As I read him, Salmon presents multiple arguments against the design hypothesis. Let us consider the logical form of each argument in turn. Note that all of the argument labels are mine; Salmon does not label any of his arguments in his essay.
This is the argument Salmon sketches in section 3 of his essay. Let A be any instance of coming-into-being; B be any instance of the operation of intelligence; and C be any instance that exhibits order or design. Finally, let Pr-F represent a probability value, as interpreted by the frequency theory.
(1) For the class of entities which come into being, non-intelligent causation is at least as plausible as intelligent design, i.e., Pr-F(~B | A) >= Pr-F(B | A).
(2) The antecedent probability that an unspecified entity exhibits order, conditional upon intelligent design, is high, i.e., Pr-F(C | A & B) > 0.5.
(3) The antecedent probability that an unspecified entity exhibits order, conditional upon the lack of intelligent design, is low, i.e., Pr-F(C | A & ~B) < 0.5.
(4) Therefore, an unspecified entity, which came into being and exhibited order, has a low posterior probability of being “produced by intelligent design,” i.e., Pr-F(B | A & C) < 0.5. [From (1), (2), and (3) by Bayes’s Theorem]
B. The Argument Against a Disembodied Designer
I am not sure whether Salmon intended for this to be treated as an independent argument or not; I have chosen to err on the side of “completeness” by treating this as an independent argument. It may be more accurate to say that Salmon merely provided an argument fragment, since his comments in section 4 seem to provide some but not all of the premises needed for a Bayesian argument against a disembodied designer. In fact, this seems to me to be the most charitable interpretation. Since Salmon does not rely upon these premises in his other arguments (see below), this interpretation is the only way I am able to see how his explicitly stated premises could be relevant to anything.
Let D be the hypothesis that the hypothesized intelligent designer is a disembodied mind; and let U be the event of the creation of the universe. Note that D entails B; hence, Pr(D) <= Pr(B).
(1) For the single case of the creation of the universe, the limit of the relative frequency of artifacts produced by a disembodied intelligence is zero, i.e., Pr-F(B & D | A) = 0.
(2) Pr-F(C | A & B & D & U) is undefined.
C. Argument for Explanatory Parity of Mechanical Causation and Intelligent Design
This argument is found in section 4 of Salmon’s essay. The focus of this argument is the hypothesis of intelligent design sans moral attributes.
Let M be any instance of coming-into-being as a result of mechanical causation.
(1) Mechanical causation is at least as plausible as intelligent design, i.e., Pr-F(M | A & U) >= Pr-F(B | A & U).
(3) Therefore, the posterior probability of the creation of the universe by mechanical causation is at least great as the posterior probability on the design hypothesis, i.e., Pr-F(U | A & M) >= Pr-F(U | A & B). [From (1) & (2) by Bayes’s Theorem]
D. Argument Against Theistic Intelligent Design
Also found in Section 4 of Salmon’s essay, the focus of this argument is an omnibenevolent intelligent designer. Since the context is the single case of the creation of the universe, I interpret such a designer to be identical to God.
Let O represent the hypothesis that the intelligent designer is omniscient, omnipotent, and perfectly benevolent; and let E represent known facts about apparently gratuitous evil.
(1) For the single case of the creation of the universe, mechanical causation is at least as plausible as intelligent design, i.e., Pr-F(M | A & U) >= Pr-F(B & O | A & U).
(2) The event of the creation of the universe conjoined with known facts about apparently gratuitous evil are more probable on the assumption that the mechanical hypothesis is true than on the assumption the theistic intelligent design hypothesis is true, i.e., Pr(U & E | A & M) > Pr(U & E | A & B & O).
(3) Therefore, the creation of the universe has a higher posterior probability of being the result of mechanical causation than the result of theistic intelligent design, i.e., Pr(M | A & E & U) > Pr(B & O | A & E & U).
E. Argument Against Theism
In section 8, Salmon describes two hypothetical epistemic situations. For each situation, he provides the premises for a Bayesian argument. These two examples suggest the following argument form.
(1) The hypothesis of theistic intelligent design is antecedently implausible because it is in direct conflict with a large body of well-established theory.
(2) The theistic intelligent design hypothesis makes the occurrence of the facts to be explained quite improbable if it is true.
(3) There is a plausible alternative hypothesis (i.e., the mechanical hypothesis) which makes the facts to be explained highly probable.
(4) Therefore, the theistic intelligent design hypothesis is very improbable.