Friday, May 2, 2014

Syllogisms and Probability

A syllogism is a systematic way of laying out a sequence of logical steps. They are very useful in formulating arguments, constructing proofs, or checking the coherence of a certain train of thought. A syllogism is a systematic way of laying out an argument. A syllogism consists of three parts: two premises (which can sometimes be construed as major and minor) and a conclusion. A syllogism is valid if the conclusion follows logically from the premises, whereas is it invalid if it does not. A syllogism is sound if both its premises are true, or at least plausibly true, and the syllogism is valid. Note that a syllogism may be invalid and yet have a true conclusion: it may even have true premises and a true conclusion and still be invalid.

We can also draw the distinction between a propositional and a categorical syllogism. In a propositional syllogism, we deal only with bare propositions connected by logical operators like "or", "and" "implies" etc. In a categorical syllogism, we deal only with statements of inclusion or exclusion in certain categories that are quantified, using such terms as "all" "some" and "no/none".

For example:
1) If Socrates is a man then Socrates is mortal.
2) Socrates is a man.
3) Therefore Socrates is mortal.

This is an example of a propositional syllogism. We can also write this in a categorical form:
1) All men are mortal.
2) Socrates is a man
3) Therefore Socrates is mortal.
We will henceforth focus on the propositional variety.

There are a number of standard propositional syllogisms (also called propositional forms)
  • Modus Ponens: If A, then B. A. Therefore B.
  • Modus Tollens: If A then B. Not B. Therefore, not A.
  • Hypothetical Syllogism: If A then B. If B then C. Therefore, if A then C.
  • Disjunctive Syllogism: Either A or B. Not A. Therefore B.
  • Conjunctive Syllogism: A. B. Therefore A and B.

We can further pare these down by noting that modus ponens and modus tollens are both actually disjunctive syllogisms when we state them using material conditionals. Remember that, in a material conditional, "if A then B" is equivalent to "Either not A, or B".

In a valid argument, if we grant that the premises are true, we must grant that the conclusion is also true. However, we typically are required to evaluate premises of which we are not certain. Namely, the premises have epistemic probabilities that are less than one. In that case, we can use our knowledge of probabilities to give bounds on the probability of the conclusion, and possibly even an estimate of the probability of the conclusion. That is, we can give limits to how plausible we are to take the conclusion to be. We will merely state it as a chart. In each case x is the probability of the first premise (major premise), y is the probability of the second premise (minor premise), and z is the probability of the conclusion. That is, \(P(p1)=x\), \(P(p2)=y\), \(P(c)=z\).

Bounds on Syllogism Probabilities

Name Form Bounds on Conclusion Estimate of Conclusion
Disjunctive Syllogism \(p1)\> A \cup B\)
\(p2)\> \sim A\)
\(c)\>\> \therefore B\)
\(x+y-1 \leq z \leq x\) \(z=x+\frac {y-1}{2}\)
Conjunctive Syllogism \(p1)\> A\)
\(p2)\> B\)
\(c)\>\> \therefore A \cap B\)
\(x+y-1 \leq z \leq \min(x,y)\) \(z=xy\)
Hypothetical Syllogism \(p1)\> A \Rightarrow B\)
\(p2)\> B \Rightarrow C\)
\(c)\>\> \therefore A \Rightarrow C\)
\(x+y-1 \leq z \leq 1\) \(z= \frac {x+y}{2}\)

For example, suppose we say "If it is raining, then Bob will bring an umbrella. It is raining. Therefore Bob will bring an umbrella." We are 80% sure that Bob will bring an umbrella if it is raining and 70% sure that it is raining. Thus, the probability that Bob does bring an umbrella is somewhere between 50% and 80%, with an estimate of 56%. That is not very confident.
Suppose we have two propositions of which we are 70% certain of each. Then the probability that both are true can be anywhere from 40% to 70% with an estimated value of 49%. Thus it is not enough that each of the premises be more plausible than their opposites: we must demand more in order that an argument be a good one.

Sometimes when people say "if A then B", they mean something different than the material conditional ("Either not A, or B"). Frequently, they mean something like "Given A, B will happen/be true". Thus the probability of "if A, then B" wouldn't be \(P(\sim A \cup B)\), but rather \(P(B|A)\). We will here give the table for the conditional probability case. That is, in every case that a conditional statement is given a probability, that probability is of the probability of the consequent given the antecedent.

Bounds on Syllogism Probabilities

Name Form Bounds on Conclusion Estimate of Conclusion
Modus Ponens \(p1)\> A \Rightarrow B\)
\(p2)\> A\)
\(c)\>\> \therefore B\)
\(xy \leq z \leq xy+1-y\) \(z=xy+\frac {1-y}{2}\)
Modus Tollens \(p1)\> A \Rightarrow B\)
\(p2)\> \sim B\)
\(c)\>\> \therefore \sim A\)
\(\frac {x+y-1}{x} \leq z < 1\) \(z=\frac {2x+y-1}{2x}\)

For a hypothetical syllogism with conditional probabilities, we can give no bounds or estimates.