PHIL 220. Introduction to Logic
We have emphasized that validity is a matter of form.
We have focused on forms that combine basic statements with the help of connectives like ‘or’, ‘and’, ‘not’, ‘if … , then …’.
Notice the use of quotation marks …
Quotation marks are used to distinguish between use and mention.
Los Angeles has many cities in its metropolitan area.
We predicate something of a city, not a linguistic expression.
‘Los Angeles’ consists of two words.
We predicate something of a name, which is itself a linguistic expression.
One statement uses a name for a city in order to denote a city.
The other statement mentions a name for a city in order to predicate something of the linguistic expression.
Quotation marks indicate that a given expression is being mentioned and not used.
Which statements below are true?
Which words does the form depend on?
This is a valid argument form because of what ‘and’ means. Replace ‘and’ with ‘or’ and we no longer have a valid form:
| \(p\) | \(q\) | \(p\) and \(q\) | \(q\) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | F | T |
| F | F | F | F |
No matter what statements we choose for \(p\) and \(q\), no argument instance will take us from true premises to a false conclusion.
| \(\neg\) | negation | it is not the case that |
| \(\wedge\) | conjunction | and |
| \(\vee\) | disjunction | or |
| \(\to\) | the conditional | if …, then … |
Propositional logic will focus on forms obtained from basic propositional letters \(p\), \(q\), \(r\), etc and the four connectives listed above.
Syntax, semantics, pragmatics.
Syntax
Semantics
Pragmatics
Only syntax and semantics will be relevant for our formal languages.
There are at least three reasons
Do these sentences differ in truth conditions?
I felt ill, and I took some syrup
I took some syrup, and I felt ill
In propositional logic, order makes no difference to the truth conditions of a conjunction: \((p \wedge q)\) and \((q \wedge p)\) are true under exactly the same circumstances.
premises
conclusion
premises
conclusion
One may be tempted to discern the same superficial form, but that would be a mistake.
The two arguments do not share a valid form: one is valid and the other is invalid.
Quantificational logic will enable to distinguish two different forms, one valid and one invalid.
There is gas in the tank or the car will start if you turn the ignition.
Unlike English, the formal language will not tolerate structural ambiguities.
The vocabulary of propositional logic contains three types of symbols:
These are the lowercase letters \(p\), \(q\), \(r\), \(s\), and \(t\) with or without numerical subscripts: \[ p, q, r, s, t \]
These are formal counterparts of ‘not’, ‘or’, ‘and’, and ‘if \(\dots\), then \(\dots\)’: \[ \neg, \vee, \wedge, \to \]
There are two parentheses: \[ ), ( \]
Nothing else is a symbol of the language.
Which expressions count as a formula of propositional logic?
Next time we will explore the use of truth tables to interpret the language of propositional logic.