collectively think it is. If a mind is strong enough, it can change the world. The world of PM is not one of natural law, but one of magic.
Since there is no lawgiver for SM, it would seem that its world is also one of magic. Nevertheless, it is held by some physicists (e.g. Bernard d'Espagnat & N. David Mermin) who would rather give up an external world than natural law. If there is any sense in which natural law can exist separately from both a divine lawgiver and an external world, then SM's world is one of science, not magic.
So, all of these theories deny that minds exist, which means that they deny T1(3) with (/\x) (~Mx). Since T1(3) claims that minds do exist, it can easily be seen how P implies its contradiction.
Although they all agree that minds don't exist, they differ on whether agents exist. FP and EP hold that agents exist, but BP denies the existence of agents. Thus, BP escapes T2 by denying T2(3) with (/\x) (~Ax). According to BP, a person is nothing more than a black box that responds to stimuli. For BP, there is nothing in this black box that makes decisions and directs actions. It just reacts to its environment like a billiard ball reacts to its environment. Both mind and agency are aspects of what B. F. Skinner calls Autonomous Man in Beyond Freedom and Dignity. For Skinner, Autonomous Man is a myth; people are completely controlled by their environment.
Although EP and FP both agree that agents exist, they disagree on whether an agent is identical with its body. For EP, an agent is its body and nothing more. The body's behavior is determined only by neural events, not by beliefs, or desires, or values. EP holds that all mental language is false and misleading. It seeks to establish a new psychological language that is rooted in neurophysiological concepts rather than mental ones. EP escapes T2 by denying that agents are self-determining. It asserts (/\x) (Ax -> ~Sx) & (/\x) (Ax), which together imply the contradiction of T2(2). It considers self-determination to be a myth along with other mental concepts.
For FP, an agent is not identical with its body. Another name for FP is Agent Dualism. According to FP, an agent is a formal system that can be considered separately from the body that it is instantiated in. The FP agent is analogous to a computer program. Just as the same computer program can work in different computers, so the same FP agent can exist in different bodies. An agent is defined entirely by the functional roles it plays, and by nothing else.
FP holds that it is possible to map an agent, and then recreate that map in another body, such as a robot body. Assuming that the mapping was complete, the robot would be that person just as much as the original body was. Also, FP holds that the same body could be different agents at different times. Conceivably, a mind-exchange device could map the "minds" of two people and switch them without switching a single bit of body tissue. Thus, FP seems to escape T2 by denying T2(3) with (/\x) (Ax -> ~Bx). FP also denies T2(2) with (/\x) (Ax -> ~Sx) & (\/x) (Ax), for it holds that an agent is an algorithm, and an algorithm simply produces certain output deterministically.
For CE, pure mind is at the top of the chain, and pure matter, or at least very impure mind, is at the bottom. As beings get further away from God, they are more material and less spiritual. CE is very much like PE, but it does not hold that forms are at the top.
Since both theories hold that some minds are far away enough from God or the Good to be physical, they deny T1(2) with (\/x) (Mx & Px). Since the theories in E are older than the mechanical conception of the universe, they escape T2 by denying that there is natural law. For E, the universe is an organism rather than a machine. Therefore, they deny T2(1) with (/\x) (Bx -> ~Nx) & (\/x) (Bx), which together imply its contradiction.
They both escape T1 by denying T1(4), the claim that the physical and non-physical can't interact. Since they hold that the physical is an emanation from the non-physical, it follows that the non-physical can have power over the physical. The physical and non-physical, for E, are not completely different things as they are for Descartes, but two variations on the same kind of stuff. Because of this commonality, they can interact.
Of all the characteristics that humans have, three are commonly offered as explanations for the mind. These are materiality, life, and the possession of a brain. The theories that adhere to each of these explanations are respectively Panpsychism (PS), Vitalism (VS), and Emergentism (ES). Panpsychism holds that everything is mental. In symbolic logic, it asserts (/\x) (Mx ª Px) & (\/x) (Mx). Vitalism holds that all life is mental. If we allow Lx to mean that x is alive, it asserts (/\x) [(Lx -> Mx) & (Mx -> Px)] & (\/x) (Lx). Strong versions of VS (SVS) assert (/\x) [(Mx -> Px) & (Lx ª Mx)] & (\/x) (Lx). Emergentism asserts that mentality is an emergent property of some living things. In symbolic logic, it asserts (/\x) (Mx -> Px) & (\/x) (Lx & Mx). Strong versions of ES (SES) add (/\x) (Mx -> Lx). Weaker versions of ES allow for the possibility of non-living minds. The common denominator of all these theories is (/\x) (Mx -> Px) & (\/x) (Mx). These together give us (\/x) (Mx & Px), which is the corresponding I statement to the E statement T1(2), (/\x) (Mx -> ~Px).
VS and PS escape T2 by claiming that agents are minds, and that minds are supernatural, which means that they are not subject to natural law. In symbolic logic, they claim (/\x) [(Ax -> Mx) & (Mx -> ~Sx)] & (\/x) (Ax), which together give us (\/x) (Ax & ~Sx), the contradiction of T2(2). For these theories, mentality is not the product of lower, non-mental properties. Rather, it is a basic property itself. PS holds that all matter has mental properties in addition to all the other physical properties. VS holds that living tissue has mental properties in addition to its other physical properties. According to VS, anything without mentality is just a puppet of outside forces. But something with mentality, for both VS and PS, is able to act on its own instead of as the puppet of natural law.
ES escapes T2 by claiming that some things which are subject to natural law are also self-determining. In symbolic logic, this is (\/x) (Nx & Sx). This contradicts the E statement T2(4), (/\x) (Nx -> ~Sx), because it is the corresponding I statement. I will discuss how this could be so later on when I compare the different theories.
PM T1(1) T2(1), T2(3)
OM, SM T1(1) T2(3)
PS, VS T1(2) T2(1)
CE, PE T1(2), T1(4) T2(1)
ES T1(2) T2(4)
ED, PD, EP T1(3) T2(2)
FP T1(3) T2(2), T2(3)
RN, BP T1(3) T2(3)
ID, OD T1(4) T2(3)