Symbolic Logic
Chapter 7:Translations in Polyadic Predicate Logic
Gary Hardegree
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Simple Polyadic Quantication
As discussed, a predicate can connect two subjects, this is called a two-place predicate:
He is good to her = x is good to y
= Gxy
Two-place
Philosophy 244: #17Intensional Objects
Intensional Objects
Entities obtained by stringing together ordinary objects, one per world and/or time,
are called intensional objects. They give us a new and more charitable take on Converse Carban. If there has to
Philosophy 244: #7Modal Metalogic: Completeness
Soundness for a system S says that its theorems are S-valid, valid in all S-frames. Completeness says that every S-valid wff is provable in S. The two together show system S is
adequate. You might think this
Philosophy 244: #10 Modal Predicate Metalogic
Soundness
Since the modal LPC models defined above all validate BF well speak of them as BF
models. For the time being well be sticking to BF models, and defining validity in terms
of these.
is valid in BF mo
Philosophy 244: Modal LogicPreliminaries
What is modal logic?
Metaphysical answer: Its the logic of modes. Modes are ways in which a condition can hold or obtain or be implemented. Descartes thought all properties were
modes either of extension or thought
Philosophy 244, #2: Modal Syntax and Semantics
The great advantage of ordinary propositional logic is that all of its connectives are
truth-functional. It was this truth-functionality that let us abstract away from propositions and define validity in term
Philosophy 244: #18Sets, Actuality, Counterparts
Substantive Modal Theories
Start with the idea of a substantive theory of some kind of objectof quarks, natural
numbers, or whatever. Theories of this sort are typically stated in nonmodal languages. Questi
Philosophy 244: #13 Shifting Domains
The BF has been shown a lot of deference so far. There is such a thing as life
without it. LPC+S is one thing, LPC+S+BF (which weve been calling S+BF) another.
The one has two axioms:
S ` for each an LPC substitution i
Philosophy 244: #5Stronger Systems
Modalities
Modalities are modal statuses, like being necessary, or possible, or not necessarily possibly necessary, or possibly necessarily necessary, or none of these. An iterated modality is a finite number of boxes an
Philosophy 244: #6Testing: Decidable & Undecidable Systems
Validity has been defined in each case as validity for every frame of an appropriate type.
Frames of an appropriate type are characterized by features of their accessibility relations:
reflexive,
Philosophy 244: #14 Existence and Identity
Existence Predicates
The problem weve been having is that (a) we want to allow models that invalidate
the CBF (xx), (b) these will have to be models in which w can see w
although Dw has members not in Dw0 , (c) m
Philosophy 244: #15 Identity and Descriptions
The last class ended with a paradox. Its a theorem of quantified modal logic with
identity that x=yx=y. But this conditional seems false when applied to
The first PM General = the inventor of bifocals.
That is
Philosophy 244: #8 Counterfactuals, Neighborhood Semantics,
Probability, Predicative Necessity, etc.
Modal operators are non-truth-functional; the truth-value of 2 at a world is not determined by s s truth-value at that world. Are modal operators X-functi
Symbolic Logic
Chapter 6: Translations in Monadic Predicate Logic
Gary Hardegree
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The Subject-Predicate Form of Atomic Statments
In Predicate logic every atomic statement consists of one predicate and one or more subjects.
i.e. John is tall. Has the pr
Translations
What is a translation? Rendering a sentence of English into the language of sentential logic.
Same as translating between two natural languages?
o Theres a sense in which we lose a bit of meaning when he render sentences in the
language of s
Translate the following story into the language of sentential logic. Use sentence letters
corresponding to the first letter of the capitalized term, and then add the first letter of the persons
name as a superscript. Ive filled in the first answer as an e
1. Circle the MAIN CONNECTIVES in the following sentences.
(P & Q) v R
(P Q) v (R & ~S)
~(P Q)
~(P Q) (R v S)
[(P & S) (Q v R)] & ~[S & ~(Q ~R)]
~[(P Q) R] S
~cfw_[(Q v ~Q) S] ~P
~(P ~Q) v (P ~Q)
2. Underline the MAIN CONNECTIVES in the following sentence
Necessary and Sufficient Conditions
sufficient
If A then B
necessary
If you are born in Kansas, then you are born in USA
If I know that X is born in Kansas, then I can conclude automatically that X is born in
USA
BUT if I know that X is born in USA,
Translation Cheat Sheet
If A then B
AB
A if B
B A
Even if A then B
AB
A even if B
B A
A only if B
~B ~A
AB
A if and only if B
(A B) & (B A)
(A B) & (~A ~B)
A unless (if not) B
A B
(B A) & (~B ~A)
~B A
A unless (except when) B
~B A
(~B A) & (A ~B)
A is suf
Philosophy 244: #4Adequacy and Extensions
Metalogic
To be a theorem of K is to be derivable from the K-axioms by the K-inference rules.
This is a purely syntactical notion, which pays no attention at all what the symbols
might mean. To be K-valid, or as w
Philosophy 244: 244 #3Basic Modal Systems
Time to start looking at some specific logical systems, starting with our base
system K the system such that theoremhood in it corresponds to absolute validity,
ie. validity in all seating arrangements whatsoever.
Philosophy 244: #16 Contingent Identity
Dispensability
Names have often been seen as linguistic outliers and philosophically problematic.
The more verbs and predicates you know, the larger your vocabulary, but it doesnt
seem to increase your vocabulary to
Philosophy 244: #9 Modal Predicate Logic
Now we turn to modal predicate logic, the result of adding modal operators to firstorder quantification theory. First-order quantification theory is itself the result of making
two additions to propositional logic.