Philosophy - Philosophy Course Notes

Predicate Logic aka Categorical Logic

Lecture 12

June 10, 2007

Predicate logic is stronger than SL.  Because SL does not capture all the valid arguments, we need a stronger system than SL.

Since the smallest blocks in SL are simple declarative sentences, the only kind of validity SL can capture is the kind between links in declarative senctences.

We need some way to handle what is going on inside the structure of a simple declarative sentence.

I. Introducing Predicate Logic

A. Grammatical Concepts

1. Singular terms.  Refer to an individual object.

2. Predicates.  In a declarative sentence, if you remove a singular term (or subject), what you are left with is the predicate.

ie. Dave is tall.

Subject = Dave

Predicate = 'is tall'

B. Structure of PL

1. Use capital letters to represent predicates.

2. Use lower case variables x,y,z to represent holes in predicates. Px

3. Use lower case variables (not x,y,z) to represent subjects/singular terms.

4. Have two special symbols called quantifiers:

i . Universal / All

(∀x) for any x

Take capital A and make it upside down. ∀

 

ii. Existential

(∃x) there is at least one x such that

Take capital E and reverse it horizontally.  ∃

5. Also use logical connectives from SL: &,v, negation (L sideways), => (if, then)

6. For (5) need a Universe of Discourse, says what we are talking about.  UD = people or UD = everything

I I. Rules of Inference

In SL, 9 simple rules were used.  In  PL, the simple rules are not sufficient to give all the valid arguments available.  See notes for showing how to do proofs in PL.

A. Showing an Argument is Invalid in PL

1. Need to give a counter-example, describe a case in which all the premises are true and the conclusion is false.

2. Need to supply an interpretation specify:

i. What things we are talking about.

ii. Have to interpret all the predicates that show up in the argument.

iii. Have to get subject/singular terms interpreted in the universe of discourse.  Assign them each to something in the UD.

3. Need to make sure the interpretation:

i . If you interpret the predicates, singular terms and quantifiers as specified by the interpretation.

ii. If the conclusion is false and the premises true then the argument is invalid.

iii. This is helpful is can cannot tell an argument is valid in English.

4. Symbolize the argument into PL, then try to construct an interpretation where (3i to 3iii)

5. This shows that the argument is not quantificationally valid.

i. In SL, if something is truth-functionally valid, it is also safe to say that it is logically valid.

ii. But just because something is not valid in SL does not mean that it is logically invalid.

iii. In PL if an argument is not quantificationally valid, we can be a little bit more sure in saying it is invalid.

There are other systems of logic that are designed to capture arguments that are clearly not valid in PL.

ie. Reasoning about causal relationships and reasoning about possibilities.

Unlike in SL, in the face of an argument that is logically invalid in PL, it is probably up to the person presenting the argument to give positive reasons for saying PL is not useful in that case and we should use another system of logic.