PHIL 220. Introduction to Logic
Propositional logic does not account for the validity of many natural language arguments.
Every US citizen can travel to the UK without a visa. You are a US citizen. Therefore, you can travel to the UK without a visa.
This appears to be an instance of a valid argument form:
translation
\(p\): Every US citizen is permitted to travel to the UK without a visa.
\(q\): You are a US citizen.
\(r\): You are permitted to travel to the UK without a visa.
validity
The argument is invalid as there is an assignment on which the premises are true and the conclusion false:
\[ TTF. \]
\[ \begin{array}{|c|c|c||c|c|c|} \hline p & q & r & p & \ q \ & \ r \\ \hline T & T & F & \color{red}{T} & \color{red}{T} & \color{red}{F} \\ \hline \end{array} \]
Propositional logic is too coarse-grained to represent the forms of a variety of valid arguments:
Designators, predicates, and quantifier phrases.
A designator is an expression whose primary function is to designate something.
Notice that the occurrence of ‘it’ in the third sentence co-refers with ‘The Griffith Observatory’. But not all pronouns act as a designator: Something is a landmark, and it is free to visit.
A predicate is an expression that combines with one or more designators to form a sentence, which says of some thing (or things) that they are a certain way (or stand in a certain relation).
Predicates come with a different number of gaps or arguments:
‘__ is a Greek philosopher’ takes one;
‘__ loves __’ takes two;
‘__ comes between __ and __’ takes three.
A quantifier phrase is an expression used to indicate generality.
Since ‘nobody’ is not a designator, the arguments below differ in form:
premises
conclusion
premises
conclusion
Quantifier phrases may bind occurrences of pronouns:
Something is a historic-cultural landmark, and it is free to visit.
Every Mercedes is expensive if it is on sale.
Some apple is red, and it is delicious.
The language of quantificational logic comes with formal counterparts for each category of expression.
To specify the syntax, we provide a vocabulary and a grammar.
The vocabulary of quantificational logic contains six types of symbols:
These are the lowercase letters \(a\), \(b\), \(c\), \(d\), and \(e\) with or without numerical subscripts: \[ a, b, c, d, e \]
These are uppercase letters \(P\), \(Q\), \(R\), \(S\), and \(T\) with or without numerical subscripts: \[ P, Q, R, S, T \]
These are the lowercase letters \(x\), \(y\), and \(z\) with or without numerical subscripts: \[ x, y, z \]
These are formal counterparts of ‘not’, ‘or’, ‘and’, and ‘if \(\dots\), then \(\dots\)’: \[ \neg, \vee, \wedge, \to \]
These are symbols \(\forall\) and \(\exists\) for the universal and existential quantifier respectively: \[ \forall, \exists \]
There are two parentheses: \[ ), ( \]
Nothing else is a symbol of the language.
Which expressions count as a formula of quantificational logic?
If \(P\) is a predicate with \(n\) argument places (or gaps) and each of \(\tau_1, \dots, \tau_n\) is a constant or a variable, then \(P\tau_1 \dots \tau_n\) is an atomic formula.
Atomic formulas are predications:
\[ Pa, \quad Qab, \quad Px, \quad Qacxb, \quad \dots \] They are formal counterparts of basic English sentences:
Sam is tall
Sam lives next to Jenna
She is tall.
Paris lies between London, it, and Berlin.
example: \((\forall x Pxy \to \exists y Qab)\)
The formula checker produces a parse tree for the formula:
Compare with another case: \(\forall x (Pxy \to \exists y Qab)\)
Identify the main connective at each stage.
Not all formulas are alike.
Each quantifier governs the formula that immediately follows it.
\[ (\forall x Px \to \exists y Qxy) \]
The last occurrence of \(x\) lies outside the scope of \(\forall x\).
\[ \forall x (Px \to \exists y Qxy) \]
The last occurrence of \(x\) lies inside the scope of \(\forall x\).
An occurrence is free if it has not been captured by a quantifier.
example: \((Px \to (Qxy \to \forall x Rxx))\)
An occurrence of a variable is bound iff it is not free.
A variable occurs freely in a formula iff some of its occurrences are free.
A formula is open iff some variable occurs freely in it. Otherwise it is closed.
A sentence of quantificational logic is a closed formula.
Build the tree, mark each occurrence free or bound, then decide: open or closed?