 Problems for Naturalism
 Why Mental States are Causally Irrelevant on Naturalism
 The Symbolic Logic
 Proving It
 The Argument is Sound, but So What?
The Symbolic Logic
Symbolic logic uses symbols and rules of inference. A quick way to introduce both is by providing some simple examples, like so:
modus ponens  

In English  In Symbolic Logic 
If p then q p Therefore, q 
p → q p ∴ q 
simplification  

In English  In Symbolic Logic 
p and q Therefore, p 
p ∧ q ∴ p 
p and q Therefore, q 
p ∧ q ∴ q 
We can use both rules of inference on premises (1) and (2) below to derive statement G being true:
 A ∧ B
 B → G
 B 1, simplification
 G 2, 3, modus ponens
In a statement like “If A, then C” the A part is called the antecedent and the C part is called the consequent. The → symbol in A → C (sometimes represented as A ⊃ C) is called a material conditional (or material implication) and is equivalent to “It is not the case that A is true and C is false.” Thus, whether the material conditional is true is determined entirely by the truth of antecedent and consequent as follows, where the truth table below exhausts all possible true/false combinations of A and C. Note the second line of the truth table where A being true and C being false makes “If A, then C” false:
A  C  If A, then C 
T  T  T 
T  F  F 
F  T  T 
F  F  T 
Notice that antecedent being true and consequent being false is the only time the material conditional is false; otherwise the material conditional is true. For example, “If 2 + 2 = 5, then life forms exist” is a true material conditional because “2 + 2 = 5” is false and “life forms exist” is true, thus fitting the third line of the truth table where A is false and C is true.
A material conditional may seem like a weak claim (in the sense that it doesn’t claim very much); after all, when “If A, then C” is a material conditional, the antecedent and the consequent don’t have to be related to each other at all for the material conditional to be true. But a material conditional is good enough for e.g. modus ponens, since in a true “If A, then C” material conditional, when the antecedent is true, then the consequent is true as well (recall that a true material conditional would prohibit C from being false when A is true, as the second line of the truth table confirms).
Another rule of inference:
conditional proof  


For example, suppose we want to prove A → D from premises 1 and 2 below:
 A → B
 B → (C ∧ D)
 A conditional proof assumption
 B 1, 3, modus ponens
 C ∧ D 2, 4, modus ponens
 D 5, simplification
 A → D 36, conditional proof
Modal Logic Symbols  

In Symbolic Logic  In English 
□p  Necessarily, p (p is true in all possible worlds). 
◊p  Possibly, p (p is true in some possible world). 
So what kind of “possibility” are we talking about? Possible world semantics per se doesn’t specify, and one can use this semantics for all sorts of possibilities. One type of possibility, called physical possibility has to do with what the physical laws permit. The term “logical possibility” in philosophy is somewhat ambiguous, and can be used in a narrow sense such that “logically impossible” means “impossible by virtue of being selfcontradictory,” e.g. married bachelors and round squares. Metaphysical possibility refers to what reality could have actually been like, and some propositions that are not selfcontradictory are still (arguably) necessary in this sense; e.g. some believe there is no metaphysically possible world in which it is morally obligatory to torture infants just for fun; even though that idea isn’t selfcontradictory like a married bachelor is. Sometimes logical possibility (sometimes under the term broadly logical possibility) is used to refer to metaphysical possibility, but in this case I’ll use “logical impossibility” to mean that narrower “impossible by virtue of being selfcontradictory” sense. For the purposes of this article, I’ll define conceivability such that something is conceivable if and only if it’s not selfcontradictory.
Modal logic (in the narrow sense of the term) is the logic of possibility and necessity. There are multiple systems of modal logic with different axioms and rules, but here we’ll use a very basic system that’s called T.[3] The rules I’ll demonstrate below are Treiteration and necessity introduction.
Treiteration, necessity introduction  


The general idea behind these rules is that anything that logically and necessarily follows from what is necessary is itself necessary. On the T system, the only propositional variables allowed to “move” into the □ region are those that begin with □ and even those have the outer □ chopped off before entering (e.g. □p becomes p when it moves into that region). An example below:
 □(L → K)
 □L
 L → K 1, Treiteration
 L 2, Treiteration
 K 3, 4, modus ponens
□ 

 □K 35, necessity introduction
Statements of the form “If P were true, then Q would be true” are called counterfactuals and the symbolization for this will be the last logic symbol I’ll introduce here.
In English  In Symbolic Logic 

If p were true, q would be true.  p □→ q 
Suppose I have been holding a hammer above the ground for a minute, and I utter the counterfactual, “If I were to have dropped the hammer thirty seconds ago, it would have fallen.” What makes this statement true? How do we evaluate whether it is true? At least part of what we would do is imagine the world as it was thirty seconds ago in which I drop the hammer but the world is otherwise similar to the actual world (same planet, same laws of physics, etc.). If it were true in some possible world that I drop the hammer thirty seconds ago, and in every possible world in which I dropped the hammer, the hammer fell, the statement “If I were to have dropped the hammer thirty seconds ago, it would have fallen” would be true. Generalizing that principle we can use this rule of inference I’ll call “nonvacuous counterfactual introduction.”
nonvacuous counterfactual introduction  

In English  In Symbolic Logic 
p is true in some possible world p entails q Therefore, If p were true, q would be true. 
◊p □(p → q) ∴ p □→ q 
This rule seems to follow from what we mean by “If P were true, then Q would be true.”[4] At least, it reflects how I use such statements.
Next I’ll use the power of symbolic logic to help show that mental states are causally irrelevant on naturalism.
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[3] This very basic system of modal logic has the following axioms in addition to the formal definitions of □ and ◊ (where they are defined in terms of each other; where ¬ symbolizes “not,” is ◊p is definitionally equivalent to ¬□¬p):
 □p implies p
 The distribution axiom: □(p → q) implies □p →□q
 The necessitation rule: Any proposition p being a theorem of propositional logic implies □p.
[4] I’ve heard one claim that this nonvacuous counterfactual introduction rule holds for metaphysical possibility but not logical possibility, but I find this claim strange and inconsistent with what we normally mean by “If P were true, then Q would be true.” To illustrate, let x be a placeholder for something that is logically possible but not metaphysically possible (a naturalist might think an example is a brain identical to that of a conscious human but with no mental state associated with it; one might also think a ghost of a computer program is metaphysically impossible). The counterfactual, “If there were seven x’s, then there would be more than two x’s” seems clearly true, and the nonvacuous counterfactual introduction rule seems like perfectly reasonable justification for that counterfactual.
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