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{{Image|Negative feedback amplifier.PNG|right|300px|Ideal negative feedback amplifier.}}
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A '''negative feedback amplifier''' (or more commonly simply a '''feedback amplifier''') is an amplifier in which a fraction of its output is combined with the signal at its input that opposes the signal in what is called [[negative feedback]]. The negative feedback improves performance (gain stability, linearity, frequency response, [[step response]]) and reduces sensitivity to parameter variations due to manufacturing or environmental uncertainties. A single feedback loop with unilateral blocks is shown in the figure. Negative feedback is used in this way in many amplifiers and control systems.<ref name=Kuo/>
In [[philosophy]] the term '''free will''' refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so.  It is an age-old concern to separate what we can do something about, choose to do, from what we cannot.  The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of ''determinism'', sometimes called ''physical reductionism'',<ref name=Ney/>  and the view that determinism precludes free will is called ''incompatibilism''.<ref name=Vihvelin/>


== Advantages and disadvantages of feedback ==
There are several ways to avoid the incompatibilst position, resulting in various ''compatibilist'' positions.<ref name=Timpe/> One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.  
Many electronic devices used to provide gain (for example, [[vacuum tube]]s, [[Bipolar transistor]]s, [[MOSFET]]s) are [[nonlinear]]Negative feedback is a circuit technique that trades gain for higher linearity (reducing [[distortion]]), amongst other things. If not designed correctly amplifiers with negative feedback can become unstable, resulting in unwanted behavior, such as [[oscillation]]. The [[Nyquist stability criterion]] developed by [[Harry Nyquist]] of [[Bell Laboratories]], or the [[Bode plot]] can be used to study the stability of feedback amplifiers.


Feedback amplifiers share these properties:<ref name=Palumbo/>
There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?


Pros:
==Science does not apply==
*Can increase or decrease input impedance (depending on type of feedback)
*Can increase or decrease output impedance (depending on type of feedback)
*Reduces distortion (increases linearity)
*Increases bandwidth (the range of frequencies for which the circuit works)
*Desensitizes gain to component variations
*Can control [[step response]] of amplifier


Cons:
One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect.  For example, [[David Hume]] suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion.  [[Immanuel Kant]] suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains.  Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.
*May lead to instability if not designed carefully
*The gain of the amplifier decreases
*The input and output impedances of the amplifier with feedback (the '''closed-loop amplifier''') become sensitive to the gain of the amplifier without feedback (the '''open-loop amplifier'''); that exposes these impedances to variations in the open loop gain, for example, due to parameter variations or due to nonlinearity of the open-loop gain


==History==
A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise.  It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience.  For example, we choose to explain phenomena like the [[Higgs boson]] found by elaborate means like a [[hadron collider]], but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention.
The negative feedback amplifier was invented by [[Harold Stephen Black]] (US patent 2,102,671, filed 1932, issued 1937) <ref name=patent/>) while a passenger on the Lackawanna Ferry (from Hoboken Terminal to Manhattan) on his way to work at [[Bell Laboratories]] on August 2, 1927. Black had been pondering how to reduce distortion in repeater amplifiers used for telephone transmission. On a blank space in his copy of The New York Times, currently on display at Bell Laboratories in Mountainside, New Jersey, he sketched a diagram equivalent to that in the figure, and derived the equations below.<ref name=Waldhauer/>
As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.  


==Classical feedback==
Although not explicitly addressing the issue of free will, it may be noted that [[Ludwig Wittgenstein]] argued that the specialized theories of science, as discussed by [[Rudolf Carnap]] for example, inevitably cover only a limited range of experience.  [[Stephen Hawking|Hawking/Mlodinow]] also noted this fact in in their [[model-dependent realism]],<ref name=Hawking/> the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.
{{quote|“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”<ref name=Davies/>}}
::: —— E.B. Davies <span style="font-size:88%">''Epistemological pluralism'', p. 4</span>


=== Voltage amplifiers ===
Still another approach to this matter is analysis of the mind-brain connection (more generally, the [[mind-body problem]]). As suggested by Northoff,<ref name=Northoff/>  there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks.  But observing our own mental activity is not possible in this way – it is a matter of subjective experiences.  The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:
{{quote|“...for each individual there is ''one'' 'mental life' but ''two'' ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the ''observational arrangements'' (the means by which a subject and an external observer access the subject's mental processes).”<ref name=Velmans/> |Max Velmans: |How could conscious experiences affect brains?,  p. 11''
}}


Below, the gain of the amplifier with feedback, the '''closed-loop gain''' ''A<sub>fb</sub>'', is derived in terms of the gain of the amplifier without feedback, the '''open-loop gain''' ''A<sub>OL</sub>'' and the '''feedback factor''' β, which governs how much of the output signal is applied to the input. See the figure, top right. The feedback  parameter β is determined by the feedback network that is connected around the amplifier.
A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by [[Niels Bohr]] and by [[Erwin Schrödinger]] in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)
{{quote|“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”<ref name=Pockett/> |Susan Pockett |The neuroscience of movement}}


Consider a voltage amplifier with voltage feedback.  Without feedback, the output voltage ''V<sub>out</sub> = A<sub>OL</sub>'' ''V<sub>in</sub>'', where the open-loop gain ''A<sub>OL</sub>'' in general may be a function of both frequency and voltage.


The open-loop gain ''A<sub>OL</sub>'' is defined by:
In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.


:<math>A_{OL} = \frac{V_{out}}{V_{in}} \ ,</math>
==Science can be accommodated==
A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do.  Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.


where ''V<sub>in</sub>'' is the input to the amplifier, with no feedback, and ''V<sub>out</sub>'' is the amplifier output, again with no feedback.
==’Will’ ''versus'' ‘action’==
There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,<ref name=Norretranders/> which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, [[addiction|drug addiction]] has been related to alteration of the mechanisms in the brain for [[dopamine]] production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.
{{quote|“Philosophers who distinguish ''freedom of action'' and ''freedom of will'' do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.]<ref name=OConnor/>| Timothy O'Connor |Free Will}}
In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?


Suppose we have a feedback loop so that a fraction β'' V<sub>out</sub>'' of the output is subtracted from the input. The input to the amplifier itself is now ''V’<sub>in</sub>'', where
==Theology==
The ancient Greeks held the view that the gods ''could'' intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:
{{quote| “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”|spoken by Cassius| Julius Caesar (I, ii, 140-141)}}
The [[Stoicism|Stoics]] wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a ''responsibility''), a sequence demanded by our natures as rational beings.<ref name=Bobzien/>  


:<math>V'_{in} = V_{in} - \beta \cdot V_{out}</math>
In [http://www.iep.utm.edu/chrysipp/ Chrysippus of Soli's] view (an apologist for Stoicism), ''fate'' precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.<ref name=Bobzien2/> The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.<ref name=Bobzien3/>


The gain of the amplifier with feedback, called the closed-loop gain, ''A<sub>fb</sub>'' still is given by,
==References==
 
{{reflist|refs=
:<math>A_{fb} = \frac{V_{out}}{V_{in}} \ ,</math>
<ref name=Bobzien>
 
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See §6.3.3 ''The cylinder and cone analogy'',  pp. 258 ''ff''.  
but now the signal increased by the gain of the amplifier is ''V’<sub>in</sub>''. That is,
</ref>
 
:<math>A_{OL}=\frac {V_{out}}{V'_{in}} \ . </math>
 
Rewriting the expression for ''V’<sub>in</sub>'':
 
:<math>\frac{V'_{in}}{V_{out}} = \frac{V_{in}}{V_{out}} - \beta \ ,</math>
 
or:
 
:<math>\frac{1}{A_{OL}} = \frac{1}{A_{fb}} - \beta \ .</math>
 
Solving for ''A<sub>fb</sub>'':
 
:<math>A_{fb} = \frac{A_{OL}}{1 + \beta \cdot A_{OL}} \ .</math>
 
If ''A''<sub>OL</sub> >> 1, then ''A''<sub>fb</sub> ≈ 1 / β and the effective amplification (or closed-loop gain) ''A''<sub>fb</sub> is set by the feedback constant β, and hence set by the feedback network, usually a simple reproducible network, thus making linearizing and stabilizing the amplification characteristics straightforward. Note also that if there are conditions where β ''A''<sub>OL</sub> = −1, the amplifier has infinite amplification – it has become an oscillator, and the system is unstable. The stability characteristics of the gain feedback product β ''A''<sub>OL</sub> are often displayed and investigated on a [[Nyquist plot]] (a polar plot of the gain/phase shift as a parametric function of frequency). A simpler, but less general technique, uses [[Bode_plot#Gain_margin_and_phase_margin|Bode plot]]s.
 
The combination ''L'' = β ''A''<sub>OL</sub> appears commonly in feedback analysis and is called the '''loop gain'''. The combination ( 1 + β ''A''<sub>OL</sub> ) also appears commonly and is variously named as the '''desensitivity factor''' or the '''improvement factor'''.
 
===Bandwidth extension===
{{Image|Bandwidth comparison.PNG|right|300px| Gain ''vs.'' frequency for a single-pole amplifier with and without feedback; corner frequencies are labeled.}}
Feedback can be used to extend the bandwidth of an amplifier (speed it up) at the cost of lowering the amplifier gain.<ref name=Broderson/> The figure at right shows such a comparison. The figure is understood as follows. Without feedback the so-called '''open-loop''' gain in this example has a single time constant frequency response given by
 
::<math> A_{OL}(f) ={A_0}  \frac {1}{ 1+ j \frac{f} {f_C} } \ , </math>
 
where ''A<sub>0</sub>'' is the zero-frequency gain of the amplifier, and ''f<sub>C</sub>'' is the [[cutoff frequency|cutoff]] or [[corner frequency]] of the amplifier. In this example, the gain at zero frequency is ''A<sub>0</sub>'' = 10<sup>5</sup> V/V and the corner frequency is ''f<sub>C</sub>'' = 10<sup>4</sup> Hz. The figure shows the gain is flat out to the corner frequency and then drops. When feedback is present the so-called '''closed-loop''' gain, as shown in the formula of the previous section, becomes,
 
::<math> A_{fb} (f) = \frac { A_{OL} } { 1 + \beta  A_{OL} } </math>
 
::::<math> = \frac { A_0/(1+jf/f_C) } { 1 + \beta A_0/(1+jf/f_C) } </math>
 
::::<math> = \frac {A_0} {1+ jf/f_C + \beta A_0} </math>
 
::::<math> = \frac {A_0} {1 + \beta A_0}\  \frac{1} {1+j \left(\frac {f} {(1+ \beta A_0) f_C }\right) }   
\ .
</math>
 
The last expression shows the feedback amplifier still has a single time constant behavior, but the corner frequency is now increased from the value ''f<sub>C</sub>'' by the improvement factor {{nowrap|(1+β ''A<sub>0</sub>'')}} to become {{nowrap|(1+β ''A<sub>0</sub>'') ''f<sub>C</sub>'',}} and the gain at zero frequency has dropped by exactly the same factor. This reciprocal behavior is called the '''gain-bandwidth tradeoff'''. In Figure 2,  (1 + β ''A<sub>0</sub>'') = 10<sup>3</sup>, so ''A<sub>fb</sub>''(0)= 10<sup>5</sup> / 10<sup>3</sup> = 100 V/V, and ''f<sub>C</sub>'' increases to 10<sup>4</sup> × 10<sup>3</sup> = 10<sup>7</sup> Hz.
 
===Multiple poles===
When the open-loop gain has several poles, rather than the single pole of the above example, feedback can result in complex poles (real and imaginary parts). In a two-pole case, the result is peaking in the frequency response of the feedback amplifier near its corner frequency, and ringing and overshoot in its its [[step response]]. In the case of more than two poles, the feedback amplifier can become unstable, and oscillate. See the discussion of [[Bode_plot#Gain_margin_and_phase_margin|gain margin and phase margin]].<ref name=Sansen/>
 
==Asymptotic gain model==
 
In the above analysis the feedback network is [[Electronic amplifier#Unilateral or bilateral|unilateral]]. However, real feedback networks often exhibit '''feed forward''' as well, that is, they feed a small portion of the input to the output, degrading performance of the feedback amplifier. A more general way to model negative feedback amplifiers including this effect is with the [[asymptotic gain model]].
 
==Feedback and amplifier type==
Amplifiers use current or voltage as input and output, so four types of amplifier are possible, choosing one of the two inputs and pairing with one of the two outputs. Any of these four choices may be the open-loop amplifier used to construct the feedback amplifier.  The objective for the feedback amplifier also may be any one of the four types of amplifier, not necessarily the the same type as the open-loop amplifier. For example, an op amp (voltage amplifier) can be arranged to make a current amplifier instead. The conversion from one type to another is implemented using different feedback connections, usually referred to as series or shunt (parallel) connections.<ref name=Goel/><ref name=Zimmer/> See the table below.
{| class="wikitable" style="background:white;text-align:center "
!Feedback amplifier type
!Input connection
!Output connection
!Ideal feedback 
!Two-port feedback
|-
|-valign="top"
| '''Current'''
| '''Shunt'''
| '''Series'''
| '''CCCS'''
| '''g-parameter'''
 
|-
|-valign="top"
| '''Transresistance'''
| '''Shunt'''
| '''Shunt'''
|  '''VCCS'''
|  '''y-parameter'''
 
|-
|-valign="top"
| '''Transconductance'''
| '''Series'''
| '''Series
| '''CCVS'''
| '''z-parameter'''
 
|-
|-valign="top"
| '''Voltage'''
| '''Series'''
| '''Shunt'''
| '''VCVS'''
| '''h-parameter'''
 
|}
The feedback can be implemented using a [[two-port network]]. There are four types of two-port network, and the selection depends upon the type of feedback. For example, for a current feedback amplifier, current at the output is sampled and combined with current at the input. Therefore, the feedback ideally is performed using an (output) current-controlled current source (CCCS), and its imperfect realization using a two-port network also must incorporate a CCCS, that is, the appropriate choice for feedback network is a g-parameter two-port.
 
==Two-port analysis of feedback==
One approach to feedback is the use of [[return ratio]]. Here an alternative method used in most textbooks is presented by means of an example treated in the article on [[Asymptotic_gain_model#Two-stage_transistor_amplifier|asymptotic gain model]].<ref name=Subramanian/><ref name=Gray-Meyer1/><ref name=Sedra1/>
[[Image:Two-transistor feedback amp.PNG|thumbnail|250px|Figure 3: A ''shunt-series'' feedback amplifier]]
Figure 3 shows a two-transistor amplifier with a feedback resistor ''R<sub>f</sub>''. The aim is to analyze this circuit to find three items: the gain, the output impedance looking into the amplifier from the load, and the input impedance looking into the amplifier from the source.
 
===Replacement of the feedback network with a two-port===
The first step is replacement of the feedback network by a two port. Just what components go into the two port?
 
On the input side of the two-port we have ''R<sub>f</sub>''. If the voltage at the right side of ''R<sub>f</sub>'' changes, it changes the current in ''R<sub>f</sub>'' that is subtracted from the current entering the base of the input transistor. That is, the input side of the two-port is a dependent current source controlled by the voltage at the top of resistor ''R<sub>2</sub>''.
 
One might say the second stage of the amplifier is just a [[voltage follower]], transmitting the voltage at the collector of the input transistor to the top of ''R<sub>2</sub>''. That is, the monitored output signal is really the voltage at the collector of the input transistor. That view is legitimate, but then the voltage follower stage becomes part of the feedback network. That makes analysis of feedback more complicated.
[[Image:G-equivalent circuit.PNG|thumbnail|250px|Figure 4: The g-parameter feedback network]]
An alternative view is that the voltage at the top of ''R<sub>2</sub>'' is set by the emitter current of the output transistor. That view leads to an entirely passive feedback network made up of ''R<sub>2</sub>'' and ''R<sub>f</sub>''. The variable controlling the feedback is the emitter current, so the feedback is a current-controlled current source (CCCS). We search through the four available [[two-port network]]s and find the only one with a CCCS is the g-parameter two-port, shown in Figure 4. The next task is to select the g-parameters so that the two-port of Figure 4 is electrically equivalent to the L-section made up of ''R<sub>2</sub>'' and ''R<sub>f</sub>''. That selection is an algebraic procedure made most simply by looking at two individual cases: the case with ''V<sub>1</sub>'' = 0, which makes the VCVS on the right side of the two-port a short-circuit; and the case with ''I<sub>2</sub>'' = 0. which makes the CCCS on the left side an open circuit. The algebra in these two cases is simple, much easier than solving for all variables at once. The choice of g-parameters that make the two-port and the L-section behave the same way are shown in the table below.
{| class="wikitable" style="background:white;text-align:center "
!g<sub>11</sub>
!g<sub>12</sub>
!g<sub>21</sub>
!g<sub>22</sub>
|-
|-valign="center"
| '''<math>\frac {1} {R_f+R_2}</math>'''
| '''<math> - \frac {R_2}{R_2+R_f}</math>''
| '''<math> \frac {R_2} {R_2+R_f} </math>'''
| '''<math>R_2\mathit{\parallel}R_f \ </math>'''
|}
[[Image:Small-signal current amplifier with feedback.PNG|thumbnail|400px|Figure 5: Small-signal circuit with two-port for feedback network; upper shaded box: main amplifier; lower shaded box: feedback two-port replacing the ''L''-section made up of ''R''<sub>f</sub> and ''R''<sub>2</sub>.]]
 
===Small-signal circuit===
The next step is to draw the small-signal schematic for the amplifier with the two-port in place using the [[hybrid-pi model]] for the transistors. Figure 5 shows the schematic with notation ''R<sub>3</sub>'' = ''R<sub>C2</sub>||R<sub>L</sub>'' and ''R<sub>11</sub>'' = 1 / ''g<sub>11</sub>'', ''R<sub>22</sub>'' = ''g<sub>22</sub>'' .
 
===Loaded open-loop gain===
Figure 3 indicates the output node, but not the choice of output variable. A useful choice is the short-circuit current output of the amplifier (leading to the short-circuit current gain). Because this variable leads simply to any of the other choices (for example, load voltage or load current), the short-circuit current gain is found below.
 
First the loaded '''open-loop gain''' is found. The feedback is turned off by setting ''g<sub>12</sub> = g<sub>21</sub>'' = 0. The idea is to find how much the amplifier gain is changed because of the resistors in the feedback network by themselves, with the feedback turned off. This calculation is pretty easy because ''R<sub>11</sub>, R<sub>B</sub>, and r<sub>π1</sub>'' all are in parallel and ''v<sub>1</sub> = v<sub>π</sub>''. Let ''R<sub>1</sub>'' = ''R<sub>11</sub> // R<sub>B</sub> // r<sub>π1</sub>''. In addition, ''i<sub>2</sub> = −(β+1) i<sub>B</sub>''.  The result for the open-loop current gain ''A<sub>OL</sub>'' is:
 
::<math> A_{OL} = \frac { \beta i_B } {i_S} = g_m R_C \left( \frac { \beta }{ \beta +1} \right)
\left(
\frac {R_1} {R_{22} +
\frac {r_{ \pi 2} + R_C } {\beta + 1 } } \right)  \ . </math>
 
===Gain with feedback===
In the classical approach to feedback, the feedforward represented by the VCVS (that is,  ''g<sub>21</sub> v<sub>1</sub>'') is neglected.<ref name=note1/> That makes the circuit of Figure 5 resemble the block diagram of Figure 1, and the gain with feedback is then:
 
::<math> A_{FB} = \frac { A_{OL} } {1 + { \beta }_{FB} A_{OL} } </math>
:::<math> = \frac {A_{OL} } {1 + \frac {R_2} {R_2+R_f}  A_{OL} } \ , </math>
 
where the feedback factor β<sub>FB</sub> = −g<sub>12</sub>. Notation β<sub>FB</sub> is introduced for the feedback factor to distinguish it from the transistor β.
 
===Input and output resistances===
[[Image:Feedback amplifier input resistance.PNG|thumb|500px|Figure 6: Circuit set-up for finding feedback amplifier input resistance]]
First, a digression on how two-port theory approaches resistance determination, and then its application to the amplifier at hand.
 
====Background on resistance determination====
Figure 6 shows a test circuit for finding the input resistance of a feedback voltage amplifier (left) and for a feedback current amplifier (right). In the case of the voltage amplifier [[Kirchhoff's circuit laws|Kirchhoff's voltage law]] provides:
 
::<math> V_x = I_x R_{in} + \beta v_{out} \ , </math>
 
where ''v''<sub>out</sub> = ''A''<sub>v</sub> ''v''<sub>in</sub> = ''A''<sub>v</sub> ''I''<sub>x</sub> ''R''<sub>in</sub>. Substituting this result in the above equation and solving for the input resistance of the feedback amplifier, the result is:
 
::<math> R_{in}(fb) = \frac {V_x} {I_x} = \left( 1 + \beta A_v \right ) R_{in} \ . </math>
 
The general conclusion to be drawn from this example and a similar example for the output resistance case is:
 
''A series feedback connection at the input (output) increase the input (output) resistance by a factor ( 1 + β ''A''<sub>OL</sub> )'', where ''A''<sub>OL</sub> = open loop gain.
 
On the other hand, for the current amplifier, which uses a shunt input connection, Kirchhoff's laws provide:
 
::<math> I_x = \frac {V_{in}} {R_{in}} + \beta i_{out} \ . </math>
 
where ''i''<sub>out</sub> = ''A''<sub>i</sub> ''i''<sub>in</sub> = ''A''<sub>i</sub> ''V''<sub>x</sub> / ''R''<sub>in</sub>. Substituting this result in the above equation and solving for the input resistance of the feedback amplifier, the result is:
 
::<math> R_{in}(fb) = \frac {V_x} {I_x} = \frac { R_{in} } { \left( 1 + \beta A_i \right ) } \ . </math>
 
The general conclusion to be drawn from this example and a similar example for the output resistance case is:
 
''A parallel feedback connection at the input (output) decreases the input (output) resistance by a factor ( 1 + β ''A''<sub>OL</sub> )'', where ''A''<sub>OL</sub> = open loop gain.
 
These conclusions can be generalized to treat cases with arbitrary [[Norton's theorem|Norton]] or [[Thevenin's theorem|Thévenin]] drives, arbitrary loads, and general [[two-port network|two-port feedback networks]]. However, the results do depend upon the main amplifier having a representation as a two-port – that is, the results depend on the ''same'' current entering and leaving the input terminals, and likewise, the same current that leaves one output terminal must enter the other output terminal.
 
A broader conclusion to be drawn, independent of the quantitative details, is that feedback can be used to increase or to decrease the input and output impedances.
 
====Application to the example amplifier====
These resistance results now are applied to the amplifier of Figure 3 and Figure 5. The ''improvement factor'' that reduces the gain, namely ( 1 + β<sub>FB</sub> A<sub>OL</sub> ), directly decides the effect of feedback upon the input and output resistances of the amplifier. In the case of a shunt connection, the input impedance is reduced by this factor; and in the case of series connection, the impedance is multiplied by this factor. However, the impedance that is modified by feedback is the impedance of the amplifier in Figure 5 with the feedback turned off, and does include the modifications to impedance caused by the resistors of the feedback network.
 
Therefore, the input impedance seen by the source with feedback turned off is ''R''<sub>in</sub> = ''R''<sub>1</sub> = ''R''<sub>11</sub> // ''R''<sub>B</sub> // ''r''<sub>π1</sub>, and with the feedback turned on (but no feedforward)
 
::<math> R_{in} = \frac {R_1} {1 + { \beta }_{FB} A_{OL} } \ , </math>
 
where ''division'' is used because the input connection is ''shunt'': the feedback two-port is in parallel with the signal source at the input side of the amplifier. A reminder: ''A''<sub>OL</sub> is the  ''loaded'' open loop gain [[Negative_feedback_amplifier#Loaded_open-loop_gain|found above]], as modified by the resistors of the feedback network.
 
The impedance seen by the load needs further discussion. The load in Figure 5 is connected to the collector of the output transistor, and therefore is separated from the body of the amplifier by the infinite impedance of the output current source. Therefore, feedback has no effect on the output impedance, which remains simply ''R<sub>C2</sub>'' as seen by the load resistor ''R<sub>L</sub>'' in Figure 3.<ref name=Jaeger/>
 
If instead we wanted to find the impedance presented at the ''emitter'' of the output transistor (instead of its collector), which is series connected to the feedback network, feedback would increase this resistance by the improvement factor ( 1 + β<sub>FB</sub> A<sub>OL</sub> ).<ref name=note2/>
 
===Load voltage and load current===
The gain derived above is the current gain at the collector of the output transistor. To relate this gain to the gain when voltage is the output of the amplifier, notice that the output voltage at the load ''R<sub>L</sub>'' is related to the collector current by [[Ohm's law]] as ''v<sub>L</sub> = i<sub>C</sub> ( R<sub>C2</sub>||R<sub>L</sub> )''. Consequently, the transresistance gain ''v<sub>L</sub> / i<sub>S</sub>'' is found by multiplying the current gain by ''R<sub>C2</sub>||R<sub>L</sub>'':
 
::<math> \frac {v_L} {i_S} = A_{FB} ( R_{C2}\mathit{\parallel}R_L ) \ . </math>


Similarly, if the output of the amplifier is taken to be the current in the load resistor ''R<sub>L</sub>'', [[current division]] determines the load current, and the gain is then:
<ref name=Bobzien2>
 
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular pp. 386 ''ff''.
::<math> \frac {i_L} {i_S} = A_{FB} \frac {R_{C2}} {R_{C2} + R_L} \ . </math>
 
=== Is the main amplifier block a two port? ===
[[Image:Two-port ground arrangement.PNG|thumbnail|400px|Figure 7: Amplifier with ground connections labeled by ''G''. The feedback network satisfies the port conditions.]]
Some complications follow, intended for the attentive reader.
 
Figure 7 shows the small-signal schematic with the main amplifier and the feedback two-port in shaded boxes. The two-port satisfies the [[Two-port_network|port conditions]]: at the input port, ''I''<sub>in</sub> enters and leaves the port, and likewise at the output, ''I''<sub>out</sub> enters and leaves. The main amplifier is shown in the upper shaded box. The ground connections are labeled.
 
Figure 7 shows the interesting fact that the main amplifier does not satisfy the port conditions at its input and output unless the ground connections are chosen to make that happen. For example, on the input side, the current entering the main amplifier is ''I''<sub>S</sub>. This current is divided three ways: to the feedback network, to the bias resistor ''R''<sub>B</sub> and to the base resistance of the input transistor ''r''<sub>π</sub>. To satisfy the port condition for the main amplifier, all three components must be returned to the input side of the main amplifier, which means all the ground leads labeled ''G''<sub>1</sub> must be connected, as well as emitter lead ''G''<sub>E1</sub>. Likewise, on the output side, all ground connections ''G''<sub>2</sub> must be connected and also ground connection ''G''<sub>E2</sub>. Then, at the bottom of the schematic, underneath the feedback two-port and outside the amplifier blocks, ''G''<sub>1</sub> is connected to ''G''<sub>2</sub>. That forces the ground currents to divide between the input and output sides as planned. Notice that this connection arrangement ''splits the emitter'' of the input transistor into a base-side and a collector-side – a physically impossible thing to do, but electrically the circuit sees all the ground connections as one node, so this fiction is permitted.
 
Of course, the way the ground leads are connected makes no difference to the amplifier (they are all one node), but it makes a difference to the port conditions. That is a weakness of this approach: the port conditions are needed to justify the method, but the circuit really is unaffected by how currents are traded among ground connections.
 
However, if there is '''no possible arrangement''' of ground conditions that will lead to the port conditions, the circuit might not behave the same way.<ref name=note3/> The improvement factors ( 1 + β<sub>FB</sub> A<sub>OL</sub> ) for determining input and output impedance might not work. This situation is awkward, because a failure to make a two-port may reflect a real problem (it just is not possible), or reflect a lack of imagination (for example, just did not think of splitting the emitter node in two). As a consequence, when the port conditions are in doubt, at least two approaches are possible to establish whether improvement factors are accurate: either simulate an example using [[SPICE|Spice]] and compare results with use of an improvement factor, or calculate the impedance using a test source and compare results.
 
A more radical choice is to drop the two-port approach altogether, and use [[return ratio]]s. That choice might be advisable if small-signal device models are complex, or are not available (for example, the devices are known only numerically, perhaps from measurement or from [[SPICE]] simulations).
 
== References  and notes ==
{{Reflist|refs=
 
<ref name=Broderson>
{{cite web |url=http://bwrc.eecs.berkeley.edu/classes/ee140/Lectures/10_stability.pdf |author=RW Brodersen |title=Analog circuit design: lectures on stability |accessdate=2011-06-24 }}
</ref>
</ref>


<ref name=Goel>
<ref name=Bobzien3>
{{cite web |url=http://www.ece.mtu.edu/faculty/goel/EE-4232/Feedback.pdf |author=Ashok K. Goel |title=Feedback topologies |accessdate=2011-06-24}}
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular p. 255.
</ref>
</ref>


<ref name=Kuo>
{{cite book
|author=Kuo, Benjamin C & Farid Golnaraghi
|title=Automatic control systems
|edition=Eighth edition
|page=p. 46
|year= 2003
|publisher=Wiley
|location=NY
|isbn=0471134767
|url=http://worldcat.org/isbn/0471134767}}
</ref>


<ref name=Davies>
{{cite web |title=Epistemological pluralism |author=E Brian Davies |url=http://philsci-archive.pitt.edu/3083/1/EP3single.doc |work=PhilSci Archive |year=2006 }}
</ref>


<ref name=Palumbo>
{{cite book
|author=Gaetano Palumbo and Salvatore Pennisi
|title=Feedback amplifiers: theory and design
|chapter=Chapter 3: Feedback
|pages=p. 64
|year= 2002
|publisher=Springer
|isbn=0792376439
|url=http://www.amazon.com/Feedback-Amplifiers-Theory-Gaetano-Palumbo/dp/0792376439/ref=sr_1_1?s=books&ie=UTF8&qid=1308924834&sr=1-1#reader_0792376439}}
</ref>


<ref name=patent>
<ref name=Hawking>
{{cite web
{{cite book |author=Hawking SW, Mlodinow L. |title=The Grand Design |isbn=0553805371 |url= http://www.amazon.com/Grand-Design-Stephen-Hawking/dp/0553805371#reader_0553805371 |pages=pp. 42-43 |chapter=Chapter 3: What is reality?|year=2010|publisher=Bantam Books}}
| title=Wave translation system
| url=http://www.google.com/patents?id=tA9EAAAAEBAJ&printsec=abstract&zoom=4&source=gbs_overview_r&cad=0#v=onepage&q&f=false
| accessdate=2011-06-24
}} Patent 2,106,671 refers to "apparatus or systems involving negative feedback".
</ref>
</ref>


<ref name=Sansen> For a complete discussion, see Sansen.
<ref name=Norretranders>
{{cite book  
{{cite book |url=http://www.google.com/search?tbo=p&tbm=bks&q=consciousness%2Bplays%2Ba%2Bsmaller%2Brole%2Bin%2Bhuman%2Blife+intitle:User+intitle:illusion&num=10 |title=The user illusion: Cutting consciousness down to size |quote=Consciousness plays a far smaller role in human life than Western culture has tended to believe |author=Tor Nørretranders |isbn=0140230122 |chapter=Preface |pages=p. ''ix'' |publisher=Penguin Books |year=1998 |edition=Jonathan Sydenham translation of ''Maerk verden'' 1991 ed }}
|author=Willy M. C. Sansen
|title=Analog design essentials
|year= 2006
|pages=§0513-§0533, p. 155-165
|publisher=Springer
|location=New York; Berlin
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


<ref name=Sedra1>
<ref name=Ney>
{{cite book
{{cite web |author=Alyssa Ney |title=Reductionism |work=Internet Encyclopedia of Philosophy |date= November 10, 2008 |url=http://www.iep.utm.edu/red-ism/}}
|author=A. S. Sedra and K.C. Smith
|title=Microelectronic Circuits
|year= 2004
|edition=Fifth Edition
|pages=Example 8.4, pp. 825-829 and PSpice simulation pp. 855-859
|publisher=Oxford
|location=New York
|isbn=0-19-514251-9
|url=http://worldcat.org/isbn/0-19-514251-9}}
</ref>
</ref>


<ref name=Subramanian>
<ref name=Northoff>
{{cite web |url=http://organics.eecs.berkeley.edu/~viveks/ee140/lectures/section10p4.pdf |author=Vivek Subramanian |title=Lectures on feedback}}
A rather extended discussion is provided in {{cite book |title=Philosophy of the Brain: The Brain Problem |author=Georg Northoff |url=http://books.google.com/books?id=r0Bf3lLys6AC&printsec=frontcover |publisher=John Benjamins Publishing |isbn=1588114171 |year=2004 |edition=Volume 52 of Advances in Consciousness Research}}
</ref>
 
<ref name=Gray-Meyer1>
{{cite book
|author=P R Gray, P J Hurst, S H Lewis, and R G Meyer
|title=Analysis and Design of Analog Integrated Circuits
|year= 2001
|page=pp. 586-587
|edition=Fourth Edition
|publisher=Wiley
|location=New York
|isbn=0-471-32168-0
|url=http://worldcat.org/isbn/0471321680}}
</ref>
</ref>


<ref name=Jaeger>
<ref name=OConnor>
The use of the improvement factor ( 1 + β<sub>FB</sub> A<sub>OL</sub> ) requires care, particularly for the case of output impedance using series feedback. See {{cite book | title = Microelectronic Circuit Design | author =R.C. Jaeger and T.N. Blalock  | publisher = McGraw-Hill Professional | year = 2006 |edition=Third Edition |page=Example 17.3 pp. 1092-1096| isbn = 978-0-07-319163-8 | url = http://worldcat.org/isbn/978-0-07-319163-8 }}
{{cite web |title=&thinsp;Free Will |date=Oct 29, 2010 |author=O'Connor, Timothy |url=http://plato.stanford.edu/archives/sum2011/entries/freewill |work=The Stanford Encyclopedia of Philosophy (Summer 2011 Edition) |editor=Edward N. Zalta, ed.}}
</ref>
</ref>


<ref name=note1>
<ref name=Pockett>
If the feedforward is included, its effect is to cause a modification of the open-loop gain, normally so small compared to the open-loop gain itself that it can be dropped. Notice also that the main amplifier block is [[Electronic_amplifier#Unilateral_or_bilateral|unilateral]].
{{cite book|title=&thinsp;Does Consciousness Cause Behavior? |chapter=The neuroscience of movement |author=Susan Pockett |url=http://books.google.com/books?id=G5CaTnNksgkC&pg=PA19&lpg=PA19 |pages= p. 19 |editor=Susan Pockett, WP Banks, Shaun Gallagher, eds. |publisher=MIT Press |date =2009 |isbn=0262512572}}
</ref>
</ref>


<ref name=note2>
<ref name=Timpe>
That is, the impedance found by turning off the signal source ''I<sub>S</sub>'' = 0,  inserting a test current in the emitter lead ''I<sub>x</sub>'',  finding the voltage across the test source ''V<sub>x</sub>'', and finding ''R<sub>out</sub> = V<sub>x</sub> / I<sub>x</sub>''.
{{cite web |author=Kevin Timpe |title=Free will |work=Internet Encyclopedia of Philosophy |date= March 31, 2006 |url=http://www.iep.utm.edu/freewill/#H5}}
</ref>
</ref>


<ref name=note3>
The equivalence of the main amplifier block to a two-port network guarantees that performance factors work, but without that equivalence they may work anyway. For example, in some cases the circuit can be shown to be equivalent to another circuit that is a two port, by "cooking up" different circuit parameters that are functions of the original ones. There is no end to creativity!</ref>


<ref name=Waldhauer>
<ref name=Velmans>
{{cite book
{{cite journal |journal=Journal of Consciousness Studies |volume=9 |issue=11 |year=2002 |pages=pp. 2-29 |author=Max Velmans  |title=How Could Conscious Experiences Affect Brains? |url=http://cogprints.org/2750/ |year=2002}}
|author=Fred F. Waldhauer
|title=Feedback
|url=http://books.google.com/books?ei=F5cETsCRFJDAsAPj9fHYDQ&ct=result&id=EesiAAAAMAAJ&dq=feedback+inauthor%3AFred+inauthor%3AD+inauthor%3AWaldhauer&q=Lackawanna+Ferry#search_anchor
|pages=p. 3, Figure 1.1 p. 4,  p. 8
|year= 1982
|publisher=Wiley
|location=NY
|isbn=0471053198
}}
</ref>
</ref>


<ref name=Zimmer>
<ref name=Vihvelin>
{{cite web |url=http://centrevirtuel.creea.u-bordeaux.fr/ELAB/docs/freebooks.php/virtual/feedback-amplifier/textbook_feedback.html#1.2 |author=Zimmer T & Geoffreoy D |title=Feedback amplifier |accessdate=2011-06-24}}  
{{cite web |author=Kadri Vihvelin |title=&thinsp;Arguments for Incompatibilism |work=The Stanford Encyclopedia of Philosophy (Spring 2011 Edition) |editor=Edward N. Zalta, ed. |url= http://plato.stanford.edu/archives/spr2011/entries/incompatibilism-arguments/ |date=Mar 1, 2011}}
</ref>
</ref>


}}
}}

Latest revision as of 04:07, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


In philosophy the term free will refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so. It is an age-old concern to separate what we can do something about, choose to do, from what we cannot. The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of determinism, sometimes called physical reductionism,[1] and the view that determinism precludes free will is called incompatibilism.[2]

There are several ways to avoid the incompatibilst position, resulting in various compatibilist positions.[3] One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.

There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?

Science does not apply

One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect. For example, David Hume suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. Immanuel Kant suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains. Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.

A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise. It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience. For example, we choose to explain phenomena like the Higgs boson found by elaborate means like a hadron collider, but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention. As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.

Although not explicitly addressing the issue of free will, it may be noted that Ludwig Wittgenstein argued that the specialized theories of science, as discussed by Rudolf Carnap for example, inevitably cover only a limited range of experience. Hawking/Mlodinow also noted this fact in in their model-dependent realism,[4] the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.

“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”[5]
—— E.B. Davies Epistemological pluralism, p. 4

Still another approach to this matter is analysis of the mind-brain connection (more generally, the mind-body problem). As suggested by Northoff,[6] there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks. But observing our own mental activity is not possible in this way – it is a matter of subjective experiences. The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:

“...for each individual there is one 'mental life' but two ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the observational arrangements (the means by which a subject and an external observer access the subject's mental processes).”[7]

—Max Velmans: , How could conscious experiences affect brains?, p. 11

A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by Niels Bohr and by Erwin Schrödinger in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)

“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”[8]

—Susan Pockett , The neuroscience of movement


In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.

Science can be accommodated

A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do. Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.

’Will’ versus ‘action’

There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,[9] which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, drug addiction has been related to alteration of the mechanisms in the brain for dopamine production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.

“Philosophers who distinguish freedom of action and freedom of will do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.][10]

— Timothy O'Connor , Free Will

In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?

Theology

The ancient Greeks held the view that the gods could intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:

“The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”

—spoken by Cassius, Julius Caesar (I, ii, 140-141)

The Stoics wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a responsibility), a sequence demanded by our natures as rational beings.[11]

In Chrysippus of Soli's view (an apologist for Stoicism), fate precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.[12] The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.[13]

References

  1. Alyssa Ney (November 10, 2008). Reductionism. Internet Encyclopedia of Philosophy.
  2. Kadri Vihvelin (Mar 1, 2011). Edward N. Zalta, ed.: Arguments for Incompatibilism. The Stanford Encyclopedia of Philosophy (Spring 2011 Edition).
  3. Kevin Timpe (March 31, 2006). Free will. Internet Encyclopedia of Philosophy.
  4. Hawking SW, Mlodinow L. (2010). “Chapter 3: What is reality?”, The Grand Design. Bantam Books, pp. 42-43. ISBN 0553805371. 
  5. E Brian Davies (2006). Epistemological pluralism. PhilSci Archive.
  6. A rather extended discussion is provided in Georg Northoff (2004). Philosophy of the Brain: The Brain Problem, Volume 52 of Advances in Consciousness Research. John Benjamins Publishing. ISBN 1588114171. 
  7. Max Velmans (2002). "How Could Conscious Experiences Affect Brains?". Journal of Consciousness Studies 9 (11): pp. 2-29.
  8. Susan Pockett (2009). “The neuroscience of movement”, Susan Pockett, WP Banks, Shaun Gallagher, eds.:  Does Consciousness Cause Behavior?. MIT Press, p. 19. ISBN 0262512572. 
  9. Tor Nørretranders (1998). “Preface”, The user illusion: Cutting consciousness down to size, Jonathan Sydenham translation of Maerk verden 1991 ed. Penguin Books, p. ix. ISBN 0140230122. “Consciousness plays a far smaller role in human life than Western culture has tended to believe” 
  10. O'Connor, Timothy (Oct 29, 2010). Edward N. Zalta, ed.: Free Will. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).
  11. Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See §6.3.3 The cylinder and cone analogy, pp. 258 ff.
  12. Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular pp. 386 ff.
  13. Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular p. 255.