Using Propositional Calculus to Define Anthropic Principle

One definition of Anthropic Principle states that if some property of the universe or the Laws of Physics was not true we couldn’t exist (The Cosmic Landscape, Susskind, p79). “The Laws of Physics have to be such that they allow life because if they weren’t, there wouldn’t be anyone to ask about the Laws of Physics.” (p197). I think it’s important to go through the definitions of the Anthropic Principle with formal logic and propositional calculus to really get to the truth of what Anthropic Principle is saying, if its saying anything at all.

With all the different forms of Anthropic reasoning, its hard to pull out what logicians call a “well formed formula” (or wffs) from the various definitions that are out there. Take for example the statement, when it rains, the ground is wet. The statement may be reduced to R -> W through modes ponens reasoning if we define the wff R to be “when it rains” and W to be “the ground is wet.”

R -> W

In these statements “when it rains” and the ground is wet” are both well formed formula, the absolute basic building blocks in forming well reasoned logical statements.

Part of the problem with Anthropic Principle is that the very basis of arguments surrounding are based on shoddy well-formed formulas. For example, in one of the previous posts (Leonard Susskind’s the Cosmic Landscape …) Anthropic Principle was defined as “the universe exists for us to observe it.” Stated this way it is hard to know exactly what well-formed formulas may be extracted for logical deduction beyond a mere tautological statements. We have choices for what our well-formed formula may be. By hypothesis, taking “the universe exists for us to observe it” to be U as a well-formed formula, allows us to make no useful deductive statements.

U -> U & U; or possibly U -> U or U;

These are tautological statements. Let’s assume that the Anthropic Principle allows us to make more precise well-formed formulas such as U, the universe exists, and O, we observe the universe. We may say U <-> O from U->O, O ->U :

Prove: U -> O, O-> U Therefore U <-> O;
1. U ->O Assumption
2. O ->U Assumption
3. U <-> O 1,2 Biconditional Introduction (<-> I)

Using some imprecise translation: 1. U->O The universe exists so we can observe it. 2. O->U We observe the universe, so it exists. However, the statement 1.,that the universe exists so that we can observe it is a very strong logical statement and is at worse a circular argument, and at best statistical syllogism whose inductive probability is extremely weak for reason stated a previous post (Leonard Susskind’s Cosmic Landscape …). The problem is that if even one other observing being existed in the universe other than us, the inductive probability, the probability that a statement is true, would go from 100 % to 50 % because the universe could only be stated to be created ½ for us and ½ for them. This hardly makes the first assumption, “the universe exists so that we can observe it” a well-formed formula.

But more problematic than the statement that “the universe exists so that we can observe it” being a statistical syllogism, one made on the 100% assumption that we are the only observing life, is that it assumes its own conclusion. To say that the universe exists so that we can observe it is to say we observe the universe so it exists. The very statement assumes its conclusion, namely , that the universe exists for a particular purpose, so that we may observe it.

So the Anthropic Principles should be defined very carefully. Cursorily phrasing the definitions will most certainly result in some sort of circular reasoning or tautology. That being said, I am not sure that Anthropic Principles are anything more than circular reasoning or tautology. Leonard Susskind and others believe some form of Anthropic reasoning is probably true, but it’s not clear what well-formed formula, or what definition of Anthropic Principle they are talking about.