Kant lecture 

Some background:
•	Principle of contradiction
•	Induction
•	Hume
•	Proposition = subject + predicate
•	Dogmatism and skepticism (certainty and fecundity)
•	A priori versus a posteriori propositions
•	Synthetic versus analytic propositions
•	Necessary versus sufficient conditions

The Distinction Between Pure and Empirical Knowledge
•	All of our knowledge begins with experience in the order of time.
•	However, it is not the case that all of our knowledge arises out of experience (alone).  So, something is logically prior but not temporally prior to experience in our knowledge.
•	“pure a priori knowledge” = “not knowledge independent of this or that experience, but knowledge absolutely independent of all experience” with no admixture of empirical knowledge.

We Are in Possession of Certain Modes of A Priori Knowledge, and Even the Common Understanding Is Never Without Them
•	Two criteria to distinguish between pure and empirical knowledge:  necessity and universality.
•	Empirical versus strict universality.   
•	We only need to look for one of these at a time; since wherever one is to be found, so is the other.  
•	Examples of necessary, universal statements:  any mathematical proposition, “every alteration must have a cause.”
•	Pure a priori principles are the necessary foundation for experience.  Experience would be unintelligible if we didn’t have pure a priori principles to “sort it out.”  Put differently, experience carries with it a kind of certainty.  How could this be the case if all the rules of experience were empirical and therefore contingent?  
•	Examples of a priori principles that ground experience in this way:  space, substance.  We can strip away all qualities of bodies in our imagination, space and substance remain.  Since these concepts press themselves upon us irresistibly, independent of all experience, they must have their origin in our faculty of a priori knowledge.

The Distinction Between Analytic and Synthetic Judgments

•	This concerns not the source of the propositions, but the relationship that holds between the subject and the predicate.
•	Analytic – explicative; synthetic – ampliative
•	“All bodies are extended” versus “all bodies are heavy.”
•	All a posteriori judgments are synthetic.
•	Are there a priori synthetic judgments?  Kant says yes.  
•	E.g., “everything which happens has a cause.”  This must be a priori because it’s necessary, but it’s got to be synthetic because the predicate isn’t contained in the subject.
•	Metaphysics must rest on a priori synthetic judgments like this.

In All Theoretical Sciences of Reason Synthetic A Priori Judgments Are Contained as Principles
•	All mathematic judgements are synthetic.  E.g., 5+7=12.  We need to count in our imagination, or draw a line, or perform a similar operation to solve mathematical equations.
•	“…the question is not what we ought to join in thought to the given concept, but what we actually think in it, even if only obscurely.”
•	In math, the intuition is required to attach the predicate to the subject.
•	The axioms of nat. sci. are similarly a priori synthetic.  
•	Metaphysics too needs to be grounded in these types of judgments.

The Highest Principle of All Analytic Judgments
•	Principle of contradiction
•	This principle is a necessary and sufficient condition of analytic judgments.
•	It is only a necessary, not a sufficient, condition of synthetic judgments.  Synthetic propositions need to conform to this principle to be true, but they also need to do something more.