Meta-Logical Themes: Soundness of Propositional Logic

We have been studying propositional logic (PL) for some time now. PL is about whole or complete propositions and their relationships to each other in terms of truth-preservation. That is, PL investigates the relationships between sets of true propositions to other propositions. How can we be sure that if we are given a set of true propositions (called the premises) that we will through a series of steps wind up with another true proposition and not a false one (i.e. how can we be sure that the relationship between the premise set and the conclusion is, in fact, a valid one). Put differently, we want to make sure that the logic we are using will never take us from truths to falsity.

We have been using our rules to show that if we start with a set of (assumed to be) true propositions we can derive another proposition that must be true if the premises are. But we have only studied a finite number of arguments. There is an infinite number of arguments. So, how can we be sure that there is not some argument lurking in the shadows that starts with true premises, uses the rules perfectly, and nevertheless winds up with a false conclusion?

Think of it like this: in the sciences, we gather data, form a hypothesis, test it, and then gather more and more data to make sure that the hypothesis is accurate. There is always a possibility that we will gather some data that falsifies the hypothesis in one way or another. A finite set of data is never enough to ensure, to guarantee, a hypothesis that is supposed to be true of ALL the data. So, the sciences can never get us to certainty or a guaranteed-to-be-true hypothesis. In our class, we have gathered a number of premise-sets, applied our rules and have thus far never wound up with something false. But we have only seen a finite number of arguments. How can we be sure that we will not discover an argument that has true premises, uses the rules perfectly, and winds up with a false conclusion? That question is one of the main questions of meta-logic.

You will be happy (or not) to know that there is, in fact, a meta-logical proof for the claim that PL is such that no set of premises, plus our rules, can ever deliver a false conclusion. That is, there is a proof in meta-logic about PL that shows that PL is sound (i.e. there is no possibility of a premise set that uses our rules and winds up with a false conclusion).

If you are interested in hearing more about this, let me know.

Comments

Post a Comment

Popular posts from this blog

Relations and Multiple Quantifiers in Predicate Logic

Thus far we have been using only one-place predicates. These are predicates that only require one name (put differently, these are properties that only require one object) to complete them. For example, _____ is red = Rx _____ is heavy = Hx _____ is a wife = Wx ( ∃ x)Rx is true if and only if there is at least one thing in the universe that is red. ( ∃ x)Hx is true iff there is at least one thing in the universe that is heavy, and so on. (x)Rx is true iff everything in the universe is red, and so on.  But, as we all know, there are predicates that are more than one place. In other words, there are predicates that require more than one name to complete them (put differently, there are properties-- better-called relations--that require more than one object). For example ____ is heavier than ____ = Hxy ____ is taller than ____ =  Txy ____ loves ____ = Lxy ____ is between ____, _____ = Bxyz (∃ x) (∃y )Hxy is true iff there is something ...
Read more

Predicates in search of nouns

According to standard categorical logic, statements of the form All S are P must have some sort of noun-phrase in the predicate.  Thus ‘All dogs are mammals’ is fine, but ‘All dogs are four-legged’ is not.  Why is this a requirement? Standard categorical logic is based on Aristotelian logic and inherits some of Aristotle’s assumptions. So one reason for the noun-requirement is that given Aristotelian metaphysics predicates without nouns are incomplete. To understand why, we first need to keep in mind that  predicates (parts of language) correspond to  properties (parts of the extra-linguistic—that is, non-linguistic—world).  Predicates without nouns are incomplete because, according to Aristotle,  properties  must inhere in objects in order to exist.  There are no free-floating properties—properties without some subject that they attach to.  Thus, statements like ‘All dogs are four-legged’ while grammatic...
Read more